Geometrical mechanics on algebroids
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1 Geometrical mechanics on algebroids Katarzyna Grabowska, Janusz Grabowski, Paweł Urbański Mathematical Institute, Polish Academy of Sciences Department of Mathematical Methods in Physics, University of Warsaw Ghent, August, 2005 p.1/17
2 Content Introduction Ghent, August, 2005 p.2/17
3 Content Introduction Notation and local coordinates Ghent, August, 2005 p.2/17
4 Content Introduction Notation and local coordinates Lagrange formalism for TQ (Tulczyjew triple) Ghent, August, 2005 p.2/17
5 Content Introduction Notation and local coordinates Lagrange formalism for TQ (Tulczyjew triple) Lie algebroids as double vector bundle morphisms Ghent, August, 2005 p.2/17
6 Content Introduction Notation and local coordinates Lagrange formalism for TQ (Tulczyjew triple) Lie algebroids as double vector bundle morphisms Lagrange formalism for general algebroids Ghent, August, 2005 p.2/17
7 Content Introduction Notation and local coordinates Lagrange formalism for TQ (Tulczyjew triple) Lie algebroids as double vector bundle morphisms Lagrange formalism for general algebroids Noether Theorem Ghent, August, 2005 p.2/17
8 References E. Martínez: Lagrangian Mechanics on Lie Algebroids, Acta Appl. Math. vol. 67, 2001 ( ). E. Martínez: Geometric formulation of Mechanics on Lie algebroids, in Proceedings of the VIII Fall Workshop on Geometry and Physics, Medina del Campo, 1999, Publicaciones de la RSME, vol. 2, 2001, ( ). M. de León, J.C. Marrero, E. Martínez: Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen. vol. 38, 2005 (R241 R308). Ghent, August, 2005 p.3/17
9 References E. Martínez: Lagrangian Mechanics on Lie Algebroids, Acta Appl. Math. vol. 67, 2001 ( ). E. Martínez: Geometric formulation of Mechanics on Lie algebroids, in Proceedings of the VIII Fall Workshop on Geometry and Physics, Medina del Campo, 1999, Publicaciones de la RSME, vol. 2, 2001, ( ). M. de León, J.C. Marrero, E. Martínez: Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen. vol. 38, 2005 (R241 R308). J. Grabowski and P. Urbański: Lie algebroids and Poisson-Nijenhuis structures, Rep. Math. Phys. vol. 40, 1997,( ). J. Grabowski and P. Urbański: Algebroids general differential calculi on vector bundles, J. Geom. Phys. vol. 31, 1999, ( ). Ghent, August, 2005 p.3/17
10 Notation and coordinates TM T M E E τ M π M M M M M (x a, ẋ b ) (x a, p b ) (x a, y i ) (x a, ξ i ) τ π Ghent, August, 2005 p.4/17
11 Notation and coordinates TM T M E E τ M π M M M M M (x a, ẋ b ) (x a, p b ) (x a, y i ) (x a, ξ i ) τ π TE (x a, y i, ẋ b, ẏ j ), TE (x a, ξ i, ẋ b, ξ j ), T E (x a, y i, p b, π j ), T E (x a, ξ i, p b, ϕ j ). } isomorphic as double vector bundles Ghent, August, 2005 p.4/17
12 Notation and coordinates TM T M E E τ M π M M M M M (x a, ẋ b ) (x a, p b ) (x a, y i ) (x a, ξ i ) τ π TE (x a, y i, ẋ b, ẏ j ), TE (x a, ξ i, ẋ b, ξ j ), T E (x a, y i, p b, π j ), T E (x a, ξ i, p b, ϕ j ). } isomorphic as double vector bundles R τ : T E T E (x a, y i, p b, π j ) R τ = (x a, ϕ i, p b, ξ j ). Ghent, August, 2005 p.4/17
13 More notation For a bundle τ : E M and its dual π : E M we denote: the tensor product over M: k M (E) = E M M E, the set of sections of the above tensor bundle: k (τ), Ghent, August, 2005 p.5/17
14 More notation For a bundle τ : E M and its dual π : E M we denote: the tensor product over M: k M (E) = E M M E, the set of sections of the above tensor bundle: k (τ), for K k (τ) we have the obvious pairing ι(k) : k (π) a K(m), a R, Ghent, August, 2005 p.5/17
15 More notation For a bundle τ : E M and its dual π : E M we denote: the tensor product over M: k M (E) = E M M E, the set of sections of the above tensor bundle: k (τ), for K k (τ) we have the obvious pairing ι(k) : k (π) a K(m), a R, for X being a section of τ we have the operator of insertion i X : k+1 (π) µ 1 µ k+1 X, µ 1 µ 2 µ k+1 k (π), Ghent, August, 2005 p.5/17
16 More notation For a bundle τ : E M and its dual π : E M we denote: the tensor product over M: k M (E) = E M M E, the set of sections of the above tensor bundle: k (τ), for K k (τ) we have the obvious pairing ι(k) : k (π) a K(m), a R, for X being a section of τ we have the operator of insertion i X : k+1 (π) µ 1 µ k+1 X, µ 1 µ 2 µ k+1 k (π), in particular, for Λ 2 (τ) we have Λ : E E, given by Λ X = i X Λ. Ghent, August, 2005 p.5/17
17 Lagrange formalism for TM (according to W.M.Tulczyjew) M - the configuration manifold, T M - the phase space, L : TM R - the Lagrangian. Ghent, August, 2005 p.6/17
18 Lagrange formalism for TM (according to W.M.Tulczyjew) M - the configuration manifold, T M - the phase space, L : TM R - the Lagrangian. T T M π T M β (T M,ωM ) T M τ T M π M TT M τ M Tπ M M α M T TM π TM TM Ghent, August, 2005 p.6/17
19 Lagrange formalism for TM (according to W.M.Tulczyjew) M - the configuration manifold, T M - the phase space, L : TM R - the Lagrangian. T T M π T M β (T M,ωM ) T M τ T M π M TT M τ M Tπ M M α M T TM π TM TM The dynamics is a first-order differential equation D TT M. In the regular case D = α 1 M (dl(tm)). Ghent, August, 2005 p.6/17
20 Lagrange formalism for TM (according to W.M.Tulczyjew) A solution γ : I T M of the dynamics D is a phase space trajectory of the system. Ghent, August, 2005 p.7/17
21 Lagrange formalism for TM (according to W.M.Tulczyjew) A solution γ : I T M of the dynamics D is a phase space trajectory of the system. Trajectories in the configuration space (i.e. π M γ) are solutions of the second order Euler-Lagrange equation. Ghent, August, 2005 p.7/17
22 Lagrange formalism for TM (according to W.M.Tulczyjew) A solution γ : I T M of the dynamics D is a phase space trajectory of the system. Trajectories in the configuration space (i.e. π M γ) are solutions of the second order Euler-Lagrange equation. The Euler-Lagrange equation can be obtained directly from D: Ghent, August, 2005 p.7/17
23 Lagrange formalism for TM (according to W.M.Tulczyjew) A solution γ : I T M of the dynamics D is a phase space trajectory of the system. Trajectories in the configuration space (i.e. π M γ) are solutions of the second order Euler-Lagrange equation. The Euler-Lagrange equation can be obtained directly from D: Take the prolongation PD = TD T 2 T M Ghent, August, 2005 p.7/17
24 Lagrange formalism for TM (according to W.M.Tulczyjew) A solution γ : I T M of the dynamics D is a phase space trajectory of the system. Trajectories in the configuration space (i.e. π M γ) are solutions of the second order Euler-Lagrange equation. The Euler-Lagrange equation can be obtained directly from D: Take the prolongation PD = TD T 2 T M Project to T 2 M by T 2 π M : T 2 T M T 2 M: E L = T 2 π M (PD). Ghent, August, 2005 p.7/17
25 Lie algebroids as double vector bundle morphisms Grabowski and Urbański proposed an alternative way of encoding an algebroid structure: Ghent, August, 2005 p.8/17
26 Lie algebroids as double vector bundle morphisms Grabowski and Urbański proposed an alternative way of encoding an algebroid structure: Traditionally: a bilinear bracket on sections of τ : E M, together with the anchor a : E TM; Ghent, August, 2005 p.8/17
27 Lie algebroids as double vector bundle morphisms Grabowski and Urbański proposed an alternative way of encoding an algebroid structure: Traditionally: a bilinear bracket on sections of τ : E M, together with the anchor a : E TM; By JG and PU: a particular double vector bundle morphism ε : T E TE Ghent, August, 2005 p.8/17
28 Lie algebroids as double vector bundle morphisms Grabowski and Urbański proposed an alternative way of encoding an algebroid structure: Traditionally: a bilinear bracket on sections of τ : E M, together with the anchor a : E TM; By JG and PU: a particular double vector bundle morphism ε : T E TE T E Λ R τ T E ε TE Λ is the double vector bundle morphism given by the 2-contravariant tensor Λ canonically associated with the bracket. Ghent, August, 2005 p.8/17
29 Algebroids as double vector bundle morphisms On the cores ε T E TE π E τ E Tπ T τ ε r E τ TM τ M E id π E π id M M ε c : T M E Ghent, August, 2005 p.9/17
30 Algebroids as double vector bundle morphisms On the cores ε T E TE π E τ E Tπ T τ ε r E τ TM τ M E id π E π id M M ε c : T M E Λ : Λ = c k ij (x)ξ k ξi ξj + ρ b i (x) ξ i x b σ a j (x) x a ξj ε : (x a, ξ i, ẋ b, ξ j ) ε = (x a, π i, ρ b k (x)yk, c k ij (x)yi π k + σ a j (x)p a) ε r : (x a, ẋ b ) ε r = (x a, ρ b k (x)yk ), ε c : (x a, ξ i ) ε c = (x a, σ b i (x)p b). Ghent, August, 2005 p.9/17
31 Algebroids as double vector bundle morphisms Remark: An algebroid (E, ε) is a Lie algebroid if and only if the corresponding tensor Λ ε is a Poisson tensor. It means in particular that ε r = (ε c ) (ρ = σ). Ghent, August, 2005 p.10/17
32 Algebroids as double vector bundle morphisms Remark: An algebroid (E, ε) is a Lie algebroid if and only if the corresponding tensor Λ ε is a Poisson tensor. It means in particular that ε r = (ε c ) (ρ = σ). Canonical example: Canonical algebroid of TM is described by the Tulczyjew α M 1 α M : T TM TT M Ghent, August, 2005 p.10/17
33 Lagrange formalism for general algebroid The algebroid analogue for the Tulczyjew triple. E replaces TM. TE T E T π τ E Tπ T τ π E π E ε r ε r E TM E τ τ M τ E id E id E π π π id M id M M T E Λ ε Ghent, August, 2005 p.11/17
34 Lagrange formalism for general algebroid The algebroid analogue for the Tulczyjew triple. E replaces TM. TE T E T π τ E Tπ T τ π E π E ε r ε r E TM E τ τ M τ E id E id E π π π id M id M M T E Λ Any function ( Lagrangian ) L : E R defines dl(e) T E and D = ε(dl(e)) TE. ε Ghent, August, 2005 p.11/17
35 Lagrange formalism for general algebroid The Lagrangian defines two smooth maps: Ghent, August, 2005 p.12/17
36 Lagrange formalism for general algebroid The Lagrangian defines two smooth maps: (1) The Legendre mapping: L eg : E E defined by L eg = τ E ε dl τ E E TE TM ε E T E dl E Ghent, August, 2005 p.12/17
37 Lagrange formalism for general algebroid The Lagrangian defines two smooth maps: (1) The Legendre mapping: L eg : E E defined by L eg = τ E ε dl τ E E TE TM ε E T E dl E (2) Leg : E TE defined by Leg = ε dl E TE TM ε E T E dl E Ghent, August, 2005 p.12/17
38 Lagrange formalism for general algebroid For general algebroid we have constructed two equations for curves γ : I E: (1) The equation E 1 L given by E 1 L = (TL eg) 1 (D). The curve γ is a solution of E 1 L if and only if L eg γ is a solution of D, i.e. T(L eg γ) D. E 1 L corresponds to the construction of de Léon and Lacomba: M. de León, E. Lacomba: Lagrangian submanifolds and higher-order mechanical systems, J.Phys. A: Math. Gen. vol. 22, 1989, ( ). Ghent, August, 2005 p.13/17
39 Lagrange formalism for general algebroid (2) The equation E 2 L given by E 2 L = {v TE : T L eg (v) T 2 E }, where T 2 E TTE is the subset of holonomic vectors w, i.e. w TTE such that τ TE (w) = Tτ E (w). The curve γ is a solution of E 2 L if the tangent prolongation T(L eg γ) is exactly L eg γ. Since L eg γ is in D by definition, we have E 2 L = E1 L. E 2 L is the equation given in works of A. Weinstein, E. Martinez,... Ghent, August, 2005 p.14/17
40 Lagrange formalism for general algebroid The coordinate expression for E 2 L is dx a dt = ρa k (x)yk, d dt ( ) L y j = c k L ij (x)yi y k (x, y)+σa j L (x) (x, y) xa Ghent, August, 2005 p.15/17
41 Lagrange formalism for general algebroid The coordinate expression for E 2 L is dx a dt = ρa k (x)yk, d dt ( ) L y j The coordinate expression for E 1 L is dx a dt = ρa k (x)yk, d dt ( ) L y j = c k L ij (x)yi y k (x, y)+σa j = c k ij (x)yi 0 L y k (x, y)+σa j for certain (x, y 0 ) E satisfying L eg (x, y 0 ) = L eg (x, y) and ε r (x, y 0 ) = ε r (x, y), i.e. L L (x, y) = yi y i (x, y 0), ρ b k (x)yk = ρ b k (x)yk 0. L (x) (x, y) xa L (x) (x, y). xa Ghent, August, 2005 p.15/17
42 Noether Theorem The vertical lift: K k (τ), v τ (K) (τ E ), in coordinates v τ (f i 1 ke i1 e ik ) = f i 1 k y i 1 y i k. The algebroid lift: ι(d ε T(K)) = d T (ι(k)) ε k. Ghent, August, 2005 p.16/17
43 Noether Theorem The vertical lift: K k (τ), v τ (K) (τ E ), in coordinates v τ (f i 1 ke i1 e ik ) = f i 1 k y i 1 y i k. The algebroid lift: ι(d ε T(K)) = d T (ι(k)) ε k. In local coordinates, for functions: d ε T(f(x)) = y i ρ a i (x) f x a (x) and for sections of τ: ) d ε T(f i (x)e i ) = f i (x)σ a i (x) x + (y i ai fk ρ (x) a x a (x) + ck ij (x)yi f j (x) y k. Ghent, August, 2005 p.16/17
44 Noether Theorem Theorem. If X is a section of E and f is a function on M, then d ε T(X)L = d ε T(f) (1) if and only if the function (ι(x) v π (f)) L eg on E is a first integral of the equation (E 2 L ). In particular, if (1) is satisfied, then (ι(x) v π (f)) L eg is a first integral of the equation (E 1 L ). Ghent, August, 2005 p.17/17
45 Noether Theorem Theorem. If X is a section of E and f is a function on M, then d ε T(X)L = d ε T(f) (1) if and only if the function (ι(x) v π (f)) L eg on E is a first integral of the equation (E 2 L ). In particular, if (1) is satisfied, then (ι(x) v π (f)) L eg is a first integral of the equation (E 1 L ). Remark. One can consider the Hamiltonian part of the Tulczyjew triple for an arbitrary algebroid and prove the standard correspondence between the Lagrangian and the Hamiltonian in the regular case. To this end, dealing with a Lie and not with an arbitrary algebroid played no role, contrary to the common opinion that geometrical mechanics is a Lie algebroid theory. No brackets have been really used! Ghent, August, 2005 p.17/17
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