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

� � � � Tulczyjew Triple Lagrangian side of the triple M - positions, T M - (kinematic) α M D � � � TT ∗ M T ∗ T M configurations, d L � ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ ❉ ❉ ❉ Λ L ❉ � ☛☛☛☛☛☛☛☛☛☛ � ☛☛☛☛☛☛☛☛☛☛ L : T M → R - Lagrangian ❉ ❉ ❉ ❉ ❉ ❉ T ∗ M - phase space T M T M � ❧❧❧❧❧❧❧❧❧❧ λ L Legendre map λ : T M → T ∗ M T ∗ M T ∗ M Tulczyjew differential Λ L : T M → TT ∗ M . M M D = α − 1 M ( d L ( T M ))) = Λ L ( T M ) is the phase dynamics q ) = ( q , ∂ L λ L : T M → T ∗ M , λ L ( v ) = ξ ( d L ( v )) , λ L ( q , ˙ q ) . ∂ ˙ � � p = ∂ L p = ∂ L D = ( q , p , ˙ q , ˙ p ) : q , ˙ ∂ ˙ ∂ q JG (IMPAN) Dirac Algebroids 26/11/2014 3 / 30

Tulczyjew Triple canonical isomorphism T ∗ T M ≃ T ∗ T ∗ M , H : T ∗ M → R D = β − 1 M ( d H ( T ∗ M ))) � � p = − ∂ H q = ∂ H D = ( q , p , ˙ q , ˙ p ) : ˙ ˙ ∂ q , ∂ p JG (IMPAN) Dirac Algebroids 26/11/2014 4 / 30

� � � � � � � � Tulczyjew Triple Hamiltonian side of the triple β M T ∗ T ∗ M TT ∗ M canonical isomorphism ❋ ❉ T π M ❋ ❉ � ✠✠✠✠✠✠✠✠✠✠✠ � ☛☛☛☛☛☛☛☛☛☛ ❋ ❉ ❉ ❋ T ∗ T M ≃ T ∗ T ∗ M , τ T ∗ M ❋ ❉ ζ π T ∗ M T M T M � ☛☛☛☛☛☛☛☛☛☛ � ☞☞☞☞☞☞☞☞☞☞ H : T ∗ M → R τ M τ M T ∗ M T ∗ M ❍ ❋ ❍ π M ❋ π M ❋ ❍ ❍ ❋ ❋ ❍ ❍ ❋ M M D = β − 1 M ( d H ( T ∗ M ))) � � p = − ∂ H q = ∂ H D = ( q , p , ˙ q , ˙ p ) : ˙ ∂ q , ˙ ∂ p JG (IMPAN) Dirac Algebroids 26/11/2014 4 / 30

� � � � � � � Tulczyjew Triple Hamiltonian side of the triple β M T ∗ T ∗ M TT ∗ M D canonical isomorphism ❋ ❉ ❋ ζ ❉ � ✠✠✠✠✠✠✠✠✠✠✠ � ☛☛☛☛☛☛☛☛☛☛ ❋ ❉ ❉ ❋ T ∗ T M ≃ T ∗ T ∗ M , ❋ ❉ d H T M T M π T ∗ M H : T ∗ M → R T ∗ M T ∗ M M M D = β − 1 M ( d H ( T ∗ M ))) � � p = − ∂ H q = ∂ H D = ( q , p , ˙ q , ˙ p ) : ˙ ∂ q , ˙ ∂ 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. π : E ∗ − τ : E − → M → M ( x a , y i ) �− → ( x a ) ( x a , ξ i ) �− → ( x a ) π M : T ∗ M − τ M : T M − → M → M ( x a , ˙ x b ) �− → ( x a ) ( x a , p b ) �− → ( x a ) x a ∂ ˙ x a + ˙ ∇ 1 = ˙ ξ i ∂ ˙ ξ i ∇ 2 = ξ i ∂ ξ i + ˙ ξ j ∂ ˙ ξ j 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. π : E ∗ − τ : E − → M → M ( x a , y i ) �− → ( x a ) ( x a , ξ i ) �− → ( x a ) π M : T ∗ M − τ M : T M − → M → M ( x a , ˙ x b ) �− → ( x a ) ( x a , p b ) �− → ( x a ) τ E ∗ : T E ∗ − → E ∗ x a + ˙ x a ∂ ˙ ∇ 1 = ˙ ξ i ∂ ˙ x b , ˙ ξ i ( x a , ξ i , ˙ → ( x a , ξ i ) ξ j ) �− T π : T E ∗ − ∇ 2 = ξ i ∂ ξ i + ˙ ξ j ∂ ˙ → T M ξ j x b , ˙ ( x a , ξ i , ˙ → ( x a , ˙ x b ) ξ j ) �− 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. π : E ∗ − τ : E − → M → M ( x a , y i ) �− → ( x a ) ( x a , ξ i ) �− → ( x a ) π M : T ∗ M − τ M : T M − → M → M ( x a , ˙ x b ) �− → ( x a ) ( x a , p b ) �− → ( x a ) T E ∗ ❋ � ②②②②②②②② ❋ ❋ τ E ∗ x a + ˙ T π x a ∂ ˙ ❋ ∇ 1 = ˙ ξ i ∂ ˙ ❋ ❋ ξ i ❋ ❋ E ∗ T M ❊ ❊ � ①①①①①①①①① ∇ 2 = ξ i ∂ ξ i + ˙ ❊ ξ j ∂ ˙ π ❊ ❊ ξ j ❊ ❊ τ M ❊ M JG (IMPAN) Dirac Algebroids 26/11/2014 5 / 30

Fundamental isomorphisms π E ∗ : T ∗ E ∗ − ζ : T ∗ E ∗ − → E ∗ → E ( x a , ξ i , p b , y j ) �− → ( x a , ξ i ) ( x a , ξ i , p b , y j ) �− → ( x a , y j ) ∇ 1 = p a ∂ p a + y i ∂ y i , ∇ 2 = p a ∂ p a + ξ i ∂ ξ i . Double vector bundle T ∗ E ∗ is (anti-)symplectically isomorphic to T ∗ E : R : T ∗ E → T ∗ E ∗ , ( x a , y i , p b , ξ j ) �− → ( x a , ξ i , − p b , y j ) . JG (IMPAN) Dirac Algebroids 26/11/2014 6 / 30

� � Fundamental isomorphisms π E ∗ : T ∗ E ∗ − ζ : T ∗ E ∗ − → E ∗ → E ( x a , ξ i , p b , y j ) �− → ( x a , ξ i ) ( x a , ξ i , p b , y j ) �− → ( x a , y j ) T ∗ E ∗ ❊ ① ❊ ① ❊ π E ∗ ζ ① ❊ ① ❊ ① ❊ ① ❊ ① ❊ ① ❊ � ① E ∗ E ❋ ❋ � ②②②②②②②② ❋ ❋ π ❋ ❋ ❋ τ ❋ ❋ M ∇ 1 = p a ∂ p a + y i ∂ y i , ∇ 2 = p a ∂ p a + ξ i ∂ ξ i . Double vector bundle T ∗ E ∗ is (anti-)symplectically isomorphic to T ∗ E : R : T ∗ E → T ∗ E ∗ , ( x a , y i , p b , ξ j ) �− → ( x a , ξ i , − p b , y j ) . 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 = T M ): ε T ∗ E T E ∗ ■ π E ▲ T π ■ ▲ � ✝✝✝✝✝✝✝✝✝ ■ � ✝✝✝✝✝✝✝✝✝ τ E ∗ ▲ ■ ▲ ■ ρ T ∗ τ E T M τ � ✞✞✞✞✞✞✞✞ � ☎☎☎☎☎☎☎☎☎ τ M id E ∗ E ∗ ❑ ❑ π π ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ id � M M ε ( x a , y i , p b , ξ j ) = ( x a , ξ i , ρ b k ( x ) y k , c k ij ( x ) y i ξ k + σ a j ( x ) p a ) Π ε = c k ij ( x ) ξ k ∂ ξ i ⊗ ∂ ξ j + ρ b i ( x ) ∂ ξ i ⊗ ∂ x b − σ a j ( x ) ∂ x a ⊗ ∂ ξ j . It is a skew algebroid (resp. Lie algebroid) if the tensor Π ε is a bivector field (resp., Poisson tensor). JG (IMPAN) Dirac Algebroids 26/11/2014 7 / 30

� � � � � � � � � � � � Algebroid setting, no constraints D L ˜ Π ε ε T ∗ E ∗ � T E ∗ T ∗ E ❋ ❍ ❊ ❋ ❍ ❊ � ✠✠✠✠✠✠✠✠✠✠✠ � ✡✡✡✡✡✡✡✡✡✡ � ✡✡✡✡✡✡✡✡✡✡ ❋ ❍ ❊ ❋ ❊ ❍ ❋ ❍ ❊ ❊ ❋ ρ ρ E T M E � ✡✡✡✡✡✡✡✡✡✡ � ✠✠✠✠✠✠✠✠✠✠✠ � ☞☞☞☞☞☞☞☞☞☞ E ∗ E ∗ E ∗ ❍ ❋ ❋ ❍ ❋ ❋ ❋ ❋ ❍ ❍ ❋ ❋ ❋ ❋ ❍ ❍ ❋ ❋ � M M M H : E ∗ − → R D = Λ L ( E ) L : E − → R D = ˜ D H = d H ( E ∗ ) ⊂ T ∗ E ∗ Π ε ( d H ( E ∗ )) D L = d L ( 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 only in particular cases there is a Hamiltonian description corresponding to a given Lagrangian (˜ Π ε ( d H ( E ∗ )) = Λ L ( E )). JG (IMPAN) Dirac Algebroids 26/11/2014 8 / 30

� � � � � � � � � � � � � � Algebroid setting, no constraints D L � � ˜ Π ε ε T ∗ E ∗ � T E ∗ T ∗ E d L � ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ ❋ ❍ ❊ Λ L ❋ ❍ ❊ � ✠✠✠✠✠✠✠✠✠✠✠ � ✡✡✡✡✡✡✡✡✡✡ � ✡✡✡✡✡✡✡✡✡✡ ❊ ❋ ❍ ❋ ❊ ❍ ❋ ❍ ❊ ❊ ❋ ρ ρ E T M E � ❧❧❧❧❧❧❧❧❧❧❧❧ λ L � ✡✡✡✡✡✡✡✡✡✡ � ✠✠✠✠✠✠✠✠✠✠✠ � ☞☞☞☞☞☞☞☞☞☞ E ∗ E ∗ E ∗ ❍ ❋ ❋ ❍ ❋ ❋ ❋ ❋ ❍ ❍ ❋ ❋ ❋ ❋ ❍ ❍ ❋ ❋ � M M M H : E ∗ − → R D = Λ L ( E ) L : E − → R D = ˜ D H = d H ( E ∗ ) ⊂ T ∗ E ∗ Π ε ( d H ( E ∗ )) D L = d L ( 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 only in particular cases there is a Hamiltonian description corresponding to a given Lagrangian (˜ Π ε ( d H ( E ∗ )) = Λ L ( E )). JG (IMPAN) Dirac Algebroids 26/11/2014 8 / 30

� � � � � � � � � � � � � � � Algebroid setting, no constraints D D L � � � � ˜ Π ε ε T ∗ E ∗ � T E ∗ T ∗ E d L � ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ ❋ ❍ ❊ Λ L ❋ ❍ ❊ � ✠✠✠✠✠✠✠✠✠✠✠ � ✡✡✡✡✡✡✡✡✡✡ � ✡✡✡✡✡✡✡✡✡✡ ❊ ❋ ❍ ❋ ❊ ❍ ❋ ❍ ❊ ❊ ❋ ρ ρ E T M E � ❧❧❧❧❧❧❧❧❧❧❧❧ λ L � ✡✡✡✡✡✡✡✡✡✡ � ✠✠✠✠✠✠✠✠✠✠✠ � ☞☞☞☞☞☞☞☞☞☞ E ∗ E ∗ E ∗ ❍ ❋ ❋ ❍ ❋ ❋ ❋ ❋ ❍ ❍ ❋ ❋ ❋ ❋ ❍ ❍ ❋ ❋ � M M M H : E ∗ − → R D = Λ L ( E ) L : E − → R D = ˜ D H = d H ( E ∗ ) ⊂ T ∗ E ∗ Π ε ( d H ( E ∗ )) D L = d L ( 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 only in particular cases there is a Hamiltonian description corresponding to a given Lagrangian (˜ Π ε ( d H ( E ∗ )) = Λ L ( E )). JG (IMPAN) Dirac Algebroids 26/11/2014 8 / 30

� � � � � � � � � � � � � � � � � Algebroid setting, no constraints D H D D L � � � � � � ˜ Π ε ε T ∗ E ∗ � T E ∗ T ∗ E d L � ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ ❋ ❍ ❊ Λ L ❋ ❍ ❊ � ✠✠✠✠✠✠✠✠✠✠✠ � ✡✡✡✡✡✡✡✡✡✡ � ✡✡✡✡✡✡✡✡✡✡ ❊ ❋ ❍ ❋ ❊ ❍ ❋ ❍ ❊ d H ❊ ❋ ρ ρ E T M E � ❧❧❧❧❧❧❧❧❧❧❧❧ λ L � ✡✡✡✡✡✡✡✡✡✡ � ✠✠✠✠✠✠✠✠✠✠✠ � ☞☞☞☞☞☞☞☞☞☞ E ∗ E ∗ E ∗ ❍ ❋ ❋ ❍ ❋ ❋ ❋ ❋ ❍ ❍ ❋ ❋ ❋ ❋ ❍ ❍ ❋ ❋ � M M M H : E ∗ − → R D = Λ L ( E ) L : E − → R D = ˜ D H = d H ( E ∗ ) ⊂ T ∗ E ∗ Π ε ( d H ( E ∗ )) D L = d L ( 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 only in particular cases there is a Hamiltonian description corresponding to a given Lagrangian (˜ Π ε ( d H ( E ∗ )) = Λ L ( E )). JG (IMPAN) Dirac Algebroids 26/11/2014 8 / 30

� � � � � � � � � Algebroids setting with constraints Consider a constraint submanifold i S : S ֒ → E . Vakonomic constraint procedure (which is actually quite ‘holonomic’): Lagrangian L : S → R defines a lagrangian submanifold D L ( S ) ⊂ T ∗ E . ε T ∗ E ⊃ D L ( S ) D ⊂ T E ∗ . ✯ � ❥ ❥ ❥ Λ L ❥ ❥ ❥ ❥ π E ❥ τ E ∗ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ λ L ✤ � E ∗ S ⊂ E Here, Λ L and λ L are relations! Similarly on the Hamiltonian side. Nonholonomic constrain procedure: S is a vector subbundle, L : E → R . T i ∗ ε S � D ⊂ T S ∗ T ∗ E T E ∗ . ❢ ❢ ❢ ❢ ❢ ❢ Λ L ❢ ❢ ❢ ❢ ❢ ❢ π E ❢ τ S ∗ ❢ d L ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ i ∗ ❢ ❢ λ L ❢ S � S ∗ E ∗ S ⊂ E 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 = T N ⊕ N T ∗ N : ( X 1 + α 1 | X 2 + α 2 ) = 1 2 ( α 1 ( X 2 ) + α 2 ( X 1 )) . Courant-Dorfman bracket on the space of Sec ( T N ): [ [ X 1 + α 1 , X 2 + α 2 ] ] = [ X 1 , X 2 ] + L X 1 α 2 − ı X 2 d α 1 . 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 N 0 ⊂ N . JG (IMPAN) Dirac Algebroids 26/11/2014 10 / 30

Dirac structures For Π ∈ Sec ( � 2 T N ), � Π : T ∗ N ∋ α �− → ı α Π ∈ T N , 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 ), � → − ı X ω ∈ T ∗ N , ω : T N ∋ X �− 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 − → T F is a morphism of double vector bundles. A two-form ω on a vector bundle F is linear if the corresponding map → T ∗ F ω : 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 τ 1 : T E ∗ → E ∗ but also a vector subbundle of the vector bundle τ 2 : T E ∗ → T M ⊕ M E . T E ∗ ⊕ E ∗ T ∗ E ∗ D ▼ τ D ▼ τ D � rrrrrrr ▼ 1 ▼ 2 ◗ τ 1 ◗ τ 2 ▼ � rrrrrrr ◗ ▼ ◗ ◗ ▼ ◗ ◗ ◗ Ph D Vel D E ∗ T M ⊕ M E ▲ π D ▲ � rrrrrr ▼ ▼ ▲ ▼ π � ♠♠♠♠♠♠♠♠♠♠ ▲ ▼ ▲ ▼ ▲ ▼ τ D ▼ ▼ ( τ M ,τ ) M D M Ph D = τ 1 ( D ) - the phase bundle. Vel D = τ 2 ( 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 2 c k ij ( x ) ξ k ∂ ξ i ∧ ∂ ξ j + ρ b c k ij ( x ) = − c k i ( x ) ∂ ξ i ∧ ∂ x b , ji ( x ) , � � x b = ρ b k ( x ) y k , ˙ ( x a , ξ i , ˙ x b , ˙ ξ j , p c , y k ) : ˙ ξ j = c k ij ( x ) y i ξ k − ρ a D Π = j ( x ) p a . The phase bundle is the whole E ∗ , the velocity bundle is the graph of � k ( x ) y k � x b = ρ b ( x a , ˙ x b , y k ) : ˙ ρ : E → T M : Vel D Π = ⊂ T M ⊕ 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 : T M → E : x b , y k ) : y k = ρ k Vel D ω = { ( x a , ˙ x a } ⊂ T M ⊕ M E . a ( x ) ˙ 3. The canonical Dirac-Lie algebroid, D M = D Π M = D ω M , corresponding to the canonical Lie algebroid E = T M , 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 ζ ) in T E ∗ such that One can find local coordinates ( x , ¯ ξ, � η, ζ, � ξ, η, � ξ ) are coordinates in E ∗ with ( x , ¯ ξ, � Ph D = { � ξ = 0 } , ( x , η, � η ) are coordinates in T M ⊕ M E with Vel D = { � η = 0 } , ζ ) are dual coordinates in T ∗ M ⊕ M E ∗ with ( x , ζ, � Vel ⊥ D = { ζ = 0 } , jk ( x ) η j ¯ D = { ( x , ¯ ξ, 0 , η, 0 , ζ, � ζ k = c i ζ ) : ξ i } , for some ‘structure coefficients’ c i 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

� � � � Dirac algebroid induced by constraints Initial data: Dirac algebroid D on E and a vector subbundle V of Vel D . D ❋ � ①①①①①①①① ❋ τ D ❋ τ D ❋ 1 2 ❋ V ⊂ Vel D ⊂ T M ⊕ M E ❋ ❋ ❋ ❋ Ph D Vel D � V = ( τ D 2 ) − 1 ( V ); ❊ ❊ � ②②②②②②②② ❊ π D ❊ ❊ V ⊥ ⊂ T ∗ M ⊕ M E ∗ ; ❊ ❊ τ D ❊ M D D V = � V + V ⊥ is a Dirac algebroid with Vel D V = V . D V ❆ � ①①①①①①①①① ❆ ❆ ❆ ❆ ❆ Definition ❆ ❆ The Dirac algebroid D V is called Ph D V V ● ⑤ ● ⑤ ● ⑤ induced from D by the subbundle V . ● ⑤ ● ⑤ ● ⑤ ● ⑤ ● ⑤ ● � ⑤ S JG (IMPAN) Dirac Algebroids 26/11/2014 17 / 30

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