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On a new affine formulation of Hamiltonian classical field theories

Juan Carlos Marrero

University of La Laguna, Spain e-mail:jcmarrer@ull.edu.es

Classical and Quantum Physics: Geometry, Dynamics and Control 60 years Alberto Ibort Fest 5-9 March 2018 ICMAT, Madrid, Spain Work in progress with F Gay-Balmaz and N Mart´ ınez

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

CONGRATULATIONS ALBERTO!!!!!!!

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

My collaboration with Alberto

Almost-Poisson brackets for non-holonomic mechanical systems and mechanical systems subjected to impulsive constraints IBORT, ALBERTO; de Le´

• n, Manuel; JCM; Mart´

ın de Diego, David: Dirac brackets in constrained dynamics. Fortschr. Phys. 47 (1999), no. 5, 459–492. IBORT, ALBERTO; de Le´

• n, Manuel; Lacomba, Ernesto A.; JCM.; de

Diego, David Mart´ ın; Pitanga, Paulo Geometric formulation of mechanical systems subjected to time-dependent one-sided constraints. J.

• Phys. A 31 (1998), no. 11, 2655–2674.

IBORT, ALBERTO; de Le´

• n, Manuel; Lacomba, Ernesto A.; JCM.; de

Diego, David Mart´ ın; Pitanga, Paulo Geometric formulation of Carnot’s

• theorem. J. Phys. A 34 (2001), no. 8, 1691–1712.

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Alberto and the geometric formulation of Hamiltonian Classical Field Theories

A multisymplectic formulation (including the formulation for manifolds with boundary) Cari˜ nena, J. F.; Crampin, M.; IBORT, L. A.: On the multisymplectic formalism for first order field theories. Differential Geom. Appl. 1 (1991),

• no. 4, 345–374.

IBORT, ALBERTO; Spivak, Amelia: Covariant Hamiltonian field theories

• n manifolds with boundary: Yang-Mills theories. J. Geom. Mech. 9

(2017), no. 1, 47–82. IBORT, ALBERTO; Spivak, Amelia: On A Covariant Hamiltonian Description of Palatini’s Gravity on Manifolds with Boundary. Preprint arXiv:1605.03492 A realization of Peierls brackets Asorey, M.; Ciaglia, M.; Di Cosmo, F.; IBORT, A: Covariant brackets for particles and fields. Modern Phys. Lett. A 32 (2017), no. 19, 1750100, 16

• pp. Erratum: Covariant brackets for particles and fields. Modern Phys.
• Lett. A 32 (2017), no. 22, 1792002, 1 p.

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

Hamilton equations in Classical Mechanics Some basic constructions on affine bundles Hamiltonian Classical Field Theories of First Order The problems

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

Hamilton equations in Classical Mechanics The phase space: Z = T ⇤Q or Z = T ⇤Q/G (a symplectic or a Poisson manifold) The observables: C 1(Z) The Poisson bracket: ] : Ω1(Z) = Γ(T ⇤Z) ! X(Z) = Γ(TZ) {', } =< d', ](d ) >, ', 2 C 1(Z) The Hamiltonian: H 2 C 1(Z) The Hamilton equations: A curve s : I ! Z is a solution of the Hamilton eqs if and only if {', H} s = s⇤(d'), 8' 2 C 1(Z) (, is an integral curve of Hamiltonian vector field Xh = ](dH)) Remark: ⌫ = dt a volume form ) Ω1(I) ' C 1(I)

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

Some basic constructions on affine bundles A an affine space which is modeled over the vector space V ; dimV = r A reference frame on A {O; e1, . . . , er}, O 2 A, {e1, . . . , er} is a basis of V + A+ = Aff (A, R) the affine dual of A: a vector space of dimension r + 1 1A+ 2 A+ the constant function equal to 1 {e0 = 1A+, e1, . . . , er} the dual basis of A+ V ⇤ = Lin(V , R) the dual space of V {e1, . . . , er} the dual basis on V ⇤ The canonical projection µ : A+ ! V ⇤ V ⇤ ' A+/ < 1A+ >' A+/R.

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

The bi-dual space of A ˜ A = (A+)⇤ = Lin(A+, R)

A is an affine subspace of ˜ A of codimension 1 V is a vector subspace of ˜ A of codimension 1

U a vector subspace of V + A/U is an affine space modelled over the vector space V /U and (A/U)+ ' U0 = {' 2 A+/'` 2 U0} Remark: All the previous constructions may be extended to affine bundles

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

Vertical lifts to affine bundles ⌧ : A ! Q an affine bundle modelled over the vector bundle V a 2 Aq, u 2 Vq ) uv(a) = d dt |t=0(a + tu) 2 Va⌧ (qi) local coordinates on Q, {O; e↵} a reference frame on A + (qi, a↵) local coordinates on A u = u↵e↵(q) ) uv(a) = u↵ @ @a↵ |a

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

Affine bundles and principal R-bundles µ : P ! Q a principal R-bundle (, it is and affine bundle of rank 1) R 2 X(P) the infinitesimal generator of the R-action The affine space of sections of µ Γ(µ) ! {F 2 C 1(P)/R(F) = 1}, h ! Fh Fh(p) = p h(µ(p)), 8p 2 P Q(qi), P(qi, p) ) R = @ @p h(qi) = (qi, H(qi)) ) Fh(qi, p) = p + H(qi) The typical example: A an affine space which is modelled over V + µ : A+ ! V ⇤ is a principal R-bundle R ⇥ A+ ! A+, (p, ') ! ' + cp

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

Hamiltonian Classical Field Theories of first order The configuration bundle: ⇡ : P ! M, dim M = m, dim P = m + n P (xi, u↵), M (xi) vol a volume form on M; vol = dx1 ^ · · · ^ dxm = dmx vol the volume m-vector; vol = @x1 ^ · · · ^ @xm = @mx The evolution space: J1⇡ = [p2P{hp : T⇡(p)M ! TpP/hp is linear and Tp⇡ hp = id} hp is a horizontal lift from T⇡(p)M to TpP ⇡1,0 : J1⇡ ! P is an affine bundle, (xi, u↵, u↵

i )

The corresponding vector bundle ⇡v

1,0 : V (J1(⇡)) ! P

V (J1⇡) = ⇡⇤(T ⇤M) ⌦ V ⇡ = [p2P{vp : T⇡(p)M ! Vp⇡/vp is linear } Local coordinates: (xi, u↵, u↵

i )

An important remark: Ehresmann connections r : X(M) ! X(P) on ⇡ : P ! M $ s : P ! J1⇡ 2 Γ(J1⇡)

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

The extended multimomentum bundle: M⇡ = (J1⇡)+ = Aff (J1(⇡), R) = Λm

2 (T ⇤P)

= {' 2 Λm(T ⇤P)/iuiv(') = 0, 8u, v 2 V (⇡)} ⌫ : M⇡ ! P a vector bundle, (xi, u↵, p, pi

↵)

' = pdmx + pi

↵du↵ ^ dm1xi,

dm1xi = i(@xi)dmx The restricted multimomentum bundle: M0⇡ = Lin(⇡⇤(T ⇤M) ⌦ V (⇡), R) ' Lin(V ⇡, ⇡⇤(Λm1T ⇤M)) ⌫0 : M0⇡ ! P a vector bundle, (xi, u↵, pi

↵)

The canonical projection: µ : M⇡ ! M0⇡ µ(xi, u↵, p, pi

↵) = (xi, u↵, pi ↵)

It is a principal R-bundle and M0⇡ = M⇡/R

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

The Hamiltonian section: h 2 Γ(µ) Γ(µ) = {h : M0⇡ ! M⇡/h is smooth and µ h = id} is an affine space which is modelled over the vector space C 1(M0⇡) Hamilton-deDonder-Weyl equations: h 2 Γ(µ) and h(xi, u↵, pi

↵) = (xi, u↵, H(xi, u↵, pi ↵), pi ↵)

s0 : U ✓ M ! M0⇡ a local section of ⇡ ⌫0, s0(xi) = (xi, u↵(x), pi

↵(x))

s0 is a solution of H-deD-W eqs , @u↵ @xi = @H @pi

↵

, @pi

↵

@xi = @H @u↵

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

A more geometric approach (Carinena, Crampin, Ibort 1991 and the Spanish group) The canonical multisymplectic structure on M⇡ !M⇡ = du↵ ^ dpi

↵ ^ dm1xi dp ^ dmx 2 Ωm+1(M⇡)

The hamiltonian section h : M0⇡ ! M⇡ + !h = h⇤(!M⇡) a multisymplectic structure on M0⇡ H-deD-W equations s : M ! M0⇡ a local section of ⇡ ⌫0 : M0⇡ ! M s satisfies the H-deD-W eqs , s⇤(iU!h) = 0, 8U 2 Γ(V (⇡ ⌫0))

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

Remark: Local sections of ⇡ ⌫0 : M0⇡ ! M ' integral sections

• f a connection on ⇡ ⌫0 : M0⇡ ! M

A connection r : X(M) ! X(M0⇡) on ⇡ ⌫0 : M0⇡ ! M is said to be Hamiltonian if i((Λmr)(vol))!h = 0 Hamiltonian connections and H-deD-W eqs The integral sections of a Hamiltonian connection are solutions of the H-deD-W eqs for h

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

Some problems: !h is not a canonical structure (it depends on h) Hamilton eqs in Classical Mechanics and in Classical Field theories have different form: Classical Mechanics: < d', XH > s0 =< d', ](dH) > s0 = s⇤

0(d'), for

' 2 C 1(Z) and s0 : I ✓ R ! Z for a curve on Z Classical Field Theories: s⇤

0(iU!h) = 0, for s0 : M ! M0⇡ a

local section and U 2 Γ(V (⇡ ⌫0)) h is not a real C 1-function. So, what is dh?. Where does take values dh? Hamiltonian connections always exist. But, they are not

• unique. However, given a Hamiltonian function H there exists

a unique Hamiltonian vector field XH associated with H.

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

A possible solution to the last problem: A suitable equivalence relation r and ¯ r connections on ⇡ ⌫0 : M0⇡ ! M r ⇠ ¯ r , r and ¯ r are Hamiltonian for the same Hamiltonian section h Remark: connections on ⇡ ⌫0 : M0⇡ ! M are sections of the affine bundle (⇡ ⌫0)1,0 : J1(⇡ ⌫0) ! M0⇡.

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

A nice result 9U a vector subbundle of V (J1(⇡ ⌫0)) ' (⇡ ⌫0)⇤(T ⇤M) ⌦ V (⇡ ⌫0) such that the equivalence class [r] of the Hamiltonian connections for a Hamiltonian section h is a section of the quotient affine bundle J1(⇡ ⌫0)/U ! M0⇡. Moreover, U doesn’t depend on h! L = U0 ✓ M(⇡ ⌫0) = Λm

2 (M0⇡) is given by

L = ['02M0⇡{ 2 Λm

2 (T ⇤ '0M0⇡)

/ i( ¯

'0)v

ϕ0 = (Λm1(T ⇤

'0(⇡ ⌫0)))( ¯

'0(u)), for u 2 V⌫0('0)⇡} L =< dmx, du↵ ^ dm1xi, dpi

↵ ^ dm1xi >) rank L = m + mn + n

Remark: J1(⇡ ⌫0)/U and the vector subbundle L ' (J1(⇡ ⌫0)/U)+ of M(⇡ ⌫0) will appear more later. They will play an important role!

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

Another important vector bundle morphism related with L µ : M⇡ ! M0⇡ = M⇡/R ) ˜ L = µ⇤(L) () L ' ˜ L/R) ˜ L is a vector subbundle of M(⇡ ⌫) = Λm

2 (M⇡) with rank

m + mn + n. ˜ L =< dmx, du↵ ^ dm1xi, dpi

↵ ^ dm1xi >

˜ [ : V (⇡⌫) ✓ T(M⇡) ! ˜ L ✓ M(⇡⌫) = Λm

2 (M⇡),

˜ u ! i˜

u(!M⇡)

is a vector bundle isomorphism

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

˜ Ψ = ˜ [1 : ˜ L ✓ M(⇡ ⌫) ! V (⇡ ⌫) Action of R on M⇡ restricts to the fibers of ⇡ ⌫ ) R ⇥ (⇡ ⌫)1(x) ! (⇡ ⌫)1(x), x 2 M + R ⇥ V (⇡ ⌫) ! V (⇡ ⌫) the induced action + V (⇡ ⌫)/R is a vector bundle over M0⇡ and we have a canonical projection c Tµ : V (⇡ ⌫)/R ! V (⇡ ⌫0)

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

The morphism Ψ : L ✓ M(⇡ ⌫0) ! V (⇡ ⌫)/R There exists a vector bundle morphism Ψ : L ' ˜ L/R ! V (⇡ ⌫)/R such that the following diagram ˜ L ✓ M(⇡ ⌫)

✏

˜ Ψ=˜ [1

/ V (⇡ ⌫) ✏

L = ˜ L/R ✓ M(⇡ ⌫0)

Ψ

/ V (⇡ ⌫)/R

is commutative Remark: ⇢ : L

Ψ

• ! V (⇡ ⌫)/R

c Tµ

• ! V (⇡ ⌫0) is the anchor map of

a Lie algebroid structure on L

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

Local expressions of Ψ and ⇢ Ψ(dmx) = [ @ @p], Ψ(du↵^dm1xi) = [ @ @pi

↵

], Ψ(dpi

↵^dm1xi) = [ @

@u↵ ] ⇢(dmx) = 0, ⇢(du↵ ^ dm1xi) = @ @pi

↵

, ⇢(dpi

↵ ^ dm1xi) =

@ @u↵

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

We can summarize the previous discussion: We have not a unique Hamiltonian connection for h, but a a section of the quotient affine bundle J1(⇡ ⌫0)/U ! M0⇡ We have a Lie algebroid structure on the vector bundle L = U0 ✓ M(⇡ ⌫0) = Λm

2 (M0⇡) with anchor map

⇢ : L ! V (⇡ ⌫0)

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

Some problems: !h is not a canonical structure (it depends on h) Hamilton eqs in Classical Mechanics and in Classical Field theories have different form h is not a real C 1-function. So, what is dh?. Where does take values dh?

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems First problem: Does there exists a higher-Poisson bracket {·, ·} associated with M0⇡ which allows to describe the H-deD-W eqs as in Classical Mechanics? Some previous approaches (it is used a structure which comes from

• utside the space or it is not canonical)

In M0⇡ using a multisymplectic structure which depends on the Hamiltonian section (Cari˜ nena, Crampin, Ibort, 1991, the usual form for the Spanish group) In M0⇡ using the vector valued fibred symplectic structure on M0⇡ and, moreover, an auxiliar connection in the configuration bundle (Castrill´

• n, Marsden, 2003)

The extended formalism in M⇡ using the canonical multisymplectic structure (Echeverr´ ıa-Enriquez, de Le´

• n, Mu˜

noz-Lecanda, Roman-Roy, 2007) Using the vector valued fibred symplectic structure on M0⇡ and,moreover, the canonical multisymplectic structure on M⇡ (Forger and collaborators)

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

We add symmetries to the picture : G ⇥ P ! P a free and proper action of G on P ⇡ g = ⇡, 8g 2 G P

⇡

✏

⇡P

/ P/G = ˆ

P

ˆ ⇡

y

M + Free and proper action of G on J1⇡, M0⇡ and M⇡

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems The reduced evolution space d J1⇡ = J1⇡/G An affine bundle d ⇡1,0 : d J1⇡ ! ˆ P The reduced restricted multimomentum bundle [ M0⇡ = M0⇡/G A vector bundle b ⌫0 : [ M0⇡ ! ˆ P The reduced extended multimomentum bundle d M⇡ = M⇡/G A vector bundle ˆ ⌫ : d M⇡ ! ˆ P A canonical projection ˆ µ : d M⇡ ! [ M0⇡ a principal R bundle

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

An equivariant Hamiltonian section h : M0⇡ ! M⇡ + A reduced Hamiltonian section ˆ h : [ M0⇡ ! d M⇡ A local section b s0 : M ! [ M0⇡ is a solution of the Hamilton-Poincar´ e field eqs for h if it is a projection of a solution s0 : M ! M0⇡ of the H-deD-W eqs for h Second problem: Does there exists a higher-Poisson bracket d {·, ·} associated with [ M0⇡ which allows to describe the Hamilton-Poincar´ e field eqs as in Classical Mechanics?

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Motivation and problems

In what follows, we will give positive answers to the previous questions {·, ·} and d {·, ·} are completely canonical ⇤ They are of affine nature

⇤ We fix a volume form on M. But all the next constructions don’t

depend on the chosen volume form

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

The two steps First step: to introduce the phase bundle P(A, ⇡) associated with an arbitrary affine bundle ⌧A : A ! P (modelled on a vector bundle ⌧V : V ! P) and a fibration ⇡ : P ! M. In fact, we will see that the differential of a section h : V ∗ ! A+ of µ : A+ ! V ∗ takes values in P(A, ⇡) Second step: we will assume the existence of a suitable Lie algebroid structure on a vector bundle L of the extended multimomentum bundle M(⇡ ⌧V ∗) = Λm

2 (V ∗) associated with the

fibration ⇡ ⌧V ∗ : V ∗ ! M

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

The first step: the phase bundle associated with an affine bundle ⌧A : A ! P and a fibration ⇡ : P ! M ⌧A : A ! P an affine bundle, ⌧V : V ! P the corresponding vector bundle ⇡ : P ! M a fibration; vol a volume form on M + ⌫ : A+ ! P ⌫0 : V ∗ ! P vector bundles µ : A+ = Aff (A, R) ! V ∗ = Lin(V , R) a principal R-bundle e R the infinitesimal generator of the R-action h : V ∗ ! A+ a section of µ (a generalized Hamiltonian section) +

Fh : A+ ! R the corresponding homogeneous function with respect to R dFh g (R) = 1

Fh ⌘ the generalized extended Hamiltonian function

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

The first step: the phase bundle associated with an affine bundle ⌧A : A ! P and a fibration ⇡ : P ! M

Local coordinates M(xi) P(xi, u↵), A, V (xi, u↵, va) A+(xi, u↵, p, pa), V ⇤(xi, u↵, pa) + µ(xi, u↵, p, pa) = (xi, u↵, pa), e R = @ @p h(xi, u↵, pa) = (xi, u↵, H(xj, u, pb), pa) + Fh(xi, u↵, p, p↵) = p + H(xi, u↵, pa)() dFh( e R) = 1)

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

The first step: the phase bundle associated with an affine bundle ⌧A : A ! P and a fibration ⇡ : P ! M

The extended phase bundle associated with A and ⇡ d(Fh|V (⇡⌫)) 2 Γ( ^ P(A, ⇡)) ^ P(A, ⇡) = [

'2A+

{e 2 V ⇤

'(⇡ ⌫)/e

( e R)(') = 1} Local coordinates (xi, u↵, p, pa; u↵, pa) ^ P(A, ⇡) is an affine sub-bundle of V ⇤(⇡ ⌫) of corank 1

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

The first step: the phase bundle associated with an affine bundle ⌧A : A ! P and a fibration ⇡ : P ! M

e R is tangent to the fibers of ⇡ ⌫ : A+ ! M + Action of R on A+ restricts to the fibers of ⇡ ⌫ R ⇥ (⇡ ⌫)1(x) ! (⇡ ⌫)1(x), x 2 M + R⇥V (⇡⌫) ! V (⇡⌫), R⇥V ⇤(⇡⌫) ! V ⇤(⇡⌫) the induced actions e R is R-invariant ) R ⇥ ^ P(A, ⇡) ! ^ P(A, ⇡) a R-action dFh|V (⇡⌫) : A+ ! ^ P(A, ⇡) is R-equivariant

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

The first step: the phase bundle associated with an affine bundle ⌧A : A ! P and a fibration ⇡ : P ! M The phase bundle associated with A and ⇡ P(A, ⇡) = ^ P(A, ⇡)/R It is an affine bundle over A+/R = V ∗ The differential of the Hamiltonian section There exists a unique section dh of ⌧P(A,π) : P(A, ⇡) ! V ∗ such that the following diagram A+

µ

✏

dFh

/ ^ P(A, ⇡)

e µ

✏ V ∗ = A+/R

dh

/ ^ P(A, ⇡)/R = P(A, ⇡) is commutative

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

The first step: the phase bundle associated with an affine bundle ⌧A : A ! P and a fibration ⇡ : P ! M

Local coordinates P(A, ⇡), (xi, u↵, pa; u↵, pa) dh(xi, u↵, pa) = (xi, u↵, pa, @H @u↵ , @H @pa ) Remark: ( ^ P(A, ⇡))+ = V (⇡ ⌫) + P(A, ⇡)+ = V (⇡ ⌫) R ! V ⇤ a Lie algebroid [ e R] a central element + P(A, ⇡) admits an affine Poisson structure

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

The second step: A Lie algebroid structure on L

Assumptions: H1) L ✓ M(⇡ ⌫0) = ^m

2 (V ⇤) a Lie algebroid on V ⇤

[ [, 0] ] = L⇢()0 L⇢(0) + d(i(⇢(0)), , 0 2 Γ(L) ⇢ : L ! V (⇡ ⌫0) the anchor map (⇡ ⌫0)⇤(vol) 2 Γ(L) is a central element H2) Ψ : L ! P(A, ⇡)+ = V (⇡⌫)

R

a Lie algebroid morphism Ψ((⇡ ⌫0)⇤(vol)) = [ e R]

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Currents, affine higher-Poisson bracket and Hamilton equations

The space of currents: O = { 2 Ωm1

1

(V ⇤)/d 2 Γ(L)} The affine higher-Poisson bracket O ⇥ Γ(µ) ! C 1(V ⇤), (, h) ! {, h} =< d, #aff (dh) > Hamilton equations for h s : M ! V ⇤ is a solution of the Hamilton eqs m < d, #aff (dh) > s = {, h} s = s⇤(d) Remark: Ωm(M) ⇠ = C 1(M)

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Hamilton equations in Classical Mechanics

The Hamilton equations: A curve s : I ! Z is a solution of the Hamilton eqs if and only if {', H} s = s⇤(d'), 8' 2 C 1(Z) Remark: ⌫ = dt a volume form ) Ω1(I) ' C 1(I)

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

We particularize for Hamiltonian Classical Field Theories of first

• rder:

The first step (the definition of the phase bundle P(A, ⇡) associated with A and ⇡ and the definition of dh as a section

• f P(A, ⇡)) is general and it may be used when A = J1⇡ (the

unreduced setting) or when A = J1⇡/G (the reduced setting) The second step for unreduced Hamiltonian Classical Field Theories of first order (A = J1⇡): OK! It is sufficient to take L = ˜ L/R, with ˜ L = ˜ [(V (⇡ ⌫)) and ˜ [ : V (⇡ ⌫) ! Λm

2 (M⇡) the monomorphism induced by the

canonical multisymplectic structure on M⇡ = Λm

2 (P)

What happens with the second step in the presence of a symmetry Lie group (that is, when A = J1⇡/G)?

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

We have two problems: The description of the Lie algebroid b L ✓ M(b ⇡ b ⌫0) = ^m

2 ( [

M0⇡)

• ver [

M0⇡ = M0⇡

G

The description of the Lie algebroid morphism b Ψ : b L ✓ M(b ⇡ b ⌫0) = ^m

2 ( [

M0⇡) ! P(d J1⇡, b ⇡)+ = V (b ⇡ b ⌫)/R

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

The first problem The definition of b L ✓ M(b ⇡ b ⌫0) = ^m

2 (T ⇤ [

M0⇡)

b '0 2 M0π

G

• ! b

L(c '0) = {b 2 ^m

2 (T ∗ c ϕ0 [

M0⇡)/(^mT ∗

ϕ0⇡M0π)(b

) 2 L('0), with '0 2 M0⇡ and ⇡M0π('0) = c '0}

L is G-invariant! = ) b L is well-defined L('0) ✓ ^m

2 (T ⇤ '0M0⇡)

^m(T ∗

ϕ0 ⇡M0π)"

b L(⇡M0⇡('0)) ✓ ^m

2 (T ⇤ ⇡M0⇡('0) [

M0⇡)

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

The second problem: The definition of b : b L ⇢ M(b ⇡ b ⌫0) ! P(d J1⇡, b ⇡)+ = V (b

⇡b ⌫) R

L('0)

Ψ('0)

/ (V (⇡ ⌫)/R)('0)

( \ T'0⇡M0⇡)

✏

b L(⇡M0⇡('0))

b Ψ(⇡M0⇡('0)) / ^m(T ⇤

'0⇡M0⇡)

O

(V (b ⇡ b ⌫)/R)(⇡M0⇡('0)) is commutative and the definition of b Ψ(⇡M0⇡('0)) doesn’t depend

• n the choosen point '0 2 M0⇡!

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Conclusions and future work

We have developed a canonical affine formulation of unreduced (reduced) Hamiltonian Classical Field Theories of first order The formulation must be applied to concrete examples. Continuous Mechanics (fluids and elasticity) and Cosserat Theories (beams or shells) are good candidates for the unreduced and reduced setting, respectively. For time-dependent Mechanics, we recover some results by J. Grabowski et al and E. Mart´ ınez et al What happens if our Hamiltonian section is only defined in a submanifold of the (reduced) restricted multimomentum bundle (the primary constraint submanifold)? This is the case when our Hamiltonian theory comes from a singular Lagrangian field theory

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

Conclusions and future work

To include boundary conditions in our theory and to discuss the relation between the resultant construction and the theory

• f covariant Peierls brackets in the space of the solutions.

It is related with Alberto’s research on Peierls brackets

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor

THANKS! CONGRATULATIONS ALBERTO!!

Juan Carlos Marrero On a new affine formulation of Hamiltonian classical field theor