Dirac Algebroids in Formalisms of Constrained Mechanics Janusz - - PowerPoint PPT Presentation

Dirac Algebroids in Formalisms of Constrained Mechanics

Janusz Grabowski

Polish Academy of Sciences

IHP-Paris, November 26, 2014

JG (IMPAN) Dirac Algebroids 26/11/2014 1 / 30

Introduction

How to deal with singular Lagrangians? → Tulczyjew triple. Implicit equations (dynamics); generation separated from solving.

• W. Tulczyjew: ”Hamiltonian systems, Lagrangian systems, and

the Legendre transformation”, Symposia Math. 14, (1974), 101–114.

• W. M. Tulczyjew: ”Les sous-vari´

et´ es lagrangiennes et la dynamique lagrangienne” (French), C. R. Acad. Sci. Paris S´ er. A-B 283 (1976), no. 8, Av, A675–A678. How to deal with nonholonomic constraints? → Dirac structures We are looking for a geometrical framework of mechanics including Reduction procedures → Tulczyjew triples for (Lie) algebroids Nonholonomic constraints → Tulczyjew triples for Dirac algebroids

• J. Grabowski, K. Grabowska: ”Dirac Algebroids in Lagrangian and

Hamiltonian Mechanics” J. Geom. Phys. 61 (2011), 2233–2253.

JG (IMPAN) Dirac Algebroids 26/11/2014 2 / 30

Introduction

How to deal with singular Lagrangians? → Tulczyjew triple. Implicit equations (dynamics); generation separated from solving.

• W. Tulczyjew: ”Hamiltonian systems, Lagrangian systems, and

the Legendre transformation”, Symposia Math. 14, (1974), 101–114.

• W. M. Tulczyjew: ”Les sous-vari´

et´ es lagrangiennes et la dynamique lagrangienne” (French), C. R. Acad. Sci. Paris S´ er. A-B 283 (1976), no. 8, Av, A675–A678. How to deal with nonholonomic constraints? → Dirac structures We are looking for a geometrical framework of mechanics including Reduction procedures → Tulczyjew triples for (Lie) algebroids Nonholonomic constraints → Tulczyjew triples for Dirac algebroids

• J. Grabowski, K. Grabowska: ”Dirac Algebroids in Lagrangian and

Hamiltonian Mechanics” J. Geom. Phys. 61 (2011), 2233–2253.

JG (IMPAN) Dirac Algebroids 26/11/2014 2 / 30

Introduction

How to deal with singular Lagrangians? → Tulczyjew triple. Implicit equations (dynamics); generation separated from solving.

• W. Tulczyjew: ”Hamiltonian systems, Lagrangian systems, and

the Legendre transformation”, Symposia Math. 14, (1974), 101–114.

• W. M. Tulczyjew: ”Les sous-vari´

et´ es lagrangiennes et la dynamique lagrangienne” (French), C. R. Acad. Sci. Paris S´ er. A-B 283 (1976), no. 8, Av, A675–A678. How to deal with nonholonomic constraints? → Dirac structures We are looking for a geometrical framework of mechanics including Reduction procedures → Tulczyjew triples for (Lie) algebroids Nonholonomic constraints → Tulczyjew triples for Dirac algebroids

• J. Grabowski, K. Grabowska: ”Dirac Algebroids in Lagrangian and

Hamiltonian Mechanics” J. Geom. Phys. 61 (2011), 2233–2253.

JG (IMPAN) Dirac Algebroids 26/11/2014 2 / 30

Introduction

How to deal with singular Lagrangians? → Tulczyjew triple. Implicit equations (dynamics); generation separated from solving.

• W. Tulczyjew: ”Hamiltonian systems, Lagrangian systems, and

the Legendre transformation”, Symposia Math. 14, (1974), 101–114.

• W. M. Tulczyjew: ”Les sous-vari´

et´ es lagrangiennes et la dynamique lagrangienne” (French), C. R. Acad. Sci. Paris S´ er. A-B 283 (1976), no. 8, Av, A675–A678. How to deal with nonholonomic constraints? → Dirac structures We are looking for a geometrical framework of mechanics including Reduction procedures → Tulczyjew triples for (Lie) algebroids Nonholonomic constraints → Tulczyjew triples for Dirac algebroids

• J. Grabowski, K. Grabowska: ”Dirac Algebroids in Lagrangian and

Hamiltonian Mechanics” J. Geom. Phys. 61 (2011), 2233–2253.

JG (IMPAN) Dirac Algebroids 26/11/2014 2 / 30

Introduction

How to deal with singular Lagrangians? → Tulczyjew triple. Implicit equations (dynamics); generation separated from solving.

• W. Tulczyjew: ”Hamiltonian systems, Lagrangian systems, and

the Legendre transformation”, Symposia Math. 14, (1974), 101–114.

• W. M. Tulczyjew: ”Les sous-vari´

et´ es lagrangiennes et la dynamique lagrangienne” (French), C. R. Acad. Sci. Paris S´ er. A-B 283 (1976), no. 8, Av, A675–A678. How to deal with nonholonomic constraints? → Dirac structures We are looking for a geometrical framework of mechanics including Reduction procedures → Tulczyjew triples for (Lie) algebroids Nonholonomic constraints → Tulczyjew triples for Dirac algebroids

• J. Grabowski, K. Grabowska: ”Dirac Algebroids in Lagrangian and

Hamiltonian Mechanics” J. Geom. Phys. 61 (2011), 2233–2253.

JG (IMPAN) Dirac Algebroids 26/11/2014 2 / 30

Introduction

How to deal with singular Lagrangians? → Tulczyjew triple. Implicit equations (dynamics); generation separated from solving.

• W. Tulczyjew: ”Hamiltonian systems, Lagrangian systems, and

the Legendre transformation”, Symposia Math. 14, (1974), 101–114.

• W. M. Tulczyjew: ”Les sous-vari´

et´ es lagrangiennes et la dynamique lagrangienne” (French), C. R. Acad. Sci. Paris S´ er. A-B 283 (1976), no. 8, Av, A675–A678. How to deal with nonholonomic constraints? → Dirac structures We are looking for a geometrical framework of mechanics including Reduction procedures → Tulczyjew triples for (Lie) algebroids Nonholonomic constraints → Tulczyjew triples for Dirac algebroids

• J. Grabowski, K. Grabowska: ”Dirac Algebroids in Lagrangian and

Hamiltonian Mechanics” J. Geom. Phys. 61 (2011), 2233–2253.

JG (IMPAN) Dirac Algebroids 26/11/2014 2 / 30

Introduction

How to deal with singular Lagrangians? → Tulczyjew triple. Implicit equations (dynamics); generation separated from solving.

• W. Tulczyjew: ”Hamiltonian systems, Lagrangian systems, and

the Legendre transformation”, Symposia Math. 14, (1974), 101–114.

• W. M. Tulczyjew: ”Les sous-vari´

et´ es lagrangiennes et la dynamique lagrangienne” (French), C. R. Acad. Sci. Paris S´ er. A-B 283 (1976), no. 8, Av, A675–A678. How to deal with nonholonomic constraints? → Dirac structures We are looking for a geometrical framework of mechanics including Reduction procedures → Tulczyjew triples for (Lie) algebroids Nonholonomic constraints → Tulczyjew triples for Dirac algebroids

• J. Grabowski, K. Grabowska: ”Dirac Algebroids in Lagrangian and

Hamiltonian Mechanics” J. Geom. Phys. 61 (2011), 2233–2253.

JG (IMPAN) Dirac Algebroids 26/11/2014 2 / 30

Tulczyjew Triple

M - positions,

TM - (kinematic)

configurations, L : TM → R - Lagrangian

T∗M - phase space

Legendre map λ : TM → T∗M Tulczyjew differential ΛL : TM → TT∗M. D = α−1

M (dL(TM))) = ΛL(TM) is the phase dynamics

λL : TM → T∗M, λL(v) = ξ(dL(v)), λL(q, ˙ q) = (q, ∂L ∂ ˙ q ). D =

• (q, p, ˙

q, ˙ p) : p = ∂L ∂ ˙ q , ˙ p = ∂L ∂q

• JG (IMPAN)

Dirac Algebroids 26/11/2014 3 / 30

Tulczyjew Triple

M - positions,

TM - (kinematic)

configurations, L : TM → R - Lagrangian

T∗M - phase space

Legendre map λ : TM → T∗M Tulczyjew differential ΛL : TM → TT∗M. D = α−1

M (dL(TM))) = ΛL(TM) is the phase dynamics

λL : TM → T∗M, λL(v) = ξ(dL(v)), λL(q, ˙ q) = (q, ∂L ∂ ˙ q ). D =

• (q, p, ˙

q, ˙ p) : p = ∂L ∂ ˙ q , ˙ p = ∂L ∂q

• JG (IMPAN)

Dirac Algebroids 26/11/2014 3 / 30

Tulczyjew Triple

M - positions,

TM - (kinematic)

configurations, L : TM → R - Lagrangian

T∗M - phase space

Legendre map λ : TM → T∗M Tulczyjew differential ΛL : TM → TT∗M. D = α−1

M (dL(TM))) = ΛL(TM) is the phase dynamics

λL : TM → T∗M, λL(v) = ξ(dL(v)), λL(q, ˙ q) = (q, ∂L ∂ ˙ q ). D =

• (q, p, ˙

q, ˙ p) : p = ∂L ∂ ˙ q , ˙ p = ∂L ∂q

• JG (IMPAN)

Dirac Algebroids 26/11/2014 3 / 30

Tulczyjew Triple

M - positions,

TM - (kinematic)

configurations, L : TM → R - Lagrangian

T∗M - phase space

Legendre map λ : TM → T∗M Tulczyjew differential ΛL : TM → TT∗M. D = α−1

M (dL(TM))) = ΛL(TM) is the phase dynamics

λL : TM → T∗M, λL(v) = ξ(dL(v)), λL(q, ˙ q) = (q, ∂L ∂ ˙ q ). D =

• (q, p, ˙

q, ˙ p) : p = ∂L ∂ ˙ q , ˙ p = ∂L ∂q

• JG (IMPAN)

Dirac Algebroids 26/11/2014 3 / 30

Tulczyjew Triple

M - positions,

TM - (kinematic)

configurations, L : TM → R - Lagrangian

T∗M - phase space

Legendre map λ : TM → T∗M Tulczyjew differential ΛL : TM → TT∗M. Lagrangian side of the triple

TT∗M

αM

• TπM
• ❉

❉ ❉ ❉ ❉

τT∗M

☛☛☛☛☛☛☛☛☛☛ T∗TM

πTM

• ❉

❉ ❉ ❉ ❉

ξ

☛☛☛☛☛☛☛☛☛☛ TM

• τM

☞☞☞☞☞☞☞☞☞☞ TM

τM

☞☞☞☞☞☞☞☞☞☞ T∗M

• πM
• ❋

❋ ❋ ❋ ❋ ❋ T∗M

πM

• ❋

❋ ❋ ❋ ❋ ❋

M

M

D = α−1

M (dL(TM))) = ΛL(TM) is the phase dynamics

λL : TM → T∗M, λL(v) = ξ(dL(v)), λL(q, ˙ q) = (q, ∂L ∂ ˙ q ). D =

• (q, p, ˙

q, ˙ p) : p = ∂L ∂ ˙ q , ˙ p = ∂L ∂q

• JG (IMPAN)

Dirac Algebroids 26/11/2014 3 / 30

Tulczyjew Triple

M - positions,

TM - (kinematic)

configurations, L : TM → R - Lagrangian

T∗M - phase space

Legendre map λ : TM → T∗M Tulczyjew differential ΛL : TM → TT∗M. Lagrangian side of the triple D

TT∗M

αM

• ❉

❉ ❉ ❉ ❉ ☛☛☛☛☛☛☛☛☛☛ T∗TM

• ❉

❉ ❉ ❉ ❉ ☛☛☛☛☛☛☛☛☛☛ TM TM

dL

• λL

❧❧❧❧❧❧❧❧❧❧

ΛL

❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ T∗M T∗M

M M D = α−1

M (dL(TM))) = ΛL(TM) is the phase dynamics

λL : TM → T∗M, λL(v) = ξ(dL(v)), λL(q, ˙ q) = (q, ∂L ∂ ˙ q ). D =

• (q, p, ˙

q, ˙ p) : p = ∂L ∂ ˙ q , ˙ p = ∂L ∂q

• JG (IMPAN)

Dirac Algebroids 26/11/2014 3 / 30

Tulczyjew Triple

M - positions,

TM - (kinematic)

configurations, L : TM → R - Lagrangian

T∗M - phase space

Legendre map λ : TM → T∗M Tulczyjew differential ΛL : TM → TT∗M. Lagrangian side of the triple D

TT∗M

αM

• ❉

❉ ❉ ❉ ❉ ☛☛☛☛☛☛☛☛☛☛ T∗TM

• ❉

❉ ❉ ❉ ❉ ☛☛☛☛☛☛☛☛☛☛ TM TM

dL

• λL

❧❧❧❧❧❧❧❧❧❧

ΛL

❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ T∗M T∗M

M M D = α−1

M (dL(TM))) = ΛL(TM) is the phase dynamics

λL : TM → T∗M, λL(v) = ξ(dL(v)), λL(q, ˙ q) = (q, ∂L ∂ ˙ q ). D =

• (q, p, ˙

q, ˙ p) : p = ∂L ∂ ˙ q , ˙ p = ∂L ∂q

• JG (IMPAN)

Dirac Algebroids 26/11/2014 3 / 30

Tulczyjew Triple

M - positions,

TM - (kinematic)

configurations, L : TM → R - Lagrangian

T∗M - phase space

Legendre map λ : TM → T∗M Tulczyjew differential ΛL : TM → TT∗M. Lagrangian side of the triple D

TT∗M

αM

• ❉

❉ ❉ ❉ ❉ ☛☛☛☛☛☛☛☛☛☛ T∗TM

• ❉

❉ ❉ ❉ ❉ ☛☛☛☛☛☛☛☛☛☛ TM TM

dL

• λL

❧❧❧❧❧❧❧❧❧❧

ΛL

❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ T∗M T∗M

M M D = α−1

M (dL(TM))) = ΛL(TM) is the phase dynamics

λL : TM → T∗M, λL(v) = ξ(dL(v)), λL(q, ˙ q) = (q, ∂L ∂ ˙ q ). D =

• (q, p, ˙

q, ˙ p) : p = ∂L ∂ ˙ q , ˙ p = ∂L ∂q

• JG (IMPAN)

Dirac Algebroids 26/11/2014 3 / 30

Tulczyjew Triple

canonical isomorphism

T∗TM ≃ T∗T∗M,

H : T∗M → R D = β−1

M (dH(T∗M)))

D =

• (q, p, ˙

q, ˙ p) : ˙ p = −∂H ∂q , ˙ q = ∂H ∂p

• JG (IMPAN)

Dirac Algebroids 26/11/2014 4 / 30

Tulczyjew Triple

canonical isomorphism

T∗TM ≃ T∗T∗M,

H : T∗M → R D = β−1

M (dH(T∗M)))

D =

• (q, p, ˙

q, ˙ p) : ˙ p = −∂H ∂q , ˙ q = ∂H ∂p

• JG (IMPAN)

Dirac Algebroids 26/11/2014 4 / 30

Tulczyjew Triple

canonical isomorphism

T∗TM ≃ T∗T∗M,

H : T∗M → R Hamiltonian side of the triple

T∗T∗M

ζ

• ❋

❋ ❋ ❋ ❋

πT∗M

✠✠✠✠✠✠✠✠✠✠✠ TT∗M

TπM

• ❉

❉ ❉ ❉ ❉

τT∗M

☛☛☛☛☛☛☛☛☛☛

βM

• TM

τM

☛☛☛☛☛☛☛☛☛☛ TM

τM

☞☞☞☞☞☞☞☞☞☞

• T∗M

πM

• ❍

❍ ❍ ❍ ❍ ❍ T∗M

πM

• ❋

❋ ❋ ❋ ❋ ❋

• M

M

• D = β−1

M (dH(T∗M)))

D =

• (q, p, ˙

q, ˙ p) : ˙ p = −∂H ∂q , ˙ q = ∂H ∂p

• JG (IMPAN)

Dirac Algebroids 26/11/2014 4 / 30

Tulczyjew Triple

canonical isomorphism

T∗TM ≃ T∗T∗M,

H : T∗M → R Hamiltonian side of the triple

T∗T∗M

ζ

• ❋

❋ ❋ ❋ ❋

πT∗M

✠✠✠✠✠✠✠✠✠✠✠ TT∗M

• ❉

❉ ❉ ❉ ❉ ☛☛☛☛☛☛☛☛☛☛

βM

• D
• TM

TM T∗M

dH

• T∗M

M M D = β−1

M (dH(T∗M)))

D =

• (q, p, ˙

q, ˙ p) : ˙ p = −∂H ∂q , ˙ q = ∂H ∂p

• JG (IMPAN)

Dirac Algebroids 26/11/2014 4 / 30

Tulczyjew Triple

canonical isomorphism

T∗TM ≃ T∗T∗M,

H : T∗M → R Hamiltonian side of the triple

T∗T∗M

ζ

• ❋

❋ ❋ ❋ ❋

πT∗M

✠✠✠✠✠✠✠✠✠✠✠ TT∗M

• ❉

❉ ❉ ❉ ❉ ☛☛☛☛☛☛☛☛☛☛

βM

• D
• TM

TM T∗M

dH

• T∗M

M M D = β−1

M (dH(T∗M)))

D =

• (q, p, ˙

q, ˙ p) : ˙ p = −∂H ∂q , ˙ q = ∂H ∂p

• JG (IMPAN)

Dirac Algebroids 26/11/2014 4 / 30

Tulczyjew Triple

canonical isomorphism

T∗TM ≃ T∗T∗M,

H : T∗M → R Hamiltonian side of the triple

T∗T∗M

ζ

• ❋

❋ ❋ ❋ ❋

πT∗M

✠✠✠✠✠✠✠✠✠✠✠ TT∗M

• ❉

❉ ❉ ❉ ❉ ☛☛☛☛☛☛☛☛☛☛

βM

• D
• TM

TM T∗M

dH

• T∗M

M M D = β−1

M (dH(T∗M)))

D =

• (q, p, ˙

q, ˙ p) : ˙ p = −∂H ∂q , ˙ q = ∂H ∂p

• JG (IMPAN)

Dirac Algebroids 26/11/2014 4 / 30

Mechanics on algebroids

Definition

A double vector bundle is a manifold with two compatible vector bundle

• structures. Compatibility means that the Euler vector fields (generators of

homotheties) associated with the two structures commute. τ : E − → M (xa, yi) − → (xa) τM : TM − → M (xa, ˙ xb) − → (xa) π : E ∗ − → M (xa, ξi) − → (xa) πM : T∗M − → M (xa, pb) − → (xa) ∇1 = ˙ xa∂ ˙

xa + ˙

ξi∂ ˙

ξi

∇2 = ξi∂ξi + ˙ ξj∂ ˙

ξj

JG (IMPAN) Dirac Algebroids 26/11/2014 5 / 30

Mechanics on algebroids

Definition

A double vector bundle is a manifold with two compatible vector bundle

• structures. Compatibility means that the Euler vector fields (generators of

homotheties) associated with the two structures commute. τ : E − → M (xa, yi) − → (xa) τM : TM − → M (xa, ˙ xb) − → (xa) π : E ∗ − → M (xa, ξi) − → (xa) πM : T∗M − → M (xa, pb) − → (xa) ∇1 = ˙ xa∂ ˙

xa + ˙

ξi∂ ˙

ξi

∇2 = ξi∂ξi + ˙ ξj∂ ˙

ξj

JG (IMPAN) Dirac Algebroids 26/11/2014 5 / 30

Mechanics on algebroids

Definition

A double vector bundle is a manifold with two compatible vector bundle

• structures. Compatibility means that the Euler vector fields (generators of

homotheties) associated with the two structures commute. τ : E − → M (xa, yi) − → (xa) τM : TM − → M (xa, ˙ xb) − → (xa) π : E ∗ − → M (xa, ξi) − → (xa) πM : T∗M − → M (xa, pb) − → (xa) ∇1 = ˙ xa∂ ˙

xa + ˙

ξi∂ ˙

ξi

∇2 = ξi∂ξi + ˙ ξj∂ ˙

ξj

JG (IMPAN) Dirac Algebroids 26/11/2014 5 / 30

Mechanics on algebroids

Definition

A double vector bundle is a manifold with two compatible vector bundle

• structures. Compatibility means that the Euler vector fields (generators of

homotheties) associated with the two structures commute. τ : E − → M (xa, yi) − → (xa) τM : TM − → M (xa, ˙ xb) − → (xa) π : E ∗ − → M (xa, ξi) − → (xa) πM : T∗M − → M (xa, pb) − → (xa) τE ∗ : TE ∗ − → E ∗ (xa, ξi, ˙ xb, ˙ ξj) − → (xa, ξi)

Tπ : TE ∗ −

→ TM (xa, ξi, ˙ xb, ˙ ξj) − → (xa, ˙ xb) ∇1 = ˙ xa∂ ˙

xa + ˙

ξi∂ ˙

ξi

∇2 = ξi∂ξi + ˙ ξj∂ ˙

ξj

JG (IMPAN) Dirac Algebroids 26/11/2014 5 / 30

Mechanics on algebroids

Definition

A double vector bundle is a manifold with two compatible vector bundle

• structures. Compatibility means that the Euler vector fields (generators of

homotheties) associated with the two structures commute. τ : E − → M (xa, yi) − → (xa) τM : TM − → M (xa, ˙ xb) − → (xa) π : E ∗ − → M (xa, ξi) − → (xa) πM : T∗M − → M (xa, pb) − → (xa)

TE ∗

Tπ

• ❋

❋ ❋ ❋ ❋ ❋ ❋ ❋

τE∗

②②②②②②②②

E ∗

π

• ❊

❊ ❊ ❊ ❊ ❊ ❊ ❊ TM

τM

①①①①①①①①①

M ∇1 = ˙ xa∂ ˙

xa + ˙

ξi∂ ˙

ξi

∇2 = ξi∂ξi + ˙ ξj∂ ˙

ξj

JG (IMPAN) Dirac Algebroids 26/11/2014 5 / 30

Mechanics on algebroids

Definition

A double vector bundle is a manifold with two compatible vector bundle

• structures. Compatibility means that the Euler vector fields (generators of

homotheties) associated with the two structures commute. τ : E − → M (xa, yi) − → (xa) τM : TM − → M (xa, ˙ xb) − → (xa) π : E ∗ − → M (xa, ξi) − → (xa) πM : T∗M − → M (xa, pb) − → (xa)

TE ∗

Tπ

• ❋

❋ ❋ ❋ ❋ ❋ ❋ ❋

τE∗

②②②②②②②②

E ∗

π

• ❊

❊ ❊ ❊ ❊ ❊ ❊ ❊ TM

τM

①①①①①①①①①

M ∇1 = ˙ xa∂ ˙

xa + ˙

ξi∂ ˙

ξi

∇2 = ξi∂ξi + ˙ ξj∂ ˙

ξj

JG (IMPAN) Dirac Algebroids 26/11/2014 5 / 30

Fundamental isomorphisms

πE ∗ : T∗E ∗ − → E ∗ ζ : T∗E ∗ − → E (xa, ξi, pb, yj) − → (xa, ξi) (xa, ξi, pb, yj) − → (xa, yj) ∇1 = pa∂pa + yi∂yi , ∇2 = pa∂pa + ξi∂ξi . Double vector bundle T∗E ∗ is (anti-)symplectically isomorphic to T∗E: R : T∗E → T∗E ∗ , (xa, yi, pb, ξj) − → (xa, ξi, −pb, yj) .

JG (IMPAN) Dirac Algebroids 26/11/2014 6 / 30

Fundamental isomorphisms

πE ∗ : T∗E ∗ − → E ∗ ζ : T∗E ∗ − → E (xa, ξi, pb, yj) − → (xa, ξi) (xa, ξi, pb, yj) − → (xa, yj) ∇1 = pa∂pa + yi∂yi , ∇2 = pa∂pa + ξi∂ξi . Double vector bundle T∗E ∗ is (anti-)symplectically isomorphic to T∗E: R : T∗E → T∗E ∗ , (xa, yi, pb, ξj) − → (xa, ξi, −pb, yj) .

JG (IMPAN) Dirac Algebroids 26/11/2014 6 / 30

Fundamental isomorphisms

πE ∗ : T∗E ∗ − → E ∗ ζ : T∗E ∗ − → E (xa, ξi, pb, yj) − → (xa, ξi) (xa, ξi, pb, yj) − → (xa, yj)

T∗E ∗

ζ

• ❊

❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊

πE∗

① ① ① ① ① ① ① ① ①

E ∗

π

• ❋

❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋

E

τ

②②②②②②②②

M ∇1 = pa∂pa + yi∂yi , ∇2 = pa∂pa + ξi∂ξi . Double vector bundle T∗E ∗ is (anti-)symplectically isomorphic to T∗E: R : T∗E → T∗E ∗ , (xa, yi, pb, ξj) − → (xa, ξi, −pb, yj) .

JG (IMPAN) Dirac Algebroids 26/11/2014 6 / 30

Fundamental isomorphisms

πE ∗ : T∗E ∗ − → E ∗ ζ : T∗E ∗ − → E (xa, ξi, pb, yj) − → (xa, ξi) (xa, ξi, pb, yj) − → (xa, yj)

T∗E ∗

ζ

• ❊

❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊

πE∗

① ① ① ① ① ① ① ① ①

E ∗

π

• ❋

❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋

E

τ

②②②②②②②②

M ∇1 = pa∂pa + yi∂yi , ∇2 = pa∂pa + ξi∂ξi . Double vector bundle T∗E ∗ is (anti-)symplectically isomorphic to T∗E: R : T∗E → T∗E ∗ , (xa, yi, pb, ξj) − → (xa, ξi, −pb, yj) .

JG (IMPAN) Dirac Algebroids 26/11/2014 6 / 30

Fundamental isomorphisms

πE ∗ : T∗E ∗ − → E ∗ ζ : T∗E ∗ − → E (xa, ξi, pb, yj) − → (xa, ξi) (xa, ξi, pb, yj) − → (xa, yj)

T∗E ∗

ζ

• ❊

❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊

πE∗

① ① ① ① ① ① ① ① ①

E ∗

π

• ❋

❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋

E

τ

②②②②②②②②

M ∇1 = pa∂pa + yi∂yi , ∇2 = pa∂pa + ξi∂ξi . Double vector bundle T∗E ∗ is (anti-)symplectically isomorphic to T∗E: R : T∗E → T∗E ∗ , (xa, yi, pb, ξj) − → (xa, ξi, −pb, yj) .

JG (IMPAN) Dirac Algebroids 26/11/2014 6 / 30

Algebroids

A general algebroid is a double vector bundle morphism covering the identity on E ∗ (ε = α−1

M for E = TM):

T∗E

ε

• πE
• ■

■ ■ ■ ■

T∗τ

✝✝✝✝✝✝✝✝✝ TE ∗

Tπ

• ▲

▲ ▲ ▲

τE∗

✝✝✝✝✝✝✝✝✝

E

ρ

• τ

✞✞✞✞✞✞✞✞ TM

τM

☎☎☎☎☎☎☎☎☎

E ∗

id

• π
• ❑

❑ ❑ ❑ ❑

E ∗

π

• ❑

❑ ❑ ❑ ❑

M

id

M

ε(xa, yi, pb, ξj) = (xa, ξi, ρb

k(x)yk, ck ij (x)yiξk + σa j (x)pa)

Πε = ck

ij (x)ξk∂ξi ⊗ ∂ξj + ρb i (x)∂ξi ⊗ ∂xb − σa j (x)∂xa ⊗ ∂ξj.

It is a skew algebroid (resp. Lie algebroid) if the tensor Πε is a bivector field (resp., Poisson tensor).

JG (IMPAN) Dirac Algebroids 26/11/2014 7 / 30

Algebroids

A general algebroid is a double vector bundle morphism covering the identity on E ∗ (ε = α−1

M for E = TM):

T∗E

ε

• πE
• ■

■ ■ ■ ■

T∗τ

✝✝✝✝✝✝✝✝✝ TE ∗

Tπ

• ▲

▲ ▲ ▲

τE∗

✝✝✝✝✝✝✝✝✝

E

ρ

• τ

✞✞✞✞✞✞✞✞ TM

τM

☎☎☎☎☎☎☎☎☎

E ∗

id

• π
• ❑

❑ ❑ ❑ ❑

E ∗

π

• ❑

❑ ❑ ❑ ❑

M

id

M

ε(xa, yi, pb, ξj) = (xa, ξi, ρb

k(x)yk, ck ij (x)yiξk + σa j (x)pa)

Πε = ck

ij (x)ξk∂ξi ⊗ ∂ξj + ρb i (x)∂ξi ⊗ ∂xb − σa j (x)∂xa ⊗ ∂ξj.

It is a skew algebroid (resp. Lie algebroid) if the tensor Πε is a bivector field (resp., Poisson tensor).

JG (IMPAN) Dirac Algebroids 26/11/2014 7 / 30

Algebroids

A general algebroid is a double vector bundle morphism covering the identity on E ∗ (ε = α−1

M for E = TM):

T∗E

ε

• πE
• ■

■ ■ ■ ■

T∗τ

✝✝✝✝✝✝✝✝✝ TE ∗

Tπ

• ▲

▲ ▲ ▲

τE∗

✝✝✝✝✝✝✝✝✝

E

ρ

• τ

✞✞✞✞✞✞✞✞ TM

τM

☎☎☎☎☎☎☎☎☎

E ∗

id

• π
• ❑

❑ ❑ ❑ ❑

E ∗

π

• ❑

❑ ❑ ❑ ❑

M

id

M

ε(xa, yi, pb, ξj) = (xa, ξi, ρb

k(x)yk, ck ij (x)yiξk + σa j (x)pa)

Πε = ck

ij (x)ξk∂ξi ⊗ ∂ξj + ρb i (x)∂ξi ⊗ ∂xb − σa j (x)∂xa ⊗ ∂ξj.

It is a skew algebroid (resp. Lie algebroid) if the tensor Πε is a bivector field (resp., Poisson tensor).

JG (IMPAN) Dirac Algebroids 26/11/2014 7 / 30

Algebroids

A general algebroid is a double vector bundle morphism covering the identity on E ∗ (ε = α−1

M for E = TM):

T∗E

ε

• πE
• ■

■ ■ ■ ■

T∗τ

✝✝✝✝✝✝✝✝✝ TE ∗

Tπ

• ▲

▲ ▲ ▲

τE∗

✝✝✝✝✝✝✝✝✝

E

ρ

• τ

✞✞✞✞✞✞✞✞ TM

τM

☎☎☎☎☎☎☎☎☎

E ∗

id

• π
• ❑

❑ ❑ ❑ ❑

E ∗

π

• ❑

❑ ❑ ❑ ❑

M

id

M

ε(xa, yi, pb, ξj) = (xa, ξi, ρb

k(x)yk, ck ij (x)yiξk + σa j (x)pa)

Πε = ck

ij (x)ξk∂ξi ⊗ ∂ξj + ρb i (x)∂ξi ⊗ ∂xb − σa j (x)∂xa ⊗ ∂ξj.

It is a skew algebroid (resp. Lie algebroid) if the tensor Πε is a bivector field (resp., Poisson tensor).

JG (IMPAN) Dirac Algebroids 26/11/2014 7 / 30

Algebroids

A general algebroid is a double vector bundle morphism covering the identity on E ∗ (ε = α−1

M for E = TM):

T∗E

ε

• πE
• ■

■ ■ ■ ■

T∗τ

✝✝✝✝✝✝✝✝✝ TE ∗

Tπ

• ▲

▲ ▲ ▲

τE∗

✝✝✝✝✝✝✝✝✝

E

ρ

• τ

✞✞✞✞✞✞✞✞ TM

τM

☎☎☎☎☎☎☎☎☎

E ∗

id

• π
• ❑

❑ ❑ ❑ ❑

E ∗

π

• ❑

❑ ❑ ❑ ❑

M

id

M

ε(xa, yi, pb, ξj) = (xa, ξi, ρb

k(x)yk, ck ij (x)yiξk + σa j (x)pa)

Πε = ck

ij (x)ξk∂ξi ⊗ ∂ξj + ρb i (x)∂ξi ⊗ ∂xb − σa j (x)∂xa ⊗ ∂ξj.

It is a skew algebroid (resp. Lie algebroid) if the tensor Πε is a bivector field (resp., Poisson tensor).

JG (IMPAN) Dirac Algebroids 26/11/2014 7 / 30

Algebroid setting, no constraints

DL

T∗E ∗

• ❋

❋ ❋ ❋ ❋ ❋ ✠✠✠✠✠✠✠✠✠✠✠

˜ Πε

TE ∗

• ❍

❍ ❍ ❍ ❍ ✡✡✡✡✡✡✡✡✡✡ T∗E

• ❊

❊ ❊ ❊ ❊ ❊ ✡✡✡✡✡✡✡✡✡✡

ε

• E

ρ

• ✡✡✡✡✡✡✡✡✡✡

TM ✠✠✠✠✠✠✠✠✠✠✠

E

☞☞☞☞☞☞☞☞☞☞

ρ

• E ∗
• ❍

❍ ❍ ❍ ❍ ❍

E ∗

• ❋

❋ ❋ ❋ ❋ ❋

E ∗

• ❋

❋ ❋ ❋ ❋ ❋

M

M

M

• H : E ∗ −

→ R DH = dH(E ∗) ⊂ T∗E ∗ D = ΛL(E) D = ˜ Πε(dH(E ∗)) L : E − → R DL = dL(E) ⊂ T∗E The left-hand side is Hamiltonian, the right-hand side is Lagrangian, the phase dynamics lives in the center. We can start equally with L or H, but

• nly in particular cases there is a Hamiltonian description corresponding to

a given Lagrangian (˜ Πε(dH(E ∗)) = ΛL(E)).

JG (IMPAN) Dirac Algebroids 26/11/2014 8 / 30

Algebroid setting, no constraints

DL

• T∗E ∗
• ❋

❋ ❋ ❋ ❋ ❋ ✠✠✠✠✠✠✠✠✠✠✠

˜ Πε

TE ∗

• ❍

❍ ❍ ❍ ❍ ✡✡✡✡✡✡✡✡✡✡ T∗E

• ❊

❊ ❊ ❊ ❊ ❊ ✡✡✡✡✡✡✡✡✡✡

ε

• E

ρ

• ✡✡✡✡✡✡✡✡✡✡

TM ✠✠✠✠✠✠✠✠✠✠✠

E

☞☞☞☞☞☞☞☞☞☞

ρ

• dL
• λL

❧❧❧❧❧❧❧❧❧❧❧❧

ΛL

❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨

E ∗

• ❍

❍ ❍ ❍ ❍ ❍

E ∗

• ❋

❋ ❋ ❋ ❋ ❋

E ∗

• ❋

❋ ❋ ❋ ❋ ❋

M

M

M

• H : E ∗ −

→ R DH = dH(E ∗) ⊂ T∗E ∗ D = ΛL(E) D = ˜ Πε(dH(E ∗)) L : E − → R DL = dL(E) ⊂ T∗E The left-hand side is Hamiltonian, the right-hand side is Lagrangian, the phase dynamics lives in the center. We can start equally with L or H, but

• nly in particular cases there is a Hamiltonian description corresponding to

a given Lagrangian (˜ Πε(dH(E ∗)) = ΛL(E)).

JG (IMPAN) Dirac Algebroids 26/11/2014 8 / 30

Algebroid setting, no constraints

D

• DL
• T∗E ∗
• ❋

❋ ❋ ❋ ❋ ❋ ✠✠✠✠✠✠✠✠✠✠✠

˜ Πε

TE ∗

• ❍

❍ ❍ ❍ ❍ ✡✡✡✡✡✡✡✡✡✡ T∗E

• ❊

❊ ❊ ❊ ❊ ❊ ✡✡✡✡✡✡✡✡✡✡

ε

• E

ρ

• ✡✡✡✡✡✡✡✡✡✡

TM ✠✠✠✠✠✠✠✠✠✠✠

E

☞☞☞☞☞☞☞☞☞☞

ρ

• dL
• λL

❧❧❧❧❧❧❧❧❧❧❧❧

ΛL

❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨

E ∗

• ❍

❍ ❍ ❍ ❍ ❍

E ∗

• ❋

❋ ❋ ❋ ❋ ❋

E ∗

• ❋

❋ ❋ ❋ ❋ ❋

M

M

M

• H : E ∗ −

→ R DH = dH(E ∗) ⊂ T∗E ∗ D = ΛL(E) D = ˜ Πε(dH(E ∗)) L : E − → R DL = dL(E) ⊂ T∗E The left-hand side is Hamiltonian, the right-hand side is Lagrangian, the phase dynamics lives in the center. We can start equally with L or H, but

• nly in particular cases there is a Hamiltonian description corresponding to

a given Lagrangian (˜ Πε(dH(E ∗)) = ΛL(E)).

JG (IMPAN) Dirac Algebroids 26/11/2014 8 / 30

Algebroid setting, no constraints

DH

• D
• DL
• T∗E ∗
• ❋

❋ ❋ ❋ ❋ ❋ ✠✠✠✠✠✠✠✠✠✠✠

˜ Πε

TE ∗

• ❍

❍ ❍ ❍ ❍ ✡✡✡✡✡✡✡✡✡✡ T∗E

• ❊

❊ ❊ ❊ ❊ ❊ ✡✡✡✡✡✡✡✡✡✡

ε

• E

ρ

• ✡✡✡✡✡✡✡✡✡✡

TM ✠✠✠✠✠✠✠✠✠✠✠

E

☞☞☞☞☞☞☞☞☞☞

ρ

• dL
• λL

❧❧❧❧❧❧❧❧❧❧❧❧

ΛL

❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨

E ∗

• ❍

❍ ❍ ❍ ❍ ❍

dH

• E ∗
• ❋

❋ ❋ ❋ ❋ ❋

E ∗

• ❋

❋ ❋ ❋ ❋ ❋

M

M

M

• H : E ∗ −

→ R DH = dH(E ∗) ⊂ T∗E ∗ D = ΛL(E) D = ˜ Πε(dH(E ∗)) L : E − → R DL = dL(E) ⊂ T∗E The left-hand side is Hamiltonian, the right-hand side is Lagrangian, the phase dynamics lives in the center. We can start equally with L or H, but

• nly in particular cases there is a Hamiltonian description corresponding to

a given Lagrangian (˜ Πε(dH(E ∗)) = ΛL(E)).

JG (IMPAN) Dirac Algebroids 26/11/2014 8 / 30

Algebroid setting, no constraints

DH

• D
• DL
• T∗E ∗
• ❋

❋ ❋ ❋ ❋ ❋ ✠✠✠✠✠✠✠✠✠✠✠

˜ Πε

TE ∗

• ❍

❍ ❍ ❍ ❍ ✡✡✡✡✡✡✡✡✡✡ T∗E

• ❊

❊ ❊ ❊ ❊ ❊ ✡✡✡✡✡✡✡✡✡✡

ε

• E

ρ

• ✡✡✡✡✡✡✡✡✡✡

TM ✠✠✠✠✠✠✠✠✠✠✠

E

☞☞☞☞☞☞☞☞☞☞

ρ

• dL
• λL

❧❧❧❧❧❧❧❧❧❧❧❧

ΛL

❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨

E ∗

• ❍

❍ ❍ ❍ ❍ ❍

dH

• E ∗
• ❋

❋ ❋ ❋ ❋ ❋

E ∗

• ❋

❋ ❋ ❋ ❋ ❋

M

M

M

• H : E ∗ −

→ R DH = dH(E ∗) ⊂ T∗E ∗ D = ΛL(E) D = ˜ Πε(dH(E ∗)) L : E − → R DL = dL(E) ⊂ T∗E The left-hand side is Hamiltonian, the right-hand side is Lagrangian, the phase dynamics lives in the center. We can start equally with L or H, but

• nly in particular cases there is a Hamiltonian description corresponding to

a given Lagrangian (˜ Πε(dH(E ∗)) = ΛL(E)).

JG (IMPAN) Dirac Algebroids 26/11/2014 8 / 30

Algebroid setting, no constraints

DH

• D
• DL
• T∗E ∗
• ❋

❋ ❋ ❋ ❋ ❋ ✠✠✠✠✠✠✠✠✠✠✠

˜ Πε

TE ∗

• ❍

❍ ❍ ❍ ❍ ✡✡✡✡✡✡✡✡✡✡ T∗E

• ❊

❊ ❊ ❊ ❊ ❊ ✡✡✡✡✡✡✡✡✡✡

ε

• E

ρ

• ✡✡✡✡✡✡✡✡✡✡

TM ✠✠✠✠✠✠✠✠✠✠✠

E

☞☞☞☞☞☞☞☞☞☞

ρ

• dL
• λL

❧❧❧❧❧❧❧❧❧❧❧❧

ΛL

❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨

E ∗

• ❍

❍ ❍ ❍ ❍ ❍

dH

• E ∗
• ❋

❋ ❋ ❋ ❋ ❋

E ∗

• ❋

❋ ❋ ❋ ❋ ❋

M

M

M

• H : E ∗ −

→ R DH = dH(E ∗) ⊂ T∗E ∗ D = ΛL(E) D = ˜ Πε(dH(E ∗)) L : E − → R DL = dL(E) ⊂ T∗E The left-hand side is Hamiltonian, the right-hand side is Lagrangian, the phase dynamics lives in the center. We can start equally with L or H, but

• nly in particular cases there is a Hamiltonian description corresponding to

a given Lagrangian (˜ Πε(dH(E ∗)) = ΛL(E)).

JG (IMPAN) Dirac Algebroids 26/11/2014 8 / 30

Algebroids setting with constraints

Consider a constraint submanifold iS : S ֒ → E. Vakonomic constraint procedure (which is actually quite ‘holonomic’): Lagrangian L : S → R defines a lagrangian submanifold DL(S) ⊂ T∗E.

T∗E ⊃ DL(S)

ε

• πE
• D ⊂ TE ∗

τE∗

• S ⊂ E

λL

✤

ΛL

✯ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥

E ∗ . Here, ΛL and λL are relations! Similarly on the Hamiltonian side. Nonholonomic constrain procedure: S is a vector subbundle, L : E → R.

T∗E

ε

• πE
• TE ∗

Ti∗

S

D ⊂ TS∗

τS∗

• S ⊂ E

dL

• λL
• ΛL
• ❢

❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢

E ∗

i∗

S

S∗

. Not exactly a Tulczyjew triple. Solution: Dirac algebroids.

JG (IMPAN) Dirac Algebroids 26/11/2014 9 / 30

Algebroids setting with constraints

Consider a constraint submanifold iS : S ֒ → E. Vakonomic constraint procedure (which is actually quite ‘holonomic’): Lagrangian L : S → R defines a lagrangian submanifold DL(S) ⊂ T∗E.

T∗E ⊃ DL(S)

ε

• πE
• D ⊂ TE ∗

τE∗

• S ⊂ E

λL

✤

ΛL

✯ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥

E ∗ . Here, ΛL and λL are relations! Similarly on the Hamiltonian side. Nonholonomic constrain procedure: S is a vector subbundle, L : E → R.

T∗E

ε

• πE
• TE ∗

Ti∗

S

D ⊂ TS∗

τS∗

• S ⊂ E

dL

• λL
• ΛL
• ❢

❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢

E ∗

i∗

S

S∗

. Not exactly a Tulczyjew triple. Solution: Dirac algebroids.

JG (IMPAN) Dirac Algebroids 26/11/2014 9 / 30

Algebroids setting with constraints

Consider a constraint submanifold iS : S ֒ → E. Vakonomic constraint procedure (which is actually quite ‘holonomic’): Lagrangian L : S → R defines a lagrangian submanifold DL(S) ⊂ T∗E.

T∗E ⊃ DL(S)

ε

• πE
• D ⊂ TE ∗

τE∗

• S ⊂ E

λL

✤

ΛL

✯ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥

E ∗ . Here, ΛL and λL are relations! Similarly on the Hamiltonian side. Nonholonomic constrain procedure: S is a vector subbundle, L : E → R.

T∗E

ε

• πE
• TE ∗

Ti∗

S

D ⊂ TS∗

τS∗

• S ⊂ E

dL

• λL
• ΛL
• ❢

❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢

E ∗

i∗

S

S∗

. Not exactly a Tulczyjew triple. Solution: Dirac algebroids.

JG (IMPAN) Dirac Algebroids 26/11/2014 9 / 30

Algebroids setting with constraints

Consider a constraint submanifold iS : S ֒ → E. Vakonomic constraint procedure (which is actually quite ‘holonomic’): Lagrangian L : S → R defines a lagrangian submanifold DL(S) ⊂ T∗E.

T∗E ⊃ DL(S)

ε

• πE
• D ⊂ TE ∗

τE∗

• S ⊂ E

λL

✤

ΛL

✯ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥

E ∗ . Here, ΛL and λL are relations! Similarly on the Hamiltonian side. Nonholonomic constrain procedure: S is a vector subbundle, L : E → R.

T∗E

ε

• πE
• TE ∗

Ti∗

S

D ⊂ TS∗

τS∗

• S ⊂ E

dL

• λL
• ΛL
• ❢

❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢

E ∗

i∗

S

S∗

. Not exactly a Tulczyjew triple. Solution: Dirac algebroids.

JG (IMPAN) Dirac Algebroids 26/11/2014 9 / 30

Algebroids setting with constraints

Consider a constraint submanifold iS : S ֒ → E. Vakonomic constraint procedure (which is actually quite ‘holonomic’): Lagrangian L : S → R defines a lagrangian submanifold DL(S) ⊂ T∗E.

T∗E ⊃ DL(S)

ε

• πE
• D ⊂ TE ∗

τE∗

• S ⊂ E

λL

✤

ΛL

✯ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥

E ∗ . Here, ΛL and λL are relations! Similarly on the Hamiltonian side. Nonholonomic constrain procedure: S is a vector subbundle, L : E → R.

T∗E

ε

• πE
• TE ∗

Ti∗

S

D ⊂ TS∗

τS∗

• S ⊂ E

dL

• λL
• ΛL
• ❢

❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢

E ∗

i∗

S

S∗

. Not exactly a Tulczyjew triple. Solution: Dirac algebroids.

JG (IMPAN) Dirac Algebroids 26/11/2014 9 / 30

Algebroids setting with constraints

Consider a constraint submanifold iS : S ֒ → E. Vakonomic constraint procedure (which is actually quite ‘holonomic’): Lagrangian L : S → R defines a lagrangian submanifold DL(S) ⊂ T∗E.

T∗E ⊃ DL(S)

ε

• πE
• D ⊂ TE ∗

τE∗

• S ⊂ E

λL

✤

ΛL

✯ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥

E ∗ . Here, ΛL and λL are relations! Similarly on the Hamiltonian side. Nonholonomic constrain procedure: S is a vector subbundle, L : E → R.

T∗E

ε

• πE
• TE ∗

Ti∗

S

D ⊂ TS∗

τS∗

• S ⊂ E

dL

• λL
• ΛL
• ❢

❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢

E ∗

i∗

S

S∗

. Not exactly a Tulczyjew triple. Solution: Dirac algebroids.

JG (IMPAN) Dirac Algebroids 26/11/2014 9 / 30

Algebroids setting with constraints

Consider a constraint submanifold iS : S ֒ → E. Vakonomic constraint procedure (which is actually quite ‘holonomic’): Lagrangian L : S → R defines a lagrangian submanifold DL(S) ⊂ T∗E.

T∗E ⊃ DL(S)

ε

• πE
• D ⊂ TE ∗

τE∗

• S ⊂ E

λL

✤

ΛL

✯ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥

E ∗ . Here, ΛL and λL are relations! Similarly on the Hamiltonian side. Nonholonomic constrain procedure: S is a vector subbundle, L : E → R.

T∗E

ε

• πE
• TE ∗

Ti∗

S

D ⊂ TS∗

τS∗

• S ⊂ E

dL

• λL
• ΛL
• ❢

❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢

E ∗

i∗

S

S∗

. Not exactly a Tulczyjew triple. Solution: Dirac algebroids.

JG (IMPAN) Dirac Algebroids 26/11/2014 9 / 30

Algebroids setting with constraints

Consider a constraint submanifold iS : S ֒ → E. Vakonomic constraint procedure (which is actually quite ‘holonomic’): Lagrangian L : S → R defines a lagrangian submanifold DL(S) ⊂ T∗E.

T∗E ⊃ DL(S)

ε

• πE
• D ⊂ TE ∗

τE∗

• S ⊂ E

λL

✤

ΛL

✯ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥

E ∗ . Here, ΛL and λL are relations! Similarly on the Hamiltonian side. Nonholonomic constrain procedure: S is a vector subbundle, L : E → R.

T∗E

ε

• πE
• TE ∗

Ti∗

S

D ⊂ TS∗

τS∗

• S ⊂ E

dL

• λL
• ΛL
• ❢

❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢

E ∗

i∗

S

S∗

. Not exactly a Tulczyjew triple. Solution: Dirac algebroids.

JG (IMPAN) Dirac Algebroids 26/11/2014 9 / 30

Algebroids setting with constraints

Consider a constraint submanifold iS : S ֒ → E. Vakonomic constraint procedure (which is actually quite ‘holonomic’): Lagrangian L : S → R defines a lagrangian submanifold DL(S) ⊂ T∗E.

T∗E ⊃ DL(S)

ε

• πE
• D ⊂ TE ∗

τE∗

• S ⊂ E

λL

✤

ΛL

✯ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥

E ∗ . Here, ΛL and λL are relations! Similarly on the Hamiltonian side. Nonholonomic constrain procedure: S is a vector subbundle, L : E → R.

T∗E

ε

• πE
• TE ∗

Ti∗

S

D ⊂ TS∗

τS∗

• S ⊂ E

dL

• λL
• ΛL
• ❢

❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢

E ∗

i∗

S

S∗

. Not exactly a Tulczyjew triple. Solution: Dirac algebroids.

JG (IMPAN) Dirac Algebroids 26/11/2014 9 / 30

Dirac structures

There is a canonical symmetric pairing on the extended tangent bundle (Pontryagin bundle) T N = TN ⊕N T∗N: (X1 + α1 | X2 + α2) = 1 2 (α1(X2) + α2(X1)) . Courant-Dorfman bracket on the space of Sec(T N): [ [X1 + α1, X2 + α2] ] = [X1, X2] + LX1α2 − ıX2dα1 .

Definition

An almost Dirac structure on the smooth manifold N is a subbundle D of T N which is maximally isotropic with respect to the symmetric pairing (·|·). If, additionally, the space of sections of D is closed under the Courant-Dorfman bracket, we speak about a Dirac structure. Note that here a subbundle D may be supported on a submanifold N0 ⊂ N.

JG (IMPAN) Dirac Algebroids 26/11/2014 10 / 30

Dirac structures

There is a canonical symmetric pairing on the extended tangent bundle (Pontryagin bundle) T N = TN ⊕N T∗N: (X1 + α1 | X2 + α2) = 1 2 (α1(X2) + α2(X1)) . Courant-Dorfman bracket on the space of Sec(T N): [ [X1 + α1, X2 + α2] ] = [X1, X2] + LX1α2 − ıX2dα1 .

Definition

An almost Dirac structure on the smooth manifold N is a subbundle D of T N which is maximally isotropic with respect to the symmetric pairing (·|·). If, additionally, the space of sections of D is closed under the Courant-Dorfman bracket, we speak about a Dirac structure. Note that here a subbundle D may be supported on a submanifold N0 ⊂ N.

JG (IMPAN) Dirac Algebroids 26/11/2014 10 / 30

Dirac structures

There is a canonical symmetric pairing on the extended tangent bundle (Pontryagin bundle) T N = TN ⊕N T∗N: (X1 + α1 | X2 + α2) = 1 2 (α1(X2) + α2(X1)) . Courant-Dorfman bracket on the space of Sec(T N): [ [X1 + α1, X2 + α2] ] = [X1, X2] + LX1α2 − ıX2dα1 .

Definition

An almost Dirac structure on the smooth manifold N is a subbundle D of T N which is maximally isotropic with respect to the symmetric pairing (·|·). If, additionally, the space of sections of D is closed under the Courant-Dorfman bracket, we speak about a Dirac structure. Note that here a subbundle D may be supported on a submanifold N0 ⊂ N.

JG (IMPAN) Dirac Algebroids 26/11/2014 10 / 30

Dirac structures

There is a canonical symmetric pairing on the extended tangent bundle (Pontryagin bundle) T N = TN ⊕N T∗N: (X1 + α1 | X2 + α2) = 1 2 (α1(X2) + α2(X1)) . Courant-Dorfman bracket on the space of Sec(T N): [ [X1 + α1, X2 + α2] ] = [X1, X2] + LX1α2 − ıX2dα1 .

Definition

An almost Dirac structure on the smooth manifold N is a subbundle D of T N which is maximally isotropic with respect to the symmetric pairing (·|·). If, additionally, the space of sections of D is closed under the Courant-Dorfman bracket, we speak about a Dirac structure. Note that here a subbundle D may be supported on a submanifold N0 ⊂ N.

JG (IMPAN) Dirac Algebroids 26/11/2014 10 / 30

Dirac structures

There is a canonical symmetric pairing on the extended tangent bundle (Pontryagin bundle) T N = TN ⊕N T∗N: (X1 + α1 | X2 + α2) = 1 2 (α1(X2) + α2(X1)) . Courant-Dorfman bracket on the space of Sec(T N): [ [X1 + α1, X2 + α2] ] = [X1, X2] + LX1α2 − ıX2dα1 .

Definition

An almost Dirac structure on the smooth manifold N is a subbundle D of T N which is maximally isotropic with respect to the symmetric pairing (·|·). If, additionally, the space of sections of D is closed under the Courant-Dorfman bracket, we speak about a Dirac structure. Note that here a subbundle D may be supported on a submanifold N0 ⊂ N.

JG (IMPAN) Dirac Algebroids 26/11/2014 10 / 30

Dirac structures

There is a canonical symmetric pairing on the extended tangent bundle (Pontryagin bundle) T N = TN ⊕N T∗N: (X1 + α1 | X2 + α2) = 1 2 (α1(X2) + α2(X1)) . Courant-Dorfman bracket on the space of Sec(T N): [ [X1 + α1, X2 + α2] ] = [X1, X2] + LX1α2 − ıX2dα1 .

Definition

An almost Dirac structure on the smooth manifold N is a subbundle D of T N which is maximally isotropic with respect to the symmetric pairing (·|·). If, additionally, the space of sections of D is closed under the Courant-Dorfman bracket, we speak about a Dirac structure. Note that here a subbundle D may be supported on a submanifold N0 ⊂ N.

JG (IMPAN) Dirac Algebroids 26/11/2014 10 / 30

Dirac structures

For Π ∈ Sec(2 TN), Π : T∗N ∋ α − → ıαΠ ∈ TN, D = graph( Π) ⊂ T N is an almost Dirac structure . Π is a Poisson ⇔ D is a Dirac structure. For ω ∈ Sec(2 T∗N), ω : TN ∋ X − → −ıXω ∈ T∗N, D = graph( ω) ⊂ T N is an almost Dirac structure . ω is a closed ⇔ D is a Dirac structure. For a distribution ∆ on N, D = ∆ ⊕ ∆⊥ ⊂ T N is an almost Dirac structure . ∆ is integrable ⇔ D is a Dirac structure.

JG (IMPAN) Dirac Algebroids 26/11/2014 11 / 30

Dirac structures

For Π ∈ Sec(2 TN), Π : T∗N ∋ α − → ıαΠ ∈ TN, D = graph( Π) ⊂ T N is an almost Dirac structure . Π is a Poisson ⇔ D is a Dirac structure. For ω ∈ Sec(2 T∗N), ω : TN ∋ X − → −ıXω ∈ T∗N, D = graph( ω) ⊂ T N is an almost Dirac structure . ω is a closed ⇔ D is a Dirac structure. For a distribution ∆ on N, D = ∆ ⊕ ∆⊥ ⊂ T N is an almost Dirac structure . ∆ is integrable ⇔ D is a Dirac structure.

JG (IMPAN) Dirac Algebroids 26/11/2014 11 / 30

Dirac structures

For Π ∈ Sec(2 TN), Π : T∗N ∋ α − → ıαΠ ∈ TN, D = graph( Π) ⊂ T N is an almost Dirac structure . Π is a Poisson ⇔ D is a Dirac structure. For ω ∈ Sec(2 T∗N), ω : TN ∋ X − → −ıXω ∈ T∗N, D = graph( ω) ⊂ T N is an almost Dirac structure . ω is a closed ⇔ D is a Dirac structure. For a distribution ∆ on N, D = ∆ ⊕ ∆⊥ ⊂ T N is an almost Dirac structure . ∆ is integrable ⇔ D is a Dirac structure.

JG (IMPAN) Dirac Algebroids 26/11/2014 11 / 30

Dirac structures

For Π ∈ Sec(2 TN), Π : T∗N ∋ α − → ıαΠ ∈ TN, D = graph( Π) ⊂ T N is an almost Dirac structure . Π is a Poisson ⇔ D is a Dirac structure. For ω ∈ Sec(2 T∗N), ω : TN ∋ X − → −ıXω ∈ T∗N, D = graph( ω) ⊂ T N is an almost Dirac structure . ω is a closed ⇔ D is a Dirac structure. For a distribution ∆ on N, D = ∆ ⊕ ∆⊥ ⊂ T N is an almost Dirac structure . ∆ is integrable ⇔ D is a Dirac structure.

JG (IMPAN) Dirac Algebroids 26/11/2014 11 / 30

Dirac structures

For Π ∈ Sec(2 TN), Π : T∗N ∋ α − → ıαΠ ∈ TN, D = graph( Π) ⊂ T N is an almost Dirac structure . Π is a Poisson ⇔ D is a Dirac structure. For ω ∈ Sec(2 T∗N), ω : TN ∋ X − → −ıXω ∈ T∗N, D = graph( ω) ⊂ T N is an almost Dirac structure . ω is a closed ⇔ D is a Dirac structure. For a distribution ∆ on N, D = ∆ ⊕ ∆⊥ ⊂ T N is an almost Dirac structure . ∆ is integrable ⇔ D is a Dirac structure.

JG (IMPAN) Dirac Algebroids 26/11/2014 11 / 30

Dirac structures

For Π ∈ Sec(2 TN), Π : T∗N ∋ α − → ıαΠ ∈ TN, D = graph( Π) ⊂ T N is an almost Dirac structure . Π is a Poisson ⇔ D is a Dirac structure. For ω ∈ Sec(2 T∗N), ω : TN ∋ X − → −ıXω ∈ T∗N, D = graph( ω) ⊂ T N is an almost Dirac structure . ω is a closed ⇔ D is a Dirac structure. For a distribution ∆ on N, D = ∆ ⊕ ∆⊥ ⊂ T N is an almost Dirac structure . ∆ is integrable ⇔ D is a Dirac structure.

JG (IMPAN) Dirac Algebroids 26/11/2014 11 / 30

Dirac algebroids

(Lie) algebroids on E are linear Poisson structures on E ∗, thus ‘linear’ Dirac structures on E ⊕ E ∗. Linearity of different geometrical structures is related to double vector bundle structures. A bivector field Π on a vector bundle F is linear if the corresponding map

• Π : T∗F −

→ TF is a morphism of double vector bundles. A two-form ω on a vector bundle F is linear if the corresponding map

• ω : TF −

→ T∗F is a morphism of double vector bundles, etc.

JG (IMPAN) Dirac Algebroids 26/11/2014 12 / 30

Dirac algebroids

(Lie) algebroids on E are linear Poisson structures on E ∗, thus ‘linear’ Dirac structures on E ⊕ E ∗. Linearity of different geometrical structures is related to double vector bundle structures. A bivector field Π on a vector bundle F is linear if the corresponding map

• Π : T∗F −

→ TF is a morphism of double vector bundles. A two-form ω on a vector bundle F is linear if the corresponding map

• ω : TF −

→ T∗F is a morphism of double vector bundles, etc.

JG (IMPAN) Dirac Algebroids 26/11/2014 12 / 30

Dirac algebroids

(Lie) algebroids on E are linear Poisson structures on E ∗, thus ‘linear’ Dirac structures on E ⊕ E ∗. Linearity of different geometrical structures is related to double vector bundle structures. A bivector field Π on a vector bundle F is linear if the corresponding map

• Π : T∗F −

→ TF is a morphism of double vector bundles. A two-form ω on a vector bundle F is linear if the corresponding map

• ω : TF −

→ T∗F is a morphism of double vector bundles, etc.

JG (IMPAN) Dirac Algebroids 26/11/2014 12 / 30

Dirac algebroids

(Lie) algebroids on E are linear Poisson structures on E ∗, thus ‘linear’ Dirac structures on E ⊕ E ∗. Linearity of different geometrical structures is related to double vector bundle structures. A bivector field Π on a vector bundle F is linear if the corresponding map

• Π : T∗F −

→ TF is a morphism of double vector bundles. A two-form ω on a vector bundle F is linear if the corresponding map

• ω : TF −

→ T∗F is a morphism of double vector bundles, etc.

JG (IMPAN) Dirac Algebroids 26/11/2014 12 / 30

Dirac algebroids

Definition

A Dirac algebroid (resp., Dirac-Lie algebroid) structure on a vector bundle E is an almost Dirac (resp., Dirac) subbundle D of T E ∗ being a double vector subbundle, i.e., D is not only a subbundle of τ1 : T E ∗ → E ∗ but also a vector subbundle of the vector bundle τ2 : T E ∗ → TM ⊕M E.

TE ∗ ⊕E ∗ T∗E ∗

τ2

• ◗

◗ ◗ ◗ ◗ ◗ ◗ ◗

τ1

rrrrrrr

E ∗

π

• ▼

▼ ▼ ▼ ▼ ▼ ▼ ▼ TM ⊕M E

(τM,τ)

♠♠♠♠♠♠♠♠♠♠

M D

τ D

2

• ▼

▼ ▼ ▼ ▼ ▼ ▼

τ D

1

rrrrrrr

PhD

πD

• ▲

▲ ▲ ▲ ▲ ▲

VelD

τ D

rrrrrr

MD PhD = τ1(D) - the phase bundle. VelD = τ2(D) - the velocity bundle (anchor relation).

JG (IMPAN) Dirac Algebroids 26/11/2014 13 / 30

Dirac algebroids

Definition

A Dirac algebroid (resp., Dirac-Lie algebroid) structure on a vector bundle E is an almost Dirac (resp., Dirac) subbundle D of T E ∗ being a double vector subbundle, i.e., D is not only a subbundle of τ1 : T E ∗ → E ∗ but also a vector subbundle of the vector bundle τ2 : T E ∗ → TM ⊕M E.

TE ∗ ⊕E ∗ T∗E ∗

τ2

• ◗

◗ ◗ ◗ ◗ ◗ ◗ ◗

τ1

rrrrrrr

E ∗

π

• ▼

▼ ▼ ▼ ▼ ▼ ▼ ▼ TM ⊕M E

(τM,τ)

♠♠♠♠♠♠♠♠♠♠

M D

τ D

2

• ▼

▼ ▼ ▼ ▼ ▼ ▼

τ D

1

rrrrrrr

PhD

πD

• ▲

▲ ▲ ▲ ▲ ▲

VelD

τ D

rrrrrr

MD PhD = τ1(D) - the phase bundle. VelD = τ2(D) - the velocity bundle (anchor relation).

JG (IMPAN) Dirac Algebroids 26/11/2014 13 / 30

Dirac algebroids

Definition

A Dirac algebroid (resp., Dirac-Lie algebroid) structure on a vector bundle E is an almost Dirac (resp., Dirac) subbundle D of T E ∗ being a double vector subbundle, i.e., D is not only a subbundle of τ1 : T E ∗ → E ∗ but also a vector subbundle of the vector bundle τ2 : T E ∗ → TM ⊕M E.

TE ∗ ⊕E ∗ T∗E ∗

τ2

• ◗

◗ ◗ ◗ ◗ ◗ ◗ ◗

τ1

rrrrrrr

E ∗

π

• ▼

▼ ▼ ▼ ▼ ▼ ▼ ▼ TM ⊕M E

(τM,τ)

♠♠♠♠♠♠♠♠♠♠

M D

τ D

2

• ▼

▼ ▼ ▼ ▼ ▼ ▼

τ D

1

rrrrrrr

PhD

πD

• ▲

▲ ▲ ▲ ▲ ▲

VelD

τ D

rrrrrr

MD PhD = τ1(D) - the phase bundle. VelD = τ2(D) - the velocity bundle (anchor relation).

JG (IMPAN) Dirac Algebroids 26/11/2014 13 / 30

Dirac algebroids

Definition

A Dirac algebroid (resp., Dirac-Lie algebroid) structure on a vector bundle E is an almost Dirac (resp., Dirac) subbundle D of T E ∗ being a double vector subbundle, i.e., D is not only a subbundle of τ1 : T E ∗ → E ∗ but also a vector subbundle of the vector bundle τ2 : T E ∗ → TM ⊕M E.

TE ∗ ⊕E ∗ T∗E ∗

τ2

• ◗

◗ ◗ ◗ ◗ ◗ ◗ ◗

τ1

rrrrrrr

E ∗

π

• ▼

▼ ▼ ▼ ▼ ▼ ▼ ▼ TM ⊕M E

(τM,τ)

♠♠♠♠♠♠♠♠♠♠

M D

τ D

2

• ▼

▼ ▼ ▼ ▼ ▼ ▼

τ D

1

rrrrrrr

PhD

πD

• ▲

▲ ▲ ▲ ▲ ▲

VelD

τ D

rrrrrr

MD PhD = τ1(D) - the phase bundle. VelD = τ2(D) - the velocity bundle (anchor relation).

JG (IMPAN) Dirac Algebroids 26/11/2014 13 / 30

Dirac algebroids

Definition

A Dirac algebroid (resp., Dirac-Lie algebroid) structure on a vector bundle E is an almost Dirac (resp., Dirac) subbundle D of T E ∗ being a double vector subbundle, i.e., D is not only a subbundle of τ1 : T E ∗ → E ∗ but also a vector subbundle of the vector bundle τ2 : T E ∗ → TM ⊕M E.

TE ∗ ⊕E ∗ T∗E ∗

τ2

• ◗

◗ ◗ ◗ ◗ ◗ ◗ ◗

τ1

rrrrrrr

E ∗

π

• ▼

▼ ▼ ▼ ▼ ▼ ▼ ▼ TM ⊕M E

(τM,τ)

♠♠♠♠♠♠♠♠♠♠

M D

τ D

2

• ▼

▼ ▼ ▼ ▼ ▼ ▼

τ D

1

rrrrrrr

PhD

πD

• ▲

▲ ▲ ▲ ▲ ▲

VelD

τ D

rrrrrr

MD PhD = τ1(D) - the phase bundle. VelD = τ2(D) - the velocity bundle (anchor relation).

JG (IMPAN) Dirac Algebroids 26/11/2014 13 / 30

Dirac algebroids – examples

• 1. The graph of any linear bivector field is a Dirac algebroid,

Π = 1 2ck

ij (x)ξk∂ξi ∧ ∂ξj + ρb i (x)∂ξi ∧ ∂xb,

ck

ij (x) = −ck ji (x) ,

DΠ =

• (xa, ξi, ˙

xb, ˙ ξj, pc, yk) : ˙ xb = ρb

k(x)yk , ˙

ξj = ck

ij (x)yiξk − ρa j (x)pa

• .

The phase bundle is the whole E ∗, the velocity bundle is the graph of ρ : E → TM: VelDΠ =

• (xa, ˙

xb, yk) : ˙ xb = ρb

k(x)yk

⊂ TM ⊕M E.

• 2. The graph of any linear two-form is a Dirac algebroid. The phase

bundle is the whole E ∗, the velocity bundle is the graph of ρ : TM → E: VelDω = {(xa, ˙ xb, yk) : yk = ρk

a(x) ˙

xa} ⊂ TM ⊕M E .

• 3. The canonical Dirac-Lie algebroid, DM = DΠM = DωM, corresponding to

the canonical Lie algebroid E = TM, belongs to the both above types.

JG (IMPAN) Dirac Algebroids 26/11/2014 14 / 30

Dirac algebroids – examples

• 1. The graph of any linear bivector field is a Dirac algebroid,

Π = 1 2ck

ij (x)ξk∂ξi ∧ ∂ξj + ρb i (x)∂ξi ∧ ∂xb,

ck

ij (x) = −ck ji (x) ,

DΠ =

• (xa, ξi, ˙

xb, ˙ ξj, pc, yk) : ˙ xb = ρb

k(x)yk , ˙

ξj = ck

ij (x)yiξk − ρa j (x)pa

• .

The phase bundle is the whole E ∗, the velocity bundle is the graph of ρ : E → TM: VelDΠ =

• (xa, ˙

xb, yk) : ˙ xb = ρb

k(x)yk

⊂ TM ⊕M E.

• 2. The graph of any linear two-form is a Dirac algebroid. The phase

bundle is the whole E ∗, the velocity bundle is the graph of ρ : TM → E: VelDω = {(xa, ˙ xb, yk) : yk = ρk

a(x) ˙

xa} ⊂ TM ⊕M E .

• 3. The canonical Dirac-Lie algebroid, DM = DΠM = DωM, corresponding to

the canonical Lie algebroid E = TM, belongs to the both above types.

JG (IMPAN) Dirac Algebroids 26/11/2014 14 / 30

Dirac algebroids – examples

• 1. The graph of any linear bivector field is a Dirac algebroid,

Π = 1 2ck

ij (x)ξk∂ξi ∧ ∂ξj + ρb i (x)∂ξi ∧ ∂xb,

ck

ij (x) = −ck ji (x) ,

DΠ =

• (xa, ξi, ˙

xb, ˙ ξj, pc, yk) : ˙ xb = ρb

k(x)yk , ˙

ξj = ck

ij (x)yiξk − ρa j (x)pa

• .

The phase bundle is the whole E ∗, the velocity bundle is the graph of ρ : E → TM: VelDΠ =

• (xa, ˙

xb, yk) : ˙ xb = ρb

k(x)yk

⊂ TM ⊕M E.

• 2. The graph of any linear two-form is a Dirac algebroid. The phase

bundle is the whole E ∗, the velocity bundle is the graph of ρ : TM → E: VelDω = {(xa, ˙ xb, yk) : yk = ρk

a(x) ˙

xa} ⊂ TM ⊕M E .

• 3. The canonical Dirac-Lie algebroid, DM = DΠM = DωM, corresponding to

the canonical Lie algebroid E = TM, belongs to the both above types.

JG (IMPAN) Dirac Algebroids 26/11/2014 14 / 30

Dirac algebroids – examples

• 1. The graph of any linear bivector field is a Dirac algebroid,

Π = 1 2ck

ij (x)ξk∂ξi ∧ ∂ξj + ρb i (x)∂ξi ∧ ∂xb,

ck

ij (x) = −ck ji (x) ,

DΠ =

• (xa, ξi, ˙

xb, ˙ ξj, pc, yk) : ˙ xb = ρb

k(x)yk , ˙

ξj = ck

ij (x)yiξk − ρa j (x)pa

• .

The phase bundle is the whole E ∗, the velocity bundle is the graph of ρ : E → TM: VelDΠ =

• (xa, ˙

xb, yk) : ˙ xb = ρb

k(x)yk

⊂ TM ⊕M E.

• 2. The graph of any linear two-form is a Dirac algebroid. The phase

bundle is the whole E ∗, the velocity bundle is the graph of ρ : TM → E: VelDω = {(xa, ˙ xb, yk) : yk = ρk

a(x) ˙

xa} ⊂ TM ⊕M E .

• 3. The canonical Dirac-Lie algebroid, DM = DΠM = DωM, corresponding to

the canonical Lie algebroid E = TM, belongs to the both above types.

JG (IMPAN) Dirac Algebroids 26/11/2014 14 / 30

Dirac algebroids – examples

• 1. The graph of any linear bivector field is a Dirac algebroid,

Π = 1 2ck

ij (x)ξk∂ξi ∧ ∂ξj + ρb i (x)∂ξi ∧ ∂xb,

ck

ij (x) = −ck ji (x) ,

DΠ =

• (xa, ξi, ˙

xb, ˙ ξj, pc, yk) : ˙ xb = ρb

k(x)yk , ˙

ξj = ck

ij (x)yiξk − ρa j (x)pa

• .

The phase bundle is the whole E ∗, the velocity bundle is the graph of ρ : E → TM: VelDΠ =

• (xa, ˙

xb, yk) : ˙ xb = ρb

k(x)yk

⊂ TM ⊕M E.

• 2. The graph of any linear two-form is a Dirac algebroid. The phase

bundle is the whole E ∗, the velocity bundle is the graph of ρ : TM → E: VelDω = {(xa, ˙ xb, yk) : yk = ρk

a(x) ˙

xa} ⊂ TM ⊕M E .

• 3. The canonical Dirac-Lie algebroid, DM = DΠM = DωM, corresponding to

the canonical Lie algebroid E = TM, belongs to the both above types.

JG (IMPAN) Dirac Algebroids 26/11/2014 14 / 30

Dirac algebroids – examples

• 1. The graph of any linear bivector field is a Dirac algebroid,

Π = 1 2ck

ij (x)ξk∂ξi ∧ ∂ξj + ρb i (x)∂ξi ∧ ∂xb,

ck

ij (x) = −ck ji (x) ,

DΠ =

• (xa, ξi, ˙

xb, ˙ ξj, pc, yk) : ˙ xb = ρb

k(x)yk , ˙

ξj = ck

ij (x)yiξk − ρa j (x)pa

• .

The phase bundle is the whole E ∗, the velocity bundle is the graph of ρ : E → TM: VelDΠ =

• (xa, ˙

xb, yk) : ˙ xb = ρb

k(x)yk

⊂ TM ⊕M E.

• 2. The graph of any linear two-form is a Dirac algebroid. The phase

bundle is the whole E ∗, the velocity bundle is the graph of ρ : TM → E: VelDω = {(xa, ˙ xb, yk) : yk = ρk

a(x) ˙

xa} ⊂ TM ⊕M E .

• 3. The canonical Dirac-Lie algebroid, DM = DΠM = DωM, corresponding to

the canonical Lie algebroid E = TM, belongs to the both above types.

JG (IMPAN) Dirac Algebroids 26/11/2014 14 / 30

Dirac algebroids – examples

• 1. The graph of any linear bivector field is a Dirac algebroid,

Π = 1 2ck

ij (x)ξk∂ξi ∧ ∂ξj + ρb i (x)∂ξi ∧ ∂xb,

ck

ij (x) = −ck ji (x) ,

DΠ =

• (xa, ξi, ˙

xb, ˙ ξj, pc, yk) : ˙ xb = ρb

k(x)yk , ˙

ξj = ck

ij (x)yiξk − ρa j (x)pa

• .

The phase bundle is the whole E ∗, the velocity bundle is the graph of ρ : E → TM: VelDΠ =

• (xa, ˙

xb, yk) : ˙ xb = ρb

k(x)yk

⊂ TM ⊕M E.

• 2. The graph of any linear two-form is a Dirac algebroid. The phase

bundle is the whole E ∗, the velocity bundle is the graph of ρ : TM → E: VelDω = {(xa, ˙ xb, yk) : yk = ρk

a(x) ˙

xa} ⊂ TM ⊕M E .

• 3. The canonical Dirac-Lie algebroid, DM = DΠM = DωM, corresponding to

the canonical Lie algebroid E = TM, belongs to the both above types.

JG (IMPAN) Dirac Algebroids 26/11/2014 14 / 30

Dirac algebroids – local structure

Assume for simplicity that D is supported on the whole E ∗.

Theorem

One can find local coordinates (x, ¯ ξ, ξ, η, η, ζ, ζ) in T E ∗ such that (x, ¯ ξ, ξ) are coordinates in E ∗ with PhD = { ξ = 0} , (x, η, η) are coordinates in TM ⊕M E with VelD = { η = 0} , (x, ζ, ζ) are dual coordinates in T∗M ⊕M E ∗ with Vel⊥

D = {ζ = 0} ,

D = {(x, ¯ ξ, 0, η, 0, ζ, ζ) : ζk = ci

jk(x)ηj ¯

ξi}, for some ‘structure coefficients’ ci

jk(x).

JG (IMPAN) Dirac Algebroids 26/11/2014 15 / 30

Dirac algebroids – local structure

Assume for simplicity that D is supported on the whole E ∗.

Theorem

One can find local coordinates (x, ¯ ξ, ξ, η, η, ζ, ζ) in T E ∗ such that (x, ¯ ξ, ξ) are coordinates in E ∗ with PhD = { ξ = 0} , (x, η, η) are coordinates in TM ⊕M E with VelD = { η = 0} , (x, ζ, ζ) are dual coordinates in T∗M ⊕M E ∗ with Vel⊥

D = {ζ = 0} ,

D = {(x, ¯ ξ, 0, η, 0, ζ, ζ) : ζk = ci

jk(x)ηj ¯

ξi}, for some ‘structure coefficients’ ci

jk(x).

JG (IMPAN) Dirac Algebroids 26/11/2014 15 / 30

Dirac algebroids – local structure

Assume for simplicity that D is supported on the whole E ∗.

Theorem

One can find local coordinates (x, ¯ ξ, ξ, η, η, ζ, ζ) in T E ∗ such that (x, ¯ ξ, ξ) are coordinates in E ∗ with PhD = { ξ = 0} , (x, η, η) are coordinates in TM ⊕M E with VelD = { η = 0} , (x, ζ, ζ) are dual coordinates in T∗M ⊕M E ∗ with Vel⊥

D = {ζ = 0} ,

D = {(x, ¯ ξ, 0, η, 0, ζ, ζ) : ζk = ci

jk(x)ηj ¯

ξi}, for some ‘structure coefficients’ ci

jk(x).

JG (IMPAN) Dirac Algebroids 26/11/2014 15 / 30

Dirac algebroids – local structure

Assume for simplicity that D is supported on the whole E ∗.

Theorem

One can find local coordinates (x, ¯ ξ, ξ, η, η, ζ, ζ) in T E ∗ such that (x, ¯ ξ, ξ) are coordinates in E ∗ with PhD = { ξ = 0} , (x, η, η) are coordinates in TM ⊕M E with VelD = { η = 0} , (x, ζ, ζ) are dual coordinates in T∗M ⊕M E ∗ with Vel⊥

D = {ζ = 0} ,

D = {(x, ¯ ξ, 0, η, 0, ζ, ζ) : ζk = ci

jk(x)ηj ¯

ξi}, for some ‘structure coefficients’ ci

jk(x).

JG (IMPAN) Dirac Algebroids 26/11/2014 15 / 30

Dirac algebroids – local structure

Assume for simplicity that D is supported on the whole E ∗.

Theorem

One can find local coordinates (x, ¯ ξ, ξ, η, η, ζ, ζ) in T E ∗ such that (x, ¯ ξ, ξ) are coordinates in E ∗ with PhD = { ξ = 0} , (x, η, η) are coordinates in TM ⊕M E with VelD = { η = 0} , (x, ζ, ζ) are dual coordinates in T∗M ⊕M E ∗ with Vel⊥

D = {ζ = 0} ,

D = {(x, ¯ ξ, 0, η, 0, ζ, ζ) : ζk = ci

jk(x)ηj ¯

ξi}, for some ‘structure coefficients’ ci

jk(x).

JG (IMPAN) Dirac Algebroids 26/11/2014 15 / 30

How to deal with (nonholonomic) constraints?

Remember that constraints are not given as just kinematic configurations but also as constraints on virtual displacements (vakonomic, nonholonomic,...). There are two main possibilities to describe a system with constraints:

1 To keep the structure of the triple unchanged and modify the way of

generating D out of L or H.

2 Keep the scheme of generating unchanged and modify the structure

(Dirac algebroid) of the triple. The latter will be our choice. No ad hoc corrections, no Poincar´ e-Cartan forms, no brackets, no junk!

JG (IMPAN) Dirac Algebroids 26/11/2014 16 / 30

How to deal with (nonholonomic) constraints?

Remember that constraints are not given as just kinematic configurations but also as constraints on virtual displacements (vakonomic, nonholonomic,...). There are two main possibilities to describe a system with constraints:

1 To keep the structure of the triple unchanged and modify the way of

generating D out of L or H.

2 Keep the scheme of generating unchanged and modify the structure

(Dirac algebroid) of the triple. The latter will be our choice. No ad hoc corrections, no Poincar´ e-Cartan forms, no brackets, no junk!

JG (IMPAN) Dirac Algebroids 26/11/2014 16 / 30

How to deal with (nonholonomic) constraints?

Remember that constraints are not given as just kinematic configurations but also as constraints on virtual displacements (vakonomic, nonholonomic,...). There are two main possibilities to describe a system with constraints:

1 To keep the structure of the triple unchanged and modify the way of

generating D out of L or H.

2 Keep the scheme of generating unchanged and modify the structure

(Dirac algebroid) of the triple. The latter will be our choice. No ad hoc corrections, no Poincar´ e-Cartan forms, no brackets, no junk!

JG (IMPAN) Dirac Algebroids 26/11/2014 16 / 30

How to deal with (nonholonomic) constraints?

Remember that constraints are not given as just kinematic configurations but also as constraints on virtual displacements (vakonomic, nonholonomic,...). There are two main possibilities to describe a system with constraints:

1 To keep the structure of the triple unchanged and modify the way of

generating D out of L or H.

2 Keep the scheme of generating unchanged and modify the structure

(Dirac algebroid) of the triple. The latter will be our choice. No ad hoc corrections, no Poincar´ e-Cartan forms, no brackets, no junk!

JG (IMPAN) Dirac Algebroids 26/11/2014 16 / 30

How to deal with (nonholonomic) constraints?

Remember that constraints are not given as just kinematic configurations but also as constraints on virtual displacements (vakonomic, nonholonomic,...). There are two main possibilities to describe a system with constraints:

1 To keep the structure of the triple unchanged and modify the way of

generating D out of L or H.

2 Keep the scheme of generating unchanged and modify the structure

(Dirac algebroid) of the triple. The latter will be our choice. No ad hoc corrections, no Poincar´ e-Cartan forms, no brackets, no junk!

JG (IMPAN) Dirac Algebroids 26/11/2014 16 / 30

How to deal with (nonholonomic) constraints?

Remember that constraints are not given as just kinematic configurations but also as constraints on virtual displacements (vakonomic, nonholonomic,...). There are two main possibilities to describe a system with constraints:

1 To keep the structure of the triple unchanged and modify the way of

generating D out of L or H.

2 Keep the scheme of generating unchanged and modify the structure

(Dirac algebroid) of the triple. The latter will be our choice. No ad hoc corrections, no Poincar´ e-Cartan forms, no brackets, no junk!

JG (IMPAN) Dirac Algebroids 26/11/2014 16 / 30

How to deal with (nonholonomic) constraints?

Remember that constraints are not given as just kinematic configurations but also as constraints on virtual displacements (vakonomic, nonholonomic,...). There are two main possibilities to describe a system with constraints:

1 To keep the structure of the triple unchanged and modify the way of

generating D out of L or H.

2 Keep the scheme of generating unchanged and modify the structure

(Dirac algebroid) of the triple. The latter will be our choice. No ad hoc corrections, no Poincar´ e-Cartan forms, no brackets, no junk!

JG (IMPAN) Dirac Algebroids 26/11/2014 16 / 30

How to deal with (nonholonomic) constraints?

Remember that constraints are not given as just kinematic configurations but also as constraints on virtual displacements (vakonomic, nonholonomic,...). There are two main possibilities to describe a system with constraints:

1 To keep the structure of the triple unchanged and modify the way of

generating D out of L or H.

2 Keep the scheme of generating unchanged and modify the structure

(Dirac algebroid) of the triple. The latter will be our choice. No ad hoc corrections, no Poincar´ e-Cartan forms, no brackets, no junk!

JG (IMPAN) Dirac Algebroids 26/11/2014 16 / 30

Dirac algebroid induced by constraints

Initial data: Dirac algebroid D on E and a vector subbundle V of VelD. D

τ D

2

• ❋

❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋

τ D

1

①①①①①①①①

PhD

πD

• ❊

❊ ❊ ❊ ❊ ❊ ❊ ❊

VelD

τ D

②②②②②②②②

MD DV

• ❆

❆ ❆ ❆ ❆ ❆ ❆ ❆ ①①①①①①①①①

PhDV

• V

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

S V ⊂ VelD ⊂ TM ⊕M E

• V = (τ D

2 )−1(V );

V ⊥ ⊂ T∗M ⊕M E ∗; DV = V + V ⊥ is a Dirac algebroid with VelDV = V .

Definition

The Dirac algebroid DV is called induced from D by the subbundle V .

JG (IMPAN) Dirac Algebroids 26/11/2014 17 / 30

Dirac algebroid induced by constraints

Initial data: Dirac algebroid D on E and a vector subbundle V of VelD. D

τ D

2

• ❋

❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋

τ D

1

①①①①①①①①

PhD

πD

• ❊

❊ ❊ ❊ ❊ ❊ ❊ ❊

VelD

τ D

②②②②②②②②

MD DV

• ❆

❆ ❆ ❆ ❆ ❆ ❆ ❆ ①①①①①①①①①

PhDV

• V

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

S V ⊂ VelD ⊂ TM ⊕M E

• V = (τ D

2 )−1(V );

V ⊥ ⊂ T∗M ⊕M E ∗; DV = V + V ⊥ is a Dirac algebroid with VelDV = V .

Definition

The Dirac algebroid DV is called induced from D by the subbundle V .

JG (IMPAN) Dirac Algebroids 26/11/2014 17 / 30

Dirac algebroid induced by constraints

Initial data: Dirac algebroid D on E and a vector subbundle V of VelD. D

τ D

2

• ❋

❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋

τ D

1

①①①①①①①①

PhD

πD

• ❊

❊ ❊ ❊ ❊ ❊ ❊ ❊

VelD

τ D

②②②②②②②②

MD DV

• ❆

❆ ❆ ❆ ❆ ❆ ❆ ❆ ①①①①①①①①①

PhDV

• V

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

S V ⊂ VelD ⊂ TM ⊕M E

• V = (τ D

2 )−1(V );

V ⊥ ⊂ T∗M ⊕M E ∗; DV = V + V ⊥ is a Dirac algebroid with VelDV = V .

Definition

The Dirac algebroid DV is called induced from D by the subbundle V .

JG (IMPAN) Dirac Algebroids 26/11/2014 17 / 30

Dirac algebroid induced by constraints

Initial data: Dirac algebroid D on E and a vector subbundle V of VelD. D

τ D

2

• ❋

❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋

τ D

1

①①①①①①①①

PhD

πD

• ❊

❊ ❊ ❊ ❊ ❊ ❊ ❊

VelD

τ D

②②②②②②②②

MD DV

• ❆

❆ ❆ ❆ ❆ ❆ ❆ ❆ ①①①①①①①①①

PhDV

• V

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

S V ⊂ VelD ⊂ TM ⊕M E

• V = (τ D

2 )−1(V );

V ⊥ ⊂ T∗M ⊕M E ∗; DV = V + V ⊥ is a Dirac algebroid with VelDV = V .

Definition

The Dirac algebroid DV is called induced from D by the subbundle V .

JG (IMPAN) Dirac Algebroids 26/11/2014 17 / 30

Dirac algebroid induced by constraints

Initial data: Dirac algebroid D on E and a vector subbundle V of VelD. D

τ D

2

• ❋

❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋

τ D

1

①①①①①①①①

PhD

πD

• ❊

❊ ❊ ❊ ❊ ❊ ❊ ❊

VelD

τ D

②②②②②②②②

MD DV

• ❆

❆ ❆ ❆ ❆ ❆ ❆ ❆ ①①①①①①①①①

PhDV

• V

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

S V ⊂ VelD ⊂ TM ⊕M E

• V = (τ D

2 )−1(V );

V ⊥ ⊂ T∗M ⊕M E ∗; DV = V + V ⊥ is a Dirac algebroid with VelDV = V .

Definition

The Dirac algebroid DV is called induced from D by the subbundle V .

JG (IMPAN) Dirac Algebroids 26/11/2014 17 / 30

Dirac algebroid induced by constraints

Initial data: Dirac algebroid D on E and a vector subbundle V of VelD. D

τ D

2

• ❋

❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋

τ D

1

①①①①①①①①

PhD

πD

• ❊

❊ ❊ ❊ ❊ ❊ ❊ ❊

VelD

τ D

②②②②②②②②

MD DV

• ❆

❆ ❆ ❆ ❆ ❆ ❆ ❆ ①①①①①①①①①

PhDV

• V

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

S V ⊂ VelD ⊂ TM ⊕M E

• V = (τ D

2 )−1(V );

V ⊥ ⊂ T∗M ⊕M E ∗; DV = V + V ⊥ is a Dirac algebroid with VelDV = V .

Definition

The Dirac algebroid DV is called induced from D by the subbundle V .

JG (IMPAN) Dirac Algebroids 26/11/2014 17 / 30

Mechanics in Dirac algebroid setting

A Dirac algebroid D ⊂

T∗E ∗ ⊕E ∗ TE ∗

can be treated as a relation ˜ ΠD : T∗E ∗− −✄TE ∗. Composing with RE : T∗E → T∗E ∗, we get another relation εD : T∗E− −✄TE ∗. The diagram is commutative in the sense of relations

T∗E ∗

• ❋

❋ ❋ ❋ ❋ ❋ ✠✠✠✠✠✠✠✠✠✠✠

˜ ΠD

✤ TE ∗

• ❍

❍ ❍ ❍ ❍ ✡✡✡✡✡✡✡✡✡✡ T∗E

• ❊

❊ ❊ ❊ ❊ ❊ ✡✡✡✡✡✡✡✡✡✡

εD

✤

• E

VelD

✤ TM

E

VelD

✤

• E ∗

PhD

✤ E ∗

E ∗

PhD

✤

• JG (IMPAN)

Dirac Algebroids 26/11/2014 18 / 30

Mechanics in Dirac algebroid setting

A Dirac algebroid D ⊂

T∗E ∗ ⊕E ∗ TE ∗

can be treated as a relation ˜ ΠD : T∗E ∗− −✄TE ∗. Composing with RE : T∗E → T∗E ∗, we get another relation εD : T∗E− −✄TE ∗. The diagram is commutative in the sense of relations

T∗E ∗

• ❋

❋ ❋ ❋ ❋ ❋ ✠✠✠✠✠✠✠✠✠✠✠

˜ ΠD

✤ TE ∗

• ❍

❍ ❍ ❍ ❍ ✡✡✡✡✡✡✡✡✡✡ T∗E

• ❊

❊ ❊ ❊ ❊ ❊ ✡✡✡✡✡✡✡✡✡✡

εD

✤

• E

VelD

✤ TM

E

VelD

✤

• E ∗

PhD

✤ E ∗

E ∗

PhD

✤

• JG (IMPAN)

Dirac Algebroids 26/11/2014 18 / 30

Mechanics in Dirac algebroid setting

A Dirac algebroid D ⊂

T∗E ∗ ⊕E ∗ TE ∗

can be treated as a relation ˜ ΠD : T∗E ∗− −✄TE ∗. Composing with RE : T∗E → T∗E ∗, we get another relation εD : T∗E− −✄TE ∗. The diagram is commutative in the sense of relations

T∗E ∗

• ❋

❋ ❋ ❋ ❋ ❋ ✠✠✠✠✠✠✠✠✠✠✠

˜ ΠD

✤ TE ∗

• ❍

❍ ❍ ❍ ❍ ✡✡✡✡✡✡✡✡✡✡ T∗E

• ❊

❊ ❊ ❊ ❊ ❊ ✡✡✡✡✡✡✡✡✡✡

εD

✤

• E

VelD

✤ TM

E

VelD

✤

• E ∗

PhD

✤ E ∗

E ∗

PhD

✤

• JG (IMPAN)

Dirac Algebroids 26/11/2014 18 / 30

Mechanics in Dirac algebroid setting

H : E ∗ − → R DH = dH(E ∗) ⊂ T∗E ∗ D = εD(dL(E)) D = ˜ ΠD(dH(E ∗)) L : E − → R DL = dL(E) ⊂ T∗E DH

• D
• DL
• T∗E ∗
• ❋

❋ ❋ ❋ ❋ ❋ ✠✠✠✠✠✠✠✠✠✠✠

˜ ΠD

✤ TE ∗

• ❍

❍ ❍ ❍ ❍ ✡✡✡✡✡✡✡✡✡✡ T∗E

• ❊

❊ ❊ ❊ ❊ ❊ ✡✡✡✡✡✡✡✡✡✡

εD

✤

• E

VelD

✤ TM

E

VelD

✤

• dL
• λ

❧❧❧❧❧❧❧❧❧❧❧❧

E ∗

PhD

✤

dH

• E ∗

E ∗

PhD

✤

• Composition of relations! Dynamics is implicit (differential relation).

In the presence of nonholonomic constraints V ⊂ VelD we just replace D with DV .

JG (IMPAN) Dirac Algebroids 26/11/2014 19 / 30

Mechanics in Dirac algebroid setting

H : E ∗ − → R DH = dH(E ∗) ⊂ T∗E ∗ D = εD(dL(E)) D = ˜ ΠD(dH(E ∗)) L : E − → R DL = dL(E) ⊂ T∗E DH

• D
• DL
• T∗E ∗
• ❋

❋ ❋ ❋ ❋ ❋ ✠✠✠✠✠✠✠✠✠✠✠

˜ ΠD

✤ TE ∗

• ❍

❍ ❍ ❍ ❍ ✡✡✡✡✡✡✡✡✡✡ T∗E

• ❊

❊ ❊ ❊ ❊ ❊ ✡✡✡✡✡✡✡✡✡✡

εD

✤

• E

VelD

✤ TM

E

VelD

✤

• dL
• λ

❧❧❧❧❧❧❧❧❧❧❧❧

E ∗

PhD

✤

dH

• E ∗

E ∗

PhD

✤

• Composition of relations! Dynamics is implicit (differential relation).

In the presence of nonholonomic constraints V ⊂ VelD we just replace D with DV .

JG (IMPAN) Dirac Algebroids 26/11/2014 19 / 30

Mechanics in Dirac algebroid setting

H : E ∗ − → R DH = dH(E ∗) ⊂ T∗E ∗ D = εD(dL(E)) D = ˜ ΠD(dH(E ∗)) L : E − → R DL = dL(E) ⊂ T∗E DH

• D
• DL
• T∗E ∗
• ❋

❋ ❋ ❋ ❋ ❋ ✠✠✠✠✠✠✠✠✠✠✠

˜ ΠD

✤ TE ∗

• ❍

❍ ❍ ❍ ❍ ✡✡✡✡✡✡✡✡✡✡ T∗E

• ❊

❊ ❊ ❊ ❊ ❊ ✡✡✡✡✡✡✡✡✡✡

εD

✤

• E

VelD

✤ TM

E

VelD

✤

• dL
• λ

❧❧❧❧❧❧❧❧❧❧❧❧

E ∗

PhD

✤

dH

• E ∗

E ∗

PhD

✤

• Composition of relations! Dynamics is implicit (differential relation).

In the presence of nonholonomic constraints V ⊂ VelD we just replace D with DV .

JG (IMPAN) Dirac Algebroids 26/11/2014 19 / 30

Mechanics in Dirac algebroid setting

H : E ∗ − → R DH = dH(E ∗) ⊂ T∗E ∗ D = εD(dL(E)) D = ˜ ΠD(dH(E ∗)) L : E − → R DL = dL(E) ⊂ T∗E DH

• D
• DL
• T∗E ∗
• ❋

❋ ❋ ❋ ❋ ❋ ✠✠✠✠✠✠✠✠✠✠✠

˜ ΠD

✤ TE ∗

• ❍

❍ ❍ ❍ ❍ ✡✡✡✡✡✡✡✡✡✡ T∗E

• ❊

❊ ❊ ❊ ❊ ❊ ✡✡✡✡✡✡✡✡✡✡

εD

✤

• E

VelD

✤ TM

E

VelD

✤

• dL
• λ

❧❧❧❧❧❧❧❧❧❧❧❧

E ∗

PhD

✤

dH

• E ∗

E ∗

PhD

✤

• Composition of relations! Dynamics is implicit (differential relation).

In the presence of nonholonomic constraints V ⊂ VelD we just replace D with DV .

JG (IMPAN) Dirac Algebroids 26/11/2014 19 / 30

Mechanics in Dirac algebroid setting

H : E ∗ − → R DH = dH(E ∗) ⊂ T∗E ∗ D = εD(dL(E)) D = ˜ ΠD(dH(E ∗)) L : E − → R DL = dL(E) ⊂ T∗E DH

• D
• DL
• T∗E ∗
• ❋

❋ ❋ ❋ ❋ ❋ ✠✠✠✠✠✠✠✠✠✠✠

˜ ΠD

✤ TE ∗

• ❍

❍ ❍ ❍ ❍ ✡✡✡✡✡✡✡✡✡✡ T∗E

• ❊

❊ ❊ ❊ ❊ ❊ ✡✡✡✡✡✡✡✡✡✡

εD

✤

• E

VelD

✤ TM

E

VelD

✤

• dL
• λ

❧❧❧❧❧❧❧❧❧❧❧❧

E ∗

PhD

✤

dH

• E ∗

E ∗

PhD

✤

• Composition of relations! Dynamics is implicit (differential relation).

In the presence of nonholonomic constraints V ⊂ VelD we just replace D with DV .

JG (IMPAN) Dirac Algebroids 26/11/2014 19 / 30

The Euler-Lagrange equations

The (implicit) Euler-Lagrange equation EL is in our setting an (implicit) first-order differential equation (exactly like Hamilton equations) on curves in E, i.e. EL ⊂ TE They are derived from the phase dynamics D as follows: EL = ε−1

D (TD ∩ T2E ∗) .

In other words, a curve γ : R → E is a solution of EL if it is εD-related to an admissible curve in E ∗. γ is then automatically an admissible curve for the Dirac algebroid D, i.e. (˙ γ, γ) ∈ VelD, where γ is the projection of γ on M. In the canonical case D = DM, E = TM, it means that γ is the tangent lift of γ. The Euler-Lagrange equation EL can be therefore viewed as a second-order differential equation on M.

JG (IMPAN) Dirac Algebroids 26/11/2014 20 / 30

The Euler-Lagrange equations

The (implicit) Euler-Lagrange equation EL is in our setting an (implicit) first-order differential equation (exactly like Hamilton equations) on curves in E, i.e. EL ⊂ TE They are derived from the phase dynamics D as follows: EL = ε−1

D (TD ∩ T2E ∗) .

In other words, a curve γ : R → E is a solution of EL if it is εD-related to an admissible curve in E ∗. γ is then automatically an admissible curve for the Dirac algebroid D, i.e. (˙ γ, γ) ∈ VelD, where γ is the projection of γ on M. In the canonical case D = DM, E = TM, it means that γ is the tangent lift of γ. The Euler-Lagrange equation EL can be therefore viewed as a second-order differential equation on M.

JG (IMPAN) Dirac Algebroids 26/11/2014 20 / 30

The Euler-Lagrange equations

The (implicit) Euler-Lagrange equation EL is in our setting an (implicit) first-order differential equation (exactly like Hamilton equations) on curves in E, i.e. EL ⊂ TE They are derived from the phase dynamics D as follows: EL = ε−1

D (TD ∩ T2E ∗) .

In other words, a curve γ : R → E is a solution of EL if it is εD-related to an admissible curve in E ∗. γ is then automatically an admissible curve for the Dirac algebroid D, i.e. (˙ γ, γ) ∈ VelD, where γ is the projection of γ on M. In the canonical case D = DM, E = TM, it means that γ is the tangent lift of γ. The Euler-Lagrange equation EL can be therefore viewed as a second-order differential equation on M.

JG (IMPAN) Dirac Algebroids 26/11/2014 20 / 30

The Euler-Lagrange equations

The (implicit) Euler-Lagrange equation EL is in our setting an (implicit) first-order differential equation (exactly like Hamilton equations) on curves in E, i.e. EL ⊂ TE They are derived from the phase dynamics D as follows: EL = ε−1

D (TD ∩ T2E ∗) .

In other words, a curve γ : R → E is a solution of EL if it is εD-related to an admissible curve in E ∗. γ is then automatically an admissible curve for the Dirac algebroid D, i.e. (˙ γ, γ) ∈ VelD, where γ is the projection of γ on M. In the canonical case D = DM, E = TM, it means that γ is the tangent lift of γ. The Euler-Lagrange equation EL can be therefore viewed as a second-order differential equation on M.

JG (IMPAN) Dirac Algebroids 26/11/2014 20 / 30

The Euler-Lagrange equations

The (implicit) Euler-Lagrange equation EL is in our setting an (implicit) first-order differential equation (exactly like Hamilton equations) on curves in E, i.e. EL ⊂ TE They are derived from the phase dynamics D as follows: EL = ε−1

D (TD ∩ T2E ∗) .

In other words, a curve γ : R → E is a solution of EL if it is εD-related to an admissible curve in E ∗. γ is then automatically an admissible curve for the Dirac algebroid D, i.e. (˙ γ, γ) ∈ VelD, where γ is the projection of γ on M. In the canonical case D = DM, E = TM, it means that γ is the tangent lift of γ. The Euler-Lagrange equation EL can be therefore viewed as a second-order differential equation on M.

JG (IMPAN) Dirac Algebroids 26/11/2014 20 / 30

The Euler-Lagrange equations

The (implicit) Euler-Lagrange equation EL is in our setting an (implicit) first-order differential equation (exactly like Hamilton equations) on curves in E, i.e. EL ⊂ TE They are derived from the phase dynamics D as follows: EL = ε−1

D (TD ∩ T2E ∗) .

In other words, a curve γ : R → E is a solution of EL if it is εD-related to an admissible curve in E ∗. γ is then automatically an admissible curve for the Dirac algebroid D, i.e. (˙ γ, γ) ∈ VelD, where γ is the projection of γ on M. In the canonical case D = DM, E = TM, it means that γ is the tangent lift of γ. The Euler-Lagrange equation EL can be therefore viewed as a second-order differential equation on M.

JG (IMPAN) Dirac Algebroids 26/11/2014 20 / 30

Vakonomic constraints. Sub-Riemannian geometry

For a constraint submanifold S ⊂ E and a Lagrangian L : S → R nothing really changes in the case of vakonomic constraints: we replace DL = dL(E) with the lagrangian submanifold DL(S). Similarly for the Hamiltonian side in case of ‘phase constraints’.

Example

In the canonical case D = DM, E = TM take a vector subbundle S ⊂ TM and a Lagrangian L : S → R coming from a metric on S (kinetic energy). These data define a sub-Riemannian manifold. Solutions of the Euler-Lagrangian equations are in this case sub-Riemannian geodesics.

JG (IMPAN) Dirac Algebroids 26/11/2014 21 / 30

Vakonomic constraints. Sub-Riemannian geometry

For a constraint submanifold S ⊂ E and a Lagrangian L : S → R nothing really changes in the case of vakonomic constraints: we replace DL = dL(E) with the lagrangian submanifold DL(S). Similarly for the Hamiltonian side in case of ‘phase constraints’.

Example

In the canonical case D = DM, E = TM take a vector subbundle S ⊂ TM and a Lagrangian L : S → R coming from a metric on S (kinetic energy). These data define a sub-Riemannian manifold. Solutions of the Euler-Lagrangian equations are in this case sub-Riemannian geodesics.

JG (IMPAN) Dirac Algebroids 26/11/2014 21 / 30

Vakonomic constraints. Sub-Riemannian geometry

For a constraint submanifold S ⊂ E and a Lagrangian L : S → R nothing really changes in the case of vakonomic constraints: we replace DL = dL(E) with the lagrangian submanifold DL(S). Similarly for the Hamiltonian side in case of ‘phase constraints’.

Example

In the canonical case D = DM, E = TM take a vector subbundle S ⊂ TM and a Lagrangian L : S → R coming from a metric on S (kinetic energy). These data define a sub-Riemannian manifold. Solutions of the Euler-Lagrangian equations are in this case sub-Riemannian geodesics.

JG (IMPAN) Dirac Algebroids 26/11/2014 21 / 30

Vakonomic constraints. Sub-Riemannian geometry

For a constraint submanifold S ⊂ E and a Lagrangian L : S → R nothing really changes in the case of vakonomic constraints: we replace DL = dL(E) with the lagrangian submanifold DL(S). Similarly for the Hamiltonian side in case of ‘phase constraints’.

Example

In the canonical case D = DM, E = TM take a vector subbundle S ⊂ TM and a Lagrangian L : S → R coming from a metric on S (kinetic energy). These data define a sub-Riemannian manifold. Solutions of the Euler-Lagrangian equations are in this case sub-Riemannian geodesics.

JG (IMPAN) Dirac Algebroids 26/11/2014 21 / 30

Vakonomic constraints. Sub-Riemannian geometry

For a constraint submanifold S ⊂ E and a Lagrangian L : S → R nothing really changes in the case of vakonomic constraints: we replace DL = dL(E) with the lagrangian submanifold DL(S). Similarly for the Hamiltonian side in case of ‘phase constraints’.

Example

In the canonical case D = DM, E = TM take a vector subbundle S ⊂ TM and a Lagrangian L : S → R coming from a metric on S (kinetic energy). These data define a sub-Riemannian manifold. Solutions of the Euler-Lagrangian equations are in this case sub-Riemannian geodesics.

JG (IMPAN) Dirac Algebroids 26/11/2014 21 / 30

Mechanics on Dirac algebroids - coordinates

Let us choose the standard adapted coordinates (x, ξ, ˙ x, y, p, ˙ ξ) in T E ∗. Starting with a Lagrangian L : E → R we can define the associated subset [[dL]] in T E ∗ as consisting of points satisfying ξ = ∂L

∂y (x, y) and

p = − ∂L

∂x (x, y).

Next, we intersect [[dL]] with D getting the (implicit) Euler-Lagrange equations defined by the following relations

• ξ
• x, ∂L

∂y (x, y)

• = 0 ,
• η(x, ˙

x, y) = 0 , ζi

• x, −∂L

∂x (x, y), d dt ∂L ∂y (x, y)

• + cj

ik(x)ηk(x, ˙

x, y) ∂L ∂yj (x, y) = 0 . Here, (x, ¯ ξ, ξ, η, η, ζ, ζ) are local coordinates for which D = {(x, ¯ ξ, 0, η, 0, ζ, ζ) : ζk = ci

jk(x)ηj ¯

ξi} , viewed as function of (x, ξ, ˙ x, y, p, ˙ ξ).

JG (IMPAN) Dirac Algebroids 26/11/2014 22 / 30

Mechanics on Dirac algebroids - coordinates

Let us choose the standard adapted coordinates (x, ξ, ˙ x, y, p, ˙ ξ) in T E ∗. Starting with a Lagrangian L : E → R we can define the associated subset [[dL]] in T E ∗ as consisting of points satisfying ξ = ∂L

∂y (x, y) and

p = − ∂L

∂x (x, y).

Next, we intersect [[dL]] with D getting the (implicit) Euler-Lagrange equations defined by the following relations

• ξ
• x, ∂L

∂y (x, y)

• = 0 ,
• η(x, ˙

x, y) = 0 , ζi

• x, −∂L

∂x (x, y), d dt ∂L ∂y (x, y)

• + cj

ik(x)ηk(x, ˙

x, y) ∂L ∂yj (x, y) = 0 . Here, (x, ¯ ξ, ξ, η, η, ζ, ζ) are local coordinates for which D = {(x, ¯ ξ, 0, η, 0, ζ, ζ) : ζk = ci

jk(x)ηj ¯

ξi} , viewed as function of (x, ξ, ˙ x, y, p, ˙ ξ).

JG (IMPAN) Dirac Algebroids 26/11/2014 22 / 30

Mechanics on Dirac algebroids - coordinates

Let us choose the standard adapted coordinates (x, ξ, ˙ x, y, p, ˙ ξ) in T E ∗. Starting with a Lagrangian L : E → R we can define the associated subset [[dL]] in T E ∗ as consisting of points satisfying ξ = ∂L

∂y (x, y) and

p = − ∂L

∂x (x, y).

Next, we intersect [[dL]] with D getting the (implicit) Euler-Lagrange equations defined by the following relations

• ξ
• x, ∂L

∂y (x, y)

• = 0 ,
• η(x, ˙

x, y) = 0 , ζi

• x, −∂L

∂x (x, y), d dt ∂L ∂y (x, y)

• + cj

ik(x)ηk(x, ˙

x, y) ∂L ∂yj (x, y) = 0 . Here, (x, ¯ ξ, ξ, η, η, ζ, ζ) are local coordinates for which D = {(x, ¯ ξ, 0, η, 0, ζ, ζ) : ζk = ci

jk(x)ηj ¯

ξi} , viewed as function of (x, ξ, ˙ x, y, p, ˙ ξ).

JG (IMPAN) Dirac Algebroids 26/11/2014 22 / 30

Mechanics on Dirac algebroids - coordinates

Let us choose the standard adapted coordinates (x, ξ, ˙ x, y, p, ˙ ξ) in T E ∗. Starting with a Lagrangian L : E → R we can define the associated subset [[dL]] in T E ∗ as consisting of points satisfying ξ = ∂L

∂y (x, y) and

p = − ∂L

∂x (x, y).

Next, we intersect [[dL]] with D getting the (implicit) Euler-Lagrange equations defined by the following relations

• ξ
• x, ∂L

∂y (x, y)

• = 0 ,
• η(x, ˙

x, y) = 0 , ζi

• x, −∂L

∂x (x, y), d dt ∂L ∂y (x, y)

• + cj

ik(x)ηk(x, ˙

x, y) ∂L ∂yj (x, y) = 0 . Here, (x, ¯ ξ, ξ, η, η, ζ, ζ) are local coordinates for which D = {(x, ¯ ξ, 0, η, 0, ζ, ζ) : ζk = ci

jk(x)ηj ¯

ξi} , viewed as function of (x, ξ, ˙ x, y, p, ˙ ξ).

JG (IMPAN) Dirac Algebroids 26/11/2014 22 / 30

Mechanics on Dirac algebroids - coordinates

The (implicit) Hamilton equations read as

• ξ = 0 ,
• η
• x, ˙

x, ∂H ∂ξ (x, ξ)

• = 0 ,

ζi

• x, ∂H

∂x (x, ξ), ˙ ξ

• + cj

ik(x)ηk

• x, ˙

x, ∂H ∂ξ (x, ξ)

• ξj = 0 .

For the canonical Dirac algebroid DM we have ξ = 0, ηa = ˙ xa − ya, ζa = ˙ ξa + pa, and ck

ij = 0, so we get the standard Euler-Lagrange

dxa dt = ya, d dt ∂L ∂ya (x, y)

• = ∂L

∂xa (x, y) and Hamilton dξa dt = − ∂H ∂xa (x, ξ) , dxb dt = ∂H ∂ξb (x, ξ) equations after changing the symbols y, ξ for velocities and momenta into the standard ones, ˙ x, p.

JG (IMPAN) Dirac Algebroids 26/11/2014 23 / 30

Mechanics on Dirac algebroids - coordinates

The (implicit) Hamilton equations read as

• ξ = 0 ,
• η
• x, ˙

x, ∂H ∂ξ (x, ξ)

• = 0 ,

ζi

• x, ∂H

∂x (x, ξ), ˙ ξ

• + cj

ik(x)ηk

• x, ˙

x, ∂H ∂ξ (x, ξ)

• ξj = 0 .

For the canonical Dirac algebroid DM we have ξ = 0, ηa = ˙ xa − ya, ζa = ˙ ξa + pa, and ck

ij = 0, so we get the standard Euler-Lagrange

dxa dt = ya, d dt ∂L ∂ya (x, y)

• = ∂L

∂xa (x, y) and Hamilton dξa dt = − ∂H ∂xa (x, ξ) , dxb dt = ∂H ∂ξb (x, ξ) equations after changing the symbols y, ξ for velocities and momenta into the standard ones, ˙ x, p.

JG (IMPAN) Dirac Algebroids 26/11/2014 23 / 30

Mechanics on Dirac algebroids - coordinates

The (implicit) Hamilton equations read as

• ξ = 0 ,
• η
• x, ˙

x, ∂H ∂ξ (x, ξ)

• = 0 ,

ζi

• x, ∂H

∂x (x, ξ), ˙ ξ

• + cj

ik(x)ηk

• x, ˙

x, ∂H ∂ξ (x, ξ)

• ξj = 0 .

For the canonical Dirac algebroid DM we have ξ = 0, ηa = ˙ xa − ya, ζa = ˙ ξa + pa, and ck

ij = 0, so we get the standard Euler-Lagrange

dxa dt = ya, d dt ∂L ∂ya (x, y)

• = ∂L

∂xa (x, y) and Hamilton dξa dt = − ∂H ∂xa (x, ξ) , dxb dt = ∂H ∂ξb (x, ξ) equations after changing the symbols y, ξ for velocities and momenta into the standard ones, ˙ x, p.

JG (IMPAN) Dirac Algebroids 26/11/2014 23 / 30

Mechanics on Dirac algebroids - coordinates

The (implicit) Hamilton equations read as

• ξ = 0 ,
• η
• x, ˙

x, ∂H ∂ξ (x, ξ)

• = 0 ,

ζi

• x, ∂H

∂x (x, ξ), ˙ ξ

• + cj

ik(x)ηk

• x, ˙

x, ∂H ∂ξ (x, ξ)

• ξj = 0 .

For the canonical Dirac algebroid DM we have ξ = 0, ηa = ˙ xa − ya, ζa = ˙ ξa + pa, and ck

ij = 0, so we get the standard Euler-Lagrange

dxa dt = ya, d dt ∂L ∂ya (x, y)

• = ∂L

∂xa (x, y) and Hamilton dξa dt = − ∂H ∂xa (x, ξ) , dxb dt = ∂H ∂ξb (x, ξ) equations after changing the symbols y, ξ for velocities and momenta into the standard ones, ˙ x, p.

JG (IMPAN) Dirac Algebroids 26/11/2014 23 / 30

Mechanics on Dirac algebroids - coordinates

The (implicit) Hamilton equations read as

• ξ = 0 ,
• η
• x, ˙

x, ∂H ∂ξ (x, ξ)

• = 0 ,

ζi

• x, ∂H

∂x (x, ξ), ˙ ξ

• + cj

ik(x)ηk

• x, ˙

x, ∂H ∂ξ (x, ξ)

• ξj = 0 .

For the canonical Dirac algebroid DM we have ξ = 0, ηa = ˙ xa − ya, ζa = ˙ ξa + pa, and ck

ij = 0, so we get the standard Euler-Lagrange

dxa dt = ya, d dt ∂L ∂ya (x, y)

• = ∂L

∂xa (x, y) and Hamilton dξa dt = − ∂H ∂xa (x, ξ) , dxb dt = ∂H ∂ξb (x, ξ) equations after changing the symbols y, ξ for velocities and momenta into the standard ones, ˙ x, p.

JG (IMPAN) Dirac Algebroids 26/11/2014 23 / 30

Mechanics on presymplectic manifolds

Consider the Dirac algebroid Dω associated with a linear 2-form ω on E ∗, ω = 1 2ck

ab(x)ξkdxa ∧ dxb + ρi b(x)dξi ∧ dxb,

ck

ab(x) = −ck ba(x) .

The implicit Euler-Lagrange equations take the form ρi

a(x)dxa

dt (x) = yi , ρi

a(x) d

dt ∂L ∂yi (x, y)

• = ck

ab(x)dxb

dt (x) ∂L ∂yk (x, y) − ∂L ∂xa (x, y) . In the case of a regular presymplectic form of rank r, ω =

• a≤r

dpa ∧ dxa , we get the equations for the presymplectic reduction by the characteristic distribution: the coordinates xa and ˙ xa with a > r are simply forgotten, d dt ∂L ∂ ˙ xa (x, ˙ x)

• = − ∂L

∂xa (x, ˙ x) , a ≤ r .

JG (IMPAN) Dirac Algebroids 26/11/2014 24 / 30

Mechanics on presymplectic manifolds

Consider the Dirac algebroid Dω associated with a linear 2-form ω on E ∗, ω = 1 2ck

ab(x)ξkdxa ∧ dxb + ρi b(x)dξi ∧ dxb,

ck

ab(x) = −ck ba(x) .

The implicit Euler-Lagrange equations take the form ρi

a(x)dxa

dt (x) = yi , ρi

a(x) d

dt ∂L ∂yi (x, y)

• = ck

ab(x)dxb

dt (x) ∂L ∂yk (x, y) − ∂L ∂xa (x, y) . In the case of a regular presymplectic form of rank r, ω =

• a≤r

dpa ∧ dxa , we get the equations for the presymplectic reduction by the characteristic distribution: the coordinates xa and ˙ xa with a > r are simply forgotten, d dt ∂L ∂ ˙ xa (x, ˙ x)

• = − ∂L

∂xa (x, ˙ x) , a ≤ r .

JG (IMPAN) Dirac Algebroids 26/11/2014 24 / 30

Mechanics on presymplectic manifolds

Consider the Dirac algebroid Dω associated with a linear 2-form ω on E ∗, ω = 1 2ck

ab(x)ξkdxa ∧ dxb + ρi b(x)dξi ∧ dxb,

ck

ab(x) = −ck ba(x) .

The implicit Euler-Lagrange equations take the form ρi

a(x)dxa

dt (x) = yi , ρi

a(x) d

dt ∂L ∂yi (x, y)

• = ck

ab(x)dxb

dt (x) ∂L ∂yk (x, y) − ∂L ∂xa (x, y) . In the case of a regular presymplectic form of rank r, ω =

• a≤r

dpa ∧ dxa , we get the equations for the presymplectic reduction by the characteristic distribution: the coordinates xa and ˙ xa with a > r are simply forgotten, d dt ∂L ∂ ˙ xa (x, ˙ x)

• = − ∂L

∂xa (x, ˙ x) , a ≤ r .

JG (IMPAN) Dirac Algebroids 26/11/2014 24 / 30

Mechanics on presymplectic manifolds

Consider the Dirac algebroid Dω associated with a linear 2-form ω on E ∗, ω = 1 2ck

ab(x)ξkdxa ∧ dxb + ρi b(x)dξi ∧ dxb,

ck

ab(x) = −ck ba(x) .

The implicit Euler-Lagrange equations take the form ρi

a(x)dxa

dt (x) = yi , ρi

a(x) d

dt ∂L ∂yi (x, y)

• = ck

ab(x)dxb

dt (x) ∂L ∂yk (x, y) − ∂L ∂xa (x, y) . In the case of a regular presymplectic form of rank r, ω =

• a≤r

dpa ∧ dxa , we get the equations for the presymplectic reduction by the characteristic distribution: the coordinates xa and ˙ xa with a > r are simply forgotten, d dt ∂L ∂ ˙ xa (x, ˙ x)

• = − ∂L

∂xa (x, ˙ x) , a ≤ r .

JG (IMPAN) Dirac Algebroids 26/11/2014 24 / 30

Mechanics on presymplectic manifolds

Consider the Dirac algebroid Dω associated with a linear 2-form ω on E ∗, ω = 1 2ck

ab(x)ξkdxa ∧ dxb + ρi b(x)dξi ∧ dxb,

ck

ab(x) = −ck ba(x) .

The implicit Euler-Lagrange equations take the form ρi

a(x)dxa

dt (x) = yi , ρi

a(x) d

dt ∂L ∂yi (x, y)

• = ck

ab(x)dxb

dt (x) ∂L ∂yk (x, y) − ∂L ∂xa (x, y) . In the case of a regular presymplectic form of rank r, ω =

• a≤r

dpa ∧ dxa , we get the equations for the presymplectic reduction by the characteristic distribution: the coordinates xa and ˙ xa with a > r are simply forgotten, d dt ∂L ∂ ˙ xa (x, ˙ x)

• = − ∂L

∂xa (x, ˙ x) , a ≤ r .

JG (IMPAN) Dirac Algebroids 26/11/2014 24 / 30

Vakonomic constraints: PMP

Assume a parametrization f : M × U → S ⊂ E, y = f (x, u). A Lagrangian L : S → R can be identified with L : M × U → R. The implicit Euler-Lagrange equations are ˆ ξ = 0 ,

• η(x, ˙

x, f (x, u)) = 0 , (1) ζi

• x, ξ ∂f

∂x − ∂L ∂x (x, y), ˙ ξ

• + cj

ik(x)ηk(x, ˙

x, f (x, u))ξ = 0 , (2) constrained additionally by ξ ∂f ∂u − ∂L ∂u = 0 . (3) Equations (1) and (2) are the Hamilton equations with the Hamiltonian H(x, u, ξ) = ξ · f (x, u) − L(x, u) (4) depending on the parameter u. Moreover, the equation (3) reads ∂H

∂u (x, u, ξ) = 0 that is an infinitesimal

form of the Pontryagin Maximum Principle (PMP). The whole picture is an obvious generalization of (PMP), this time for Dirac algebroids (of course in its smooth and infinitesimal version).

JG (IMPAN) Dirac Algebroids 26/11/2014 25 / 30

Vakonomic constraints: PMP

Assume a parametrization f : M × U → S ⊂ E, y = f (x, u). A Lagrangian L : S → R can be identified with L : M × U → R. The implicit Euler-Lagrange equations are ˆ ξ = 0 ,

• η(x, ˙

x, f (x, u)) = 0 , (1) ζi

• x, ξ ∂f

∂x − ∂L ∂x (x, y), ˙ ξ

• + cj

ik(x)ηk(x, ˙

x, f (x, u))ξ = 0 , (2) constrained additionally by ξ ∂f ∂u − ∂L ∂u = 0 . (3) Equations (1) and (2) are the Hamilton equations with the Hamiltonian H(x, u, ξ) = ξ · f (x, u) − L(x, u) (4) depending on the parameter u. Moreover, the equation (3) reads ∂H

∂u (x, u, ξ) = 0 that is an infinitesimal

form of the Pontryagin Maximum Principle (PMP). The whole picture is an obvious generalization of (PMP), this time for Dirac algebroids (of course in its smooth and infinitesimal version).

JG (IMPAN) Dirac Algebroids 26/11/2014 25 / 30

Vakonomic constraints: PMP

Assume a parametrization f : M × U → S ⊂ E, y = f (x, u). A Lagrangian L : S → R can be identified with L : M × U → R. The implicit Euler-Lagrange equations are ˆ ξ = 0 ,

• η(x, ˙

x, f (x, u)) = 0 , (1) ζi

• x, ξ ∂f

∂x − ∂L ∂x (x, y), ˙ ξ

• + cj

ik(x)ηk(x, ˙

x, f (x, u))ξ = 0 , (2) constrained additionally by ξ ∂f ∂u − ∂L ∂u = 0 . (3) Equations (1) and (2) are the Hamilton equations with the Hamiltonian H(x, u, ξ) = ξ · f (x, u) − L(x, u) (4) depending on the parameter u. Moreover, the equation (3) reads ∂H

∂u (x, u, ξ) = 0 that is an infinitesimal

form of the Pontryagin Maximum Principle (PMP). The whole picture is an obvious generalization of (PMP), this time for Dirac algebroids (of course in its smooth and infinitesimal version).

JG (IMPAN) Dirac Algebroids 26/11/2014 25 / 30

Vakonomic constraints: PMP

Assume a parametrization f : M × U → S ⊂ E, y = f (x, u). A Lagrangian L : S → R can be identified with L : M × U → R. The implicit Euler-Lagrange equations are ˆ ξ = 0 ,

• η(x, ˙

x, f (x, u)) = 0 , (1) ζi

• x, ξ ∂f

∂x − ∂L ∂x (x, y), ˙ ξ

• + cj

ik(x)ηk(x, ˙

x, f (x, u))ξ = 0 , (2) constrained additionally by ξ ∂f ∂u − ∂L ∂u = 0 . (3) Equations (1) and (2) are the Hamilton equations with the Hamiltonian H(x, u, ξ) = ξ · f (x, u) − L(x, u) (4) depending on the parameter u. Moreover, the equation (3) reads ∂H

∂u (x, u, ξ) = 0 that is an infinitesimal

form of the Pontryagin Maximum Principle (PMP). The whole picture is an obvious generalization of (PMP), this time for Dirac algebroids (of course in its smooth and infinitesimal version).

JG (IMPAN) Dirac Algebroids 26/11/2014 25 / 30

Nonholonomic constraints

For a linear constraint V0 ⊂ E defined by yI = 0 for the splitting yi = (yι, yI) in a skew algebroid DΠ =

• (xa, ξi, ˙

xb, ˙ ξj, pc, yk) : ˙ xb = ρb

k(x)yk , ˙

ξj = ck

ij (x)yiξk − ρa j (x)pa

• we get DV

Π (V is the graph of ρ : V0 → TM) defined by equations

˙ xb = ρb

ι (x)yι , ˙

ξκ = cj

ικ(x)yιξj − ρa κ(x)pa , yI = 0 .

The constrained E-L equations are yI = 0 , dxa dt (x) = ρa

ι (x)yι ,

d dt ∂L ∂yι (x, y)

• − ck

νι(x)yν ∂L

∂yk (x, y) − ρa

ι (x) ∂L

∂xa (x, y) = 0 . You recognize D’Alembert’s principle, as there are no Is in the latter equation.

JG (IMPAN) Dirac Algebroids 26/11/2014 26 / 30

Nonholonomic constraints

For a linear constraint V0 ⊂ E defined by yI = 0 for the splitting yi = (yι, yI) in a skew algebroid DΠ =

• (xa, ξi, ˙

xb, ˙ ξj, pc, yk) : ˙ xb = ρb

k(x)yk , ˙

ξj = ck

ij (x)yiξk − ρa j (x)pa

• we get DV

Π (V is the graph of ρ : V0 → TM) defined by equations

˙ xb = ρb

ι (x)yι , ˙

ξκ = cj

ικ(x)yιξj − ρa κ(x)pa , yI = 0 .

The constrained E-L equations are yI = 0 , dxa dt (x) = ρa

ι (x)yι ,

d dt ∂L ∂yι (x, y)

• − ck

νι(x)yν ∂L

∂yk (x, y) − ρa

ι (x) ∂L

∂xa (x, y) = 0 . You recognize D’Alembert’s principle, as there are no Is in the latter equation.

JG (IMPAN) Dirac Algebroids 26/11/2014 26 / 30

Nonholonomic constraints

For a linear constraint V0 ⊂ E defined by yI = 0 for the splitting yi = (yι, yI) in a skew algebroid DΠ =

• (xa, ξi, ˙

xb, ˙ ξj, pc, yk) : ˙ xb = ρb

k(x)yk , ˙

ξj = ck

ij (x)yiξk − ρa j (x)pa

• we get DV

Π (V is the graph of ρ : V0 → TM) defined by equations

˙ xb = ρb

ι (x)yι , ˙

ξκ = cj

ικ(x)yιξj − ρa κ(x)pa , yI = 0 .

The constrained E-L equations are yI = 0 , dxa dt (x) = ρa

ι (x)yι ,

d dt ∂L ∂yι (x, y)

• − ck

νι(x)yν ∂L

∂yk (x, y) − ρa

ι (x) ∂L

∂xa (x, y) = 0 . You recognize D’Alembert’s principle, as there are no Is in the latter equation.

JG (IMPAN) Dirac Algebroids 26/11/2014 26 / 30

Nonholonomic constraints

For a linear constraint V0 ⊂ E defined by yI = 0 for the splitting yi = (yι, yI) in a skew algebroid DΠ =

• (xa, ξi, ˙

xb, ˙ ξj, pc, yk) : ˙ xb = ρb

k(x)yk , ˙

ξj = ck

ij (x)yiξk − ρa j (x)pa

• we get DV

Π (V is the graph of ρ : V0 → TM) defined by equations

˙ xb = ρb

ι (x)yι , ˙

ξκ = cj

ικ(x)yιξj − ρa κ(x)pa , yI = 0 .

The constrained E-L equations are yI = 0 , dxa dt (x) = ρa

ι (x)yι ,

d dt ∂L ∂yι (x, y)

• − ck

νι(x)yν ∂L

∂yk (x, y) − ρa

ι (x) ∂L

∂xa (x, y) = 0 . You recognize D’Alembert’s principle, as there are no Is in the latter equation.

JG (IMPAN) Dirac Algebroids 26/11/2014 26 / 30

Rolling disc

b b (x, y)

ϕ ϑ

N = R2 × S1 × S1 ∋ (x, y, ϕ, θ) L : TN − → R L(v) = m

2 ( ˙

x2 + ˙ y2) + J1

2 ˙

ϕ2 + J2

2 ˙

θ2 constraints: ˙ x = R ˙ θ cos ϕ ˙ y = R ˙ θ sin ϕ ⇓ reduction System on a Lie algebroid E = TS1 × R3

τ

• M = S1

, (ϕ, ˙ ϕ, ˙ x, ˙ y, ˙ θ)

• (ϕ)

ρ : E ∋ (ϕ, ˙ ϕ, ˙ x, ˙ y, ˙ θ) → (ϕ, ˙ ϕ) ∈ TS1

JG (IMPAN) Dirac Algebroids 26/11/2014 27 / 30

Rolling disc

b b (x, y)

ϕ ϑ

N = R2 × S1 × S1 ∋ (x, y, ϕ, θ) L : TN − → R L(v) = m

2 ( ˙

x2 + ˙ y2) + J1

2 ˙

ϕ2 + J2

2 ˙

θ2 constraints: ˙ x = R ˙ θ cos ϕ ˙ y = R ˙ θ sin ϕ ⇓ reduction System on a Lie algebroid E = TS1 × R3

τ

• M = S1

, (ϕ, ˙ ϕ, ˙ x, ˙ y, ˙ θ)

• (ϕ)

ρ : E ∋ (ϕ, ˙ ϕ, ˙ x, ˙ y, ˙ θ) → (ϕ, ˙ ϕ) ∈ TS1

JG (IMPAN) Dirac Algebroids 26/11/2014 27 / 30

Rolling disc

b b (x, y)

ϕ ϑ

N = R2 × S1 × S1 ∋ (x, y, ϕ, θ) L : TN − → R L(v) = m

2 ( ˙

x2 + ˙ y2) + J1

2 ˙

ϕ2 + J2

2 ˙

θ2 constraints: ˙ x = R ˙ θ cos ϕ ˙ y = R ˙ θ sin ϕ ⇓ reduction System on a Lie algebroid E = TS1 × R3

τ

• M = S1

, (ϕ, ˙ ϕ, ˙ x, ˙ y, ˙ θ)

• (ϕ)

ρ : E ∋ (ϕ, ˙ ϕ, ˙ x, ˙ y, ˙ θ) → (ϕ, ˙ ϕ) ∈ TS1

JG (IMPAN) Dirac Algebroids 26/11/2014 27 / 30

Rolling disc

New coordinates (ϕ, yi) in E associated to global sections e1 = ∂ϕ, e2 = ∂θ + R cos ϕ∂x + R cos ϕ∂y, e3 = ∂x, e4 = ∂y [e1, e2] = R cos ϕe4 − R sin ϕe3 This gives a Dirac algebroid D ⊂ TE ∗ ⊕E ∗ T∗E ∗. The constraint is now V0 = {(ϕ, y) : y3 = y4 = 0} ⊂ E. It induces the linear constraint V = {(ϕ, y, ˙ ϕ) : y3 = y4 = 0, y1 = ˙ ϕ}, so that DV ⊂ TE ∗ ⊕E ∗ T∗E ∗ , DV = {(ϕ, ξi, ˙ ϕ, ˙ ξj, p, yk) : ˙ ϕ = y1 , ˙ ξ1 = y2R(ξ3 sin ϕ − ξ4 cos ϕ) − p , ˙ ξ2 = −R(ξ3 sin ϕ + ξ4 cos ϕ) , ˙ ξ3, ˙ ξ4 arbitrary , y3 = y4 = 0} . The phase dynamics D ⊂ TE ∗ is D = {(ϕ, ξi, ˙ ϕ, ˙ ξj) : ξ3 = mR mR2 + J2 ξ2 cos ϕ , ˙ ϕ = 1 J1 ξ1 , ξ4 = mR mR2 + J2 ξ2 sin ϕ , ˙ ξ1 = ˙ ξ2 = 0} .

JG (IMPAN) Dirac Algebroids 26/11/2014 28 / 30

Rolling disc

New coordinates (ϕ, yi) in E associated to global sections e1 = ∂ϕ, e2 = ∂θ + R cos ϕ∂x + R cos ϕ∂y, e3 = ∂x, e4 = ∂y [e1, e2] = R cos ϕe4 − R sin ϕe3 This gives a Dirac algebroid D ⊂ TE ∗ ⊕E ∗ T∗E ∗. The constraint is now V0 = {(ϕ, y) : y3 = y4 = 0} ⊂ E. It induces the linear constraint V = {(ϕ, y, ˙ ϕ) : y3 = y4 = 0, y1 = ˙ ϕ}, so that DV ⊂ TE ∗ ⊕E ∗ T∗E ∗ , DV = {(ϕ, ξi, ˙ ϕ, ˙ ξj, p, yk) : ˙ ϕ = y1 , ˙ ξ1 = y2R(ξ3 sin ϕ − ξ4 cos ϕ) − p , ˙ ξ2 = −R(ξ3 sin ϕ + ξ4 cos ϕ) , ˙ ξ3, ˙ ξ4 arbitrary , y3 = y4 = 0} . The phase dynamics D ⊂ TE ∗ is D = {(ϕ, ξi, ˙ ϕ, ˙ ξj) : ξ3 = mR mR2 + J2 ξ2 cos ϕ , ˙ ϕ = 1 J1 ξ1 , ξ4 = mR mR2 + J2 ξ2 sin ϕ , ˙ ξ1 = ˙ ξ2 = 0} .

JG (IMPAN) Dirac Algebroids 26/11/2014 28 / 30

Rolling disc

New coordinates (ϕ, yi) in E associated to global sections e1 = ∂ϕ, e2 = ∂θ + R cos ϕ∂x + R cos ϕ∂y, e3 = ∂x, e4 = ∂y [e1, e2] = R cos ϕe4 − R sin ϕe3 This gives a Dirac algebroid D ⊂ TE ∗ ⊕E ∗ T∗E ∗. The constraint is now V0 = {(ϕ, y) : y3 = y4 = 0} ⊂ E. It induces the linear constraint V = {(ϕ, y, ˙ ϕ) : y3 = y4 = 0, y1 = ˙ ϕ}, so that DV ⊂ TE ∗ ⊕E ∗ T∗E ∗ , DV = {(ϕ, ξi, ˙ ϕ, ˙ ξj, p, yk) : ˙ ϕ = y1 , ˙ ξ1 = y2R(ξ3 sin ϕ − ξ4 cos ϕ) − p , ˙ ξ2 = −R(ξ3 sin ϕ + ξ4 cos ϕ) , ˙ ξ3, ˙ ξ4 arbitrary , y3 = y4 = 0} . The phase dynamics D ⊂ TE ∗ is D = {(ϕ, ξi, ˙ ϕ, ˙ ξj) : ξ3 = mR mR2 + J2 ξ2 cos ϕ , ˙ ϕ = 1 J1 ξ1 , ξ4 = mR mR2 + J2 ξ2 sin ϕ , ˙ ξ1 = ˙ ξ2 = 0} .

JG (IMPAN) Dirac Algebroids 26/11/2014 28 / 30

Rolling disc

New coordinates (ϕ, yi) in E associated to global sections e1 = ∂ϕ, e2 = ∂θ + R cos ϕ∂x + R cos ϕ∂y, e3 = ∂x, e4 = ∂y [e1, e2] = R cos ϕe4 − R sin ϕe3 This gives a Dirac algebroid D ⊂ TE ∗ ⊕E ∗ T∗E ∗. The constraint is now V0 = {(ϕ, y) : y3 = y4 = 0} ⊂ E. It induces the linear constraint V = {(ϕ, y, ˙ ϕ) : y3 = y4 = 0, y1 = ˙ ϕ}, so that DV ⊂ TE ∗ ⊕E ∗ T∗E ∗ , DV = {(ϕ, ξi, ˙ ϕ, ˙ ξj, p, yk) : ˙ ϕ = y1 , ˙ ξ1 = y2R(ξ3 sin ϕ − ξ4 cos ϕ) − p , ˙ ξ2 = −R(ξ3 sin ϕ + ξ4 cos ϕ) , ˙ ξ3, ˙ ξ4 arbitrary , y3 = y4 = 0} . The phase dynamics D ⊂ TE ∗ is D = {(ϕ, ξi, ˙ ϕ, ˙ ξj) : ξ3 = mR mR2 + J2 ξ2 cos ϕ , ˙ ϕ = 1 J1 ξ1 , ξ4 = mR mR2 + J2 ξ2 sin ϕ , ˙ ξ1 = ˙ ξ2 = 0} .

JG (IMPAN) Dirac Algebroids 26/11/2014 28 / 30

Rolling disc

New coordinates (ϕ, yi) in E associated to global sections e1 = ∂ϕ, e2 = ∂θ + R cos ϕ∂x + R cos ϕ∂y, e3 = ∂x, e4 = ∂y [e1, e2] = R cos ϕe4 − R sin ϕe3 This gives a Dirac algebroid D ⊂ TE ∗ ⊕E ∗ T∗E ∗. The constraint is now V0 = {(ϕ, y) : y3 = y4 = 0} ⊂ E. It induces the linear constraint V = {(ϕ, y, ˙ ϕ) : y3 = y4 = 0, y1 = ˙ ϕ}, so that DV ⊂ TE ∗ ⊕E ∗ T∗E ∗ , DV = {(ϕ, ξi, ˙ ϕ, ˙ ξj, p, yk) : ˙ ϕ = y1 , ˙ ξ1 = y2R(ξ3 sin ϕ − ξ4 cos ϕ) − p , ˙ ξ2 = −R(ξ3 sin ϕ + ξ4 cos ϕ) , ˙ ξ3, ˙ ξ4 arbitrary , y3 = y4 = 0} . The phase dynamics D ⊂ TE ∗ is D = {(ϕ, ξi, ˙ ϕ, ˙ ξj) : ξ3 = mR mR2 + J2 ξ2 cos ϕ , ˙ ϕ = 1 J1 ξ1 , ξ4 = mR mR2 + J2 ξ2 sin ϕ , ˙ ξ1 = ˙ ξ2 = 0} .

JG (IMPAN) Dirac Algebroids 26/11/2014 28 / 30

Rolling disc

Here, ˙ ξ3, ˙ ξ4 are formally arbitrary, but they are determined by the integrability condition. The phase space of the system PhL

D ⊂ E ∗ is

described by ξ3 = µ cos ϕ · ξ2, ξ4 = µ sin ϕ · ξ2, where µ =

mR mR2+J2 . The constrained Euler-Lagrange equations are

y3 = y4 = 0 , dϕ dt = y1 , (mR2 + J2)dy2 dt = 0 , J1 dy1 dt = 0 . that can be rewritten as ˙ x1 = R ˙ θ cos ϕ , ˙ x2 = R ˙ θ sin ϕ , ¨ ϕ = 0 , ¨ θ = 0 . The phase dynamics D ⊂ TE ∗ is Hamiltonian ! H(ϕ, ξ) = 1 2J1 (ξ1)2 + 1 2J2 (ξ2 −Rξ3 cos ϕ−Rξ4 sin ϕ)2 + 1 2m((ξ3)2 +(ξ4)2)

JG (IMPAN) Dirac Algebroids 26/11/2014 29 / 30

Rolling disc

Here, ˙ ξ3, ˙ ξ4 are formally arbitrary, but they are determined by the integrability condition. The phase space of the system PhL

D ⊂ E ∗ is

described by ξ3 = µ cos ϕ · ξ2, ξ4 = µ sin ϕ · ξ2, where µ =

mR mR2+J2 . The constrained Euler-Lagrange equations are

y3 = y4 = 0 , dϕ dt = y1 , (mR2 + J2)dy2 dt = 0 , J1 dy1 dt = 0 . that can be rewritten as ˙ x1 = R ˙ θ cos ϕ , ˙ x2 = R ˙ θ sin ϕ , ¨ ϕ = 0 , ¨ θ = 0 . The phase dynamics D ⊂ TE ∗ is Hamiltonian ! H(ϕ, ξ) = 1 2J1 (ξ1)2 + 1 2J2 (ξ2 −Rξ3 cos ϕ−Rξ4 sin ϕ)2 + 1 2m((ξ3)2 +(ξ4)2)

JG (IMPAN) Dirac Algebroids 26/11/2014 29 / 30

Rolling disc

Here, ˙ ξ3, ˙ ξ4 are formally arbitrary, but they are determined by the integrability condition. The phase space of the system PhL

D ⊂ E ∗ is

described by ξ3 = µ cos ϕ · ξ2, ξ4 = µ sin ϕ · ξ2, where µ =

mR mR2+J2 . The constrained Euler-Lagrange equations are

y3 = y4 = 0 , dϕ dt = y1 , (mR2 + J2)dy2 dt = 0 , J1 dy1 dt = 0 . that can be rewritten as ˙ x1 = R ˙ θ cos ϕ , ˙ x2 = R ˙ θ sin ϕ , ¨ ϕ = 0 , ¨ θ = 0 . The phase dynamics D ⊂ TE ∗ is Hamiltonian ! H(ϕ, ξ) = 1 2J1 (ξ1)2 + 1 2J2 (ξ2 −Rξ3 cos ϕ−Rξ4 sin ϕ)2 + 1 2m((ξ3)2 +(ξ4)2)

JG (IMPAN) Dirac Algebroids 26/11/2014 29 / 30

Rolling disc

Here, ˙ ξ3, ˙ ξ4 are formally arbitrary, but they are determined by the integrability condition. The phase space of the system PhL

D ⊂ E ∗ is

described by ξ3 = µ cos ϕ · ξ2, ξ4 = µ sin ϕ · ξ2, where µ =

mR mR2+J2 . The constrained Euler-Lagrange equations are

y3 = y4 = 0 , dϕ dt = y1 , (mR2 + J2)dy2 dt = 0 , J1 dy1 dt = 0 . that can be rewritten as ˙ x1 = R ˙ θ cos ϕ , ˙ x2 = R ˙ θ sin ϕ , ¨ ϕ = 0 , ¨ θ = 0 . The phase dynamics D ⊂ TE ∗ is Hamiltonian ! H(ϕ, ξ) = 1 2J1 (ξ1)2 + 1 2J2 (ξ2 −Rξ3 cos ϕ−Rξ4 sin ϕ)2 + 1 2m((ξ3)2 +(ξ4)2)

JG (IMPAN) Dirac Algebroids 26/11/2014 29 / 30

Rolling disc

Here, ˙ ξ3, ˙ ξ4 are formally arbitrary, but they are determined by the integrability condition. The phase space of the system PhL

D ⊂ E ∗ is

described by ξ3 = µ cos ϕ · ξ2, ξ4 = µ sin ϕ · ξ2, where µ =

mR mR2+J2 . The constrained Euler-Lagrange equations are

y3 = y4 = 0 , dϕ dt = y1 , (mR2 + J2)dy2 dt = 0 , J1 dy1 dt = 0 . that can be rewritten as ˙ x1 = R ˙ θ cos ϕ , ˙ x2 = R ˙ θ sin ϕ , ¨ ϕ = 0 , ¨ θ = 0 . The phase dynamics D ⊂ TE ∗ is Hamiltonian ! H(ϕ, ξ) = 1 2J1 (ξ1)2 + 1 2J2 (ξ2 −Rξ3 cos ϕ−Rξ4 sin ϕ)2 + 1 2m((ξ3)2 +(ξ4)2)

JG (IMPAN) Dirac Algebroids 26/11/2014 29 / 30

Rolling disc

Here, ˙ ξ3, ˙ ξ4 are formally arbitrary, but they are determined by the integrability condition. The phase space of the system PhL

D ⊂ E ∗ is

described by ξ3 = µ cos ϕ · ξ2, ξ4 = µ sin ϕ · ξ2, where µ =

mR mR2+J2 . The constrained Euler-Lagrange equations are

y3 = y4 = 0 , dϕ dt = y1 , (mR2 + J2)dy2 dt = 0 , J1 dy1 dt = 0 . that can be rewritten as ˙ x1 = R ˙ θ cos ϕ , ˙ x2 = R ˙ θ sin ϕ , ¨ ϕ = 0 , ¨ θ = 0 . The phase dynamics D ⊂ TE ∗ is Hamiltonian ! H(ϕ, ξ) = 1 2J1 (ξ1)2 + 1 2J2 (ξ2 −Rξ3 cos ϕ−Rξ4 sin ϕ)2 + 1 2m((ξ3)2 +(ξ4)2)

JG (IMPAN) Dirac Algebroids 26/11/2014 29 / 30

Conclusions

Linear (Poisson) bivector fields (thus (Lie) algebroids), linear (closed) two-forms, and linear Dirac constraints are all special examples of Dirac

• algebroid. The category of Dirac algebroids is closed with respect to

reductions and constraints. The developed Lagrangian and Hamiltonian formalisms on Dirac algebroids are extremally general and describe all main types of mechanical systems, including singular Lagrangians, constraints, and reductions. One can also develop a rigorous optimal control theory in this setting:

• J. Grabowski and M. J´
• ´

zwikowski: ”Pontryagin Maximum Principle

• n almost Lie algebroids”, SIAM J. Control Optim. 49 (2011),

1306-1357. THANK YOU FOR YOUR ATTENTION!

JG (IMPAN) Dirac Algebroids 26/11/2014 30 / 30

Conclusions

Linear (Poisson) bivector fields (thus (Lie) algebroids), linear (closed) two-forms, and linear Dirac constraints are all special examples of Dirac

• algebroid. The category of Dirac algebroids is closed with respect to

reductions and constraints. The developed Lagrangian and Hamiltonian formalisms on Dirac algebroids are extremally general and describe all main types of mechanical systems, including singular Lagrangians, constraints, and reductions. One can also develop a rigorous optimal control theory in this setting:

• J. Grabowski and M. J´
• ´

zwikowski: ”Pontryagin Maximum Principle

• n almost Lie algebroids”, SIAM J. Control Optim. 49 (2011),

1306-1357. THANK YOU FOR YOUR ATTENTION!

JG (IMPAN) Dirac Algebroids 26/11/2014 30 / 30

Conclusions

Linear (Poisson) bivector fields (thus (Lie) algebroids), linear (closed) two-forms, and linear Dirac constraints are all special examples of Dirac

• algebroid. The category of Dirac algebroids is closed with respect to

reductions and constraints. The developed Lagrangian and Hamiltonian formalisms on Dirac algebroids are extremally general and describe all main types of mechanical systems, including singular Lagrangians, constraints, and reductions. One can also develop a rigorous optimal control theory in this setting:

• J. Grabowski and M. J´
• ´

zwikowski: ”Pontryagin Maximum Principle

• n almost Lie algebroids”, SIAM J. Control Optim. 49 (2011),

1306-1357. THANK YOU FOR YOUR ATTENTION!

JG (IMPAN) Dirac Algebroids 26/11/2014 30 / 30

Conclusions

Linear (Poisson) bivector fields (thus (Lie) algebroids), linear (closed) two-forms, and linear Dirac constraints are all special examples of Dirac

• algebroid. The category of Dirac algebroids is closed with respect to

reductions and constraints. The developed Lagrangian and Hamiltonian formalisms on Dirac algebroids are extremally general and describe all main types of mechanical systems, including singular Lagrangians, constraints, and reductions. One can also develop a rigorous optimal control theory in this setting:

• J. Grabowski and M. J´
• ´

zwikowski: ”Pontryagin Maximum Principle

• n almost Lie algebroids”, SIAM J. Control Optim. 49 (2011),

1306-1357. THANK YOU FOR YOUR ATTENTION!

JG (IMPAN) Dirac Algebroids 26/11/2014 30 / 30

Conclusions

Linear (Poisson) bivector fields (thus (Lie) algebroids), linear (closed) two-forms, and linear Dirac constraints are all special examples of Dirac

• algebroid. The category of Dirac algebroids is closed with respect to

reductions and constraints. The developed Lagrangian and Hamiltonian formalisms on Dirac algebroids are extremally general and describe all main types of mechanical systems, including singular Lagrangians, constraints, and reductions. One can also develop a rigorous optimal control theory in this setting:

• J. Grabowski and M. J´
• ´

zwikowski: ”Pontryagin Maximum Principle

• n almost Lie algebroids”, SIAM J. Control Optim. 49 (2011),

1306-1357. THANK YOU FOR YOUR ATTENTION!

JG (IMPAN) Dirac Algebroids 26/11/2014 30 / 30

Conclusions

Linear (Poisson) bivector fields (thus (Lie) algebroids), linear (closed) two-forms, and linear Dirac constraints are all special examples of Dirac

• algebroid. The category of Dirac algebroids is closed with respect to

reductions and constraints. The developed Lagrangian and Hamiltonian formalisms on Dirac algebroids are extremally general and describe all main types of mechanical systems, including singular Lagrangians, constraints, and reductions. One can also develop a rigorous optimal control theory in this setting:

• J. Grabowski and M. J´
• ´

zwikowski: ”Pontryagin Maximum Principle

• n almost Lie algebroids”, SIAM J. Control Optim. 49 (2011),

1306-1357. THANK YOU FOR YOUR ATTENTION!

JG (IMPAN) Dirac Algebroids 26/11/2014 30 / 30