Jets and Fields on Lie Algebroids Geometry of Jets and Fields - PowerPoint PPT Presentation

Jets and Fields on Lie Algebroids —— ◦ —— Geometry of Jets and Fields —— ◦ —— Eduardo Martínez University of Zaragoza emf@unizar.es Będlewo (Poland), 10-15 May 2005

Mechanics on Lie algebroids (Weinstein 1996, Martínez 2001, ...) Lie algebroid E → M . L ∈ C ∞ ( E ) or H ∈ C ∞ ( E ∗ ) � � E = TM → M Standard classical Mechanics � � E = D ⊂ TM → M (integrable) System with holonomic constraints � � � � � E = TQ/G → M = Q/G System with symmetry � E = g → { e } System on Lie algebras � � � � � E = M × g → M System on a semidirect products (ej. heavy top) 1

Symplectic and variational The theory is symplectic: i Γ ω L = dE L with ω L = − dθ L , θ L = S ( dL ) and E L = d ∆ L − L . Here d is the differential on the Lie algebroid τ E E : T E E → E . It is also a variational theory: - Admissible curves or E -paths - Variations are E -homotopies - Infinitesimal variations are δx i = ρ i α σ α δy α = ˙ σ α + C α βγ y β σ γ 2

Time dependent systems (Martínez, Mestdag and Sarlet 2002) With suitable modifications one can describe time-dependent systems. Cartan form Θ L = S ( dL ) + Ldt. Dynamical equation i Γ d Θ L = 0 and � Γ , dt � = 1 . Field theory in 1-d space-time Affgebroids Martinez, Mestdag and Sarlet 2002 Grabowska, Grabowski and Urbanski 2003 3

� � Example: standard case T π � TN TM � N M π m ∈ M and n = π ( m ) 0 − → Ver m − → T m M − → T n N − → 0 Set of splittings: J m π = { φ : T n N → T m M | Tπ ◦ φ = id T n N } . Lagrangian: L : Jπ → R 4

� � Example: principal bundle [ T π ] � TM TQ/G � M Q/G = M id m ∈ M 0 − → Ad m − → ( TQ/G ) m − → T m M − → 0 Set of splittings: C m ( π ) . Lagrangian: L : C ( π ) → R 5

� � General case Consider π � F E � N M π with π = ( π, π ) epimorphism. Consider the subbundle K = ker( π ) → M . For m ∈ M and n = π ( m ) we have 0 − → K m − → E m − → F n − → 0 and we can consider the set of splittings of this sequence. 6

We define the sets L m π = { w : F n → E m | w is linear } J m π = { φ ∈ L m π | π ◦ φ = id F n } V m π = { ψ ∈ L m π | π ◦ ψ = 0 } . Projections ˜ π 10 : L π → M vector bundle π 10 : J π → M affine subbundle π 10 : V π → M vector subbundle 7

Local expressions Take { e a , e α } adapted basis of Sec( E ) , i.e. { π ( e a ) = ¯ e a } is a basis of Sec( F ) and { e α } basis of Sec( K ) . Also take adapted coordinates ( x i , u A ) to the bundle π : M → N . An element of L π is of the form w = ( y b a e b + y α a e α ) ⊗ e a Thus we have coordinates ( x i , u A , y b a , y α a ) on L π . An element of J π is of the form φ = ( e a + y α a e α ) ⊗ e a Thus we have coordinates ( x i , u A , y α a ) on J π . 8

Anchor We will assume that F and E are Lie algebroids and π is a morphism of Lie algebroids. ∂ e a ) = ρ i ρ (¯ a ∂x i ∂ ∂ ρ ( e a ) = ρ i ∂x i + ρ A a a ∂u A ∂ ρ ( e α ) = ρ A α ∂u A Total derivative with respect to a section η ∈ Sec( F ) � df ⊗ η = ´ f | a η a . where ∂f a ) ∂f f | a = ρ i ´ ∂x i + ( ρ A a + ρ A α y α ∂u A . a 9

Bracket Since π is a morphism e b ] = C a [¯ e a , ¯ bc ¯ e a [ e a , e b ] = C γ ab e γ + C a bc e a [ e a , e β ] = C γ aβ e γ [ e α , e β ] = C γ αβ e γ Affine structure functions: � Z α e a = C α aγ + C α βγ y β ( d e γ e α ) ⊗ ¯ aγ = a � Z α e a = C α ac + C α βc y β ac = ( d e c e α ) ⊗ ¯ a � Z b ( d e γ e b ) ⊗ ¯ aγ = e a = 0 � Z b e a = C b ac = ( d e c e b ) ⊗ ¯ ac 10

Variational Calculus Only for F = TN . Let ω be a fixed volume form on N . Variational problem: Given a function L ∈ C ∞ ( J π ) find those morphisms Φ: F → E of Lie algebroids which are sections of π and are critical points of the action functional � S (Φ) = L (Φ) ω N 11

� � Variations A homotopy is a morphism of Lie algebroids , Ψ � E TI × F � M I × N ϕ where I = [0 , 1] , such that π ◦ Ψ = pr 2 , satisfying some boundary conditions. For every s ∈ I = [0 , 1] define the maps � � � ϕ s : N → M by ϕ s ( n ) = ϕ ( s, n ) . � φ s : N → J π , section of π 1 : J π → N along ϕ s by � � φ s ( n )( a ) = Ψ(0 s , a ) for all n ∈ N and all a ∈ F n . � � σ s : N → E , section of E → N along ϕ s by � � ∂ � � � σ s ( n ) = Ψ s , 0 n � ∂s 12

In this way � Ψ( λ ∂ � � s , a n ) = φ s ( a n ) + λσ s ( n ) . ∂s � Interpretation: � � � φ s is a 1-parameter family of jets, and we say that φ 0 is homotopic to φ 1 � � σ s is the section that controls the variation φ s � � Boundary conditions: � � � σ s with compact support. � Variational vector field: � � aγ σ γ � d α σ α ∂ ∂ � s =0 = ρ A σ α ,a + Z α ds φ s ( n ) � ∂u A + . ∂y α a 13

Two consequences � Variations are of the form δu A = ρ A α σ α δy α a = σ α ,a + Z α aβ σ β . where σ α have compact support. � φ s is a morphism of Lie algebroids for every s ∈ [0 , 1] . 14

Variational problem Only for F = TN . Let ω be a fixed volume form on N . Variational problem: Given a function L ∈ C ∞ ( J π ) find those sections Φ: F → E of π which are a morphism of Lie algebroids and are critical points of the action � S (Φ) = L (Φ) ω N 15

Euler-Lagrange equations Infinitesimal admissible variations are δu A = ρ A α σ α δy α a = σ α ,a + Z α aβ σ β . Integrating by parts we get the Euler-Lagrange equations � ∂L � d = ∂L aα + ∂L Z γ ∂u A ρ A α , ∂y γ dx a ∂y α a a u A ,a = ρ A a + ρ A α y α a � � � � y α a,b + C α bγ y γ y α b,a + C α aγ y γ + C α βγ y β b y γ a + y α c C c ab + C α − ab = 0 . a b 16

Prolongation Given a Lie algebroid τ : E → M and a submersion µ : P → M we can construct the E -tangent to P (the prolongation of P with respect to E ). It is the vector bundle τ E P : T E P → P where the fibre over p ∈ P is T E p P = { ( b, v ) ∈ E m × T p P | Tµ ( v ) = ρ ( b ) } where m = µ ( p ) . Redundant notation: ( p, b, v ) for the element ( b, v ) ∈ T E p P . The bundle T E P can be endowed with a structure of Lie algebroid. The anchor ρ 1 : T E P → TP is just the projection onto the third factor ρ 1 ( p, b, v ) = v . The bracket is given in terms of projectable sections ( σ, X ) , ( η, Y ) [( σ, X ) , ( η, Y )] = ([ σ, η ] , [ X, Y ]) . 17

Local basis Local coordinates ( x i , u A ) on P and a local basis { e α } of sections of E , define a local basis { X α , V A } of sections of T E P by � � � � � � ∂ ∂ � � p, e α ( π ( p )) , ρ i X α ( p ) = and V A ( p ) = p, 0 , . � � α ∂x i ∂u A p p The Lie brackets of the elements of the basis are [ X α , X β ] = C γ αβ X γ , [ X α , V B ] = 0 and [ V A , V B ] = 0 , and the exterior differential is determined by dx i = ρ i du A = V A , α X α , d X γ = − 1 αβ X α ∧ X β , d V A = 0 , 2 C γ where { X α , V A } is the dual basis corresponding to { X α , V A } . 18

Prolongation of maps If Ψ: P → P ′ is a bundle map over ϕ : M → M ′ and Φ: E → E ′ is a morphism over the same map ϕ then we can define a morphism T Φ Ψ: T E P → T E ′ P ′ by means of T Φ Ψ( p, b, v ) = (Ψ( p ) , Φ( b ) , T p Ψ( v )) . In particular, for P = E we have the E -tangent to E T E a E = { ( b, v ) ∈ E m × T a E | Tτ ( v ) = ρ ( b ) } . 19

� � Repeated jets � E -tangent to J π . Consider τ E J π : T E J π → J π � � � � T φ π 10 ( V ) = ρ ( a ) T E J π = ( φ, a, V ) ∈ J π × E × T J π and the projection π 1 = π ◦ π 10 = ( π ◦ π 10 , π ◦ π 10 ) π 1 � F T E J π � N J π π 1 20

A repeated jet ψ ∈ J π 1 at the point φ ∈ J π is a map ψ : F n → T E φ J π such that π 1 ◦ ψ = id F n . Explicitely ψ is of the form Ψ = ( φ, ζ, V ) with � � � φ, ζ ∈ J π and V ∈ T φ J π , � � π 10 ( φ ) = π 10 ( ζ ) , � � V : F n → T φ J π satisfying � � Tπ 10 ◦ V = ρ ◦ ζ. Locally ψ = ( X a + Ψ α a X α + Ψ α ab V b e a . α ) ⊗ ¯ 21

Contact forms An element ( φ, a, V ) ∈ T E J π is horizontal if a = φ ( π ( a )) ; Z = a b ( X b + y β b X β ) + V β b V b β . An element µ ∈ T ∗ E J π is vertical if it vanishes on horizontal elements. A contact 1-form is a section of T ∗ E J π which is vertical at every point. They are spanned by θ α = X α − y α a X a . 22

The module generated by contact 1-forms is the contact module M c M c = � θ α � . The differential ideal generated by contact 1-forms is the contact ideal I c . I c = � θ α , dθ α � 23

Second order jets � A jet ψ ∈ J φ π 1 is semiholonomic if ψ ⋆ θ = 0 for every θ in M c . The jet ψ = ( φ, ζ, V ) is semiholonomic ifand only if φ = ζ . � A jet ψ ∈ J φ π 1 is holonomic if ψ ⋆ θ = 0 for every θ in I c . The jet ψ = ( φ, ζ, V ) is semiholonomic if and only if φ = ζ and M γ ab = 0 , where M γ ab = y γ ab − y γ ba + C γ bα y α a − C γ aβ y β b − C γ αβ y α a y β b + y γ c C c ab + C γ ab . 24