Tulczyjew s Triple in Classical Field Theories: Lagrangian submanifolds of premultisymplectic manifolds.

Transcription

1 Tulczyjew s Triple in Classical Field Theories: Lagrangian submanifolds of premultisymplectic manifolds. E. Guzmán ICMAT- University of La Laguna eguzman@ull.es Workshop on Rough Paths and Combinatorics in Control Theory University of California, San Diego Center for Computational Mathematics Department of Mathematics July 25-27, Joint work with C. Campos and J.C. Marrero

2 Motivation W. Tulczyjew (1976) succeeded in formulating Classical Mechanics in terms of Lagrangian submanifolds in special symplectic manifolds. This approach permits to deal with singular Lagrangians. Question: Is it possible to formulate a similar result in Classical Field Theory? In the last years, some attempts have been made in this direction. However, although accurate and interesting, they all exhibit some defects, when comparing with Tulczyjew s original work. In our approach, we propose a solution to the previous problems and deficiencies. In a natural way, the symplectic structures in Tulczyjew s work are replaced by premultisymplectic structures.

3 The Lagrangian Formalism Consider a mechanical system moving in a configuration space Q whose tangent bundle TQ describes the states positions and velocities of the system. The dynamics of the system is typically governed by a function L : TQ R of the form Variational formulation: L(v q ) = 1 2 g(v q, v q ) U(q). S L (c(.)) = t1 t 0 L(c i (t), ċ i (t))dt. The extremals of the S L are characterized by the equations L q i d L dt q i = 0.

4 Symplectic formulation: X X (TQ) SODE, i X Ω L = de L. In the case, L is regular (that is Ω L is symplectic over TQ. Since, ( ) 2 L q i q j is a regular matrix), then! ξ L X (TQ) SODE and i ξl Ω L = de L.

5 The Hamiltonian Formalism The motion of the previous system is governed by a function H on the phase space of the system - positions and momentas -, H : T Q R. The Hamiltonian function is the total energy of the system, typically, Equations of motion H(p q ) = K(p q ) + U(q). Given a Hamiltonian vector field X H. That is, X H X (T Q) such that i XH Ω Q = dh. The integral curves of X H are characterized by the equations q i = H p i, ṗ i = H q i.

6 The Legendre transformation and equivalence Given a Lagrangian L : TQ R, the Legendre transformation is the fibered map leg L : TQ T Q given by < leg L (v), w >:= d dt L(v + tw) t=0, leg L (q i, q i ) = ( q i, p i = L q i ). If L is regular, then leg L is a local diffeomorphism. If L is hiper-regular (that is, leg L is a global diffeomorphism), we may define h := E L leg 1 L. Then, ξ L and X H are leg L - related.

7 Tulczyjew s Triple for Classical Mechanics W. Tulczyjew (1976) succeeded in formulating classical mechanics in terms of Lagrangian submanifolds in special symplectic manifolds. Submanifold of what manifold? and Lagrangian with respect to what structure? (T (T Q), Ω c Q ) where Ω c Q is the complete lift of the canonical symplectic structure Ω Q of T Q. S L = A 1 Q (dl(tq)) S H = (b ΩQ ) 1 (dh(t Q)) A Q T (TQ) T (T Q) T (T Q) b ΩQ dl TQ T Q dh

8 The isomorphisms A Q and b ΩQ A Q : T (T Q) T (TQ) is defined by The natural pairing <, >: T Q TQ Q R. The involution map K : T (TQ) T (TQ). This isomorphism relates the two different vector bundle structures of T (TQ) over TQ. b ΩQ : T (T Q) T (T Q) is defined by The canonical symplectic structure of T Q. A Q (q i, p i, q i, ṗ i ) = (q i, q i, ṗ i, p i ) is a symplectomorphism b ΩQ (q i, p i, q i, ṗ i ) = (q i, p i, ṗ i, q i ) is an anti-symplectomorphism A Q the Tulczyjew canonical diffeomorphism

9 Tulczyjew s triple for Classical Mechanics S L = A 1 π (dl(tq)) S h = b 1 Ω Q (dh(t Q)) S L = S H T (TQ) π TQ A Q T π Q T (T Q) τ T Q b ΩQ π T Q T (T Q) d dt (leg L σ) dl TQ leg L T Q dh τ σ Q τ σ R

10 Classical Field Theory WHAT HAPPENS IN CLASSICAL FIELD THEORY?

11 Setting J 1 π (x i,u α,u α i ) π 10 E(x i, u α ) π M(x i ) ν J 1 π + (x i,u α,p,p i α ) Base manifold: oriented manifold M, dimm = m, with fixed volume form η. Bundle configuration space: fibre bundle π : E M, dime = n + m. Space of velocities : the first jet manifold J 1 π = {j 1 x φ} where φ Γ(π) s.t. T π T φ = Id TM. Space of momenta : the dual jet manifold J 1 π + = Λ m 2 E.

12 The canonical multisymplectic form on Λ m 2 E Remark: A multisymplectic form Ω on a vector space V is a closed (m + 1)-form on V such that it is non-degenerate. That is the linear mapping is injective. Λ m 2 v V i v Ω Λ m V E admits a canonical (m+1) multisymplectic structure. Θ(ω)(X 1,..., X m ) = ω(t ω ν(x 1 ),..., T ω ν(x m )) where ω Λ m 2 E, X i X (Λ m 2 E) and ν : Λm 2 E E. Ω := d Θ.

13 The Lagrangian Formalism Let L : J 1 π Λ m M be a fibered mapping, i.e. Lagrangian density, L = Lη where L : J 1 π R. Variational formulation: Let σ Γ(π) S L (j 1 σ) = L(x i, u α, ui α ). The extremals of S L are characterized by the equations Multisymplectic formulation U L u α d ( L ) dx i ui α = 0. (j 1 σ) (i X Ω L ) = 0, X X (J 1 π).

14 The Hamiltonian formalism Consider the canonical projection µ : (J 1 π) + = Λ m 2 E Λm 2 E Λ m 1 E = J1 π. µ is a principal R-bundle. We have a one to one correspondence between F h : (J 1 π) + R h : J 1 π (J 1 π) + λ λ h(u(α)) h(x i, u α, pα) i = (x i, u α, H(x i, u α, pα), i pα) i F h (x i, u α, pα) i = p + H(x i, u α, pα). i Equations of motion: τ : M J 1 π a section of π1 : J1 π M is a solution of the Hamilton equations if i T (h τ)(xn) Ω = ( 1) m+1 df h u α x i = H pα i ; p i α x i = H u α

15 Equivalence The extended Legendre map Leg L : J 1 π J 1 π + Leg L (j 1 x φ)(x 1,..., X m ) = (Θ L ) j 1 x φ( X 1,..., X m ) where j 1 φ J 1 π and X i T φ(x) E, where X i T j 1 x φj 1 π are such that (T π 10 )( X i ) = X i. The Legendre transformation Leg L (x i, u α, u α i ) = leg L : J 1 π J 1 π leg L = µ Leg L leg L (x i, u α, u α i ) = ( x i, u α, L L ui α ui α, ( x i, u α, L u α ). L ) u α.

16 Equivalence The Legendre transformation leg L is a local diffeomorphism, if and only if the ( Lagrangian ) function L is regular (that is, the Hessian matrix 2 L is regular.) u α i u β j Whenever leg L is a global diffeomorphism, we say that the Lagrangian L is hyper-regular. In this case, we may define the Hamiltonian section h = Leg L (Leg L ) 1.

17 Tulczyjew s triple for Classical Field Theory In this work, we describe the Classical Field Theory of first order in terms of Lagrangian submanifolds in a premultisymplectic manifold. A premultisymplectic structure Ω on M is a closed (m + 1)-form on M. This premultisymplectic manifold will be the first jet of the fibration (π ν) : J 1 π + = Λ m 2 E E M. J 1 (π ν) (x i, u α, p, p i α, u α j, p j, p i αj).

18 How are the Lagrangian submanifolds defined? S L = (A π ) 1 (dl(j 1 π)) S H = (b Ω) 1 (d(f h η)(j 1 π + )) 2 J 1 π J 1 b Ω (π ν) Λ m+1 2 J 1 π + Λ m+1 A π dl J 1 π J 1 π + d(f h η)

19 Construction of the mapping A π A π : J 1 (π ν) Λ m+1 2 J 1 π is defined by The natural pairing <, >: J 1 π J 1 π + M R. (j 1 x φ) (λ) =< λ, j 1 x φ > η x. An involution map: Consider the first jet prolongation of the fibration π 1 : J 1 π M. J 1 π 1 (x i, u α, u α i, ū α j, u α ij ). J 1 π 1 admits two different affine structures over J 1 π. J 1 π (π 1) 10 J 1 π 1 π 1 M π 10 j 1 (π 10) π E π 10 J 1 π

20 Construction of the mapping A π Let be a linear connection on M. ex (x i, u α, u α i, ū α j, u α ij ) = (x i, u α, ū α i, u α j, u α ji Λ l ji(u α l ū α l )) After some operations A π : J 1 (π ν) Λ m+1 2 J 1 π A π (x i, u α, p, p i α, u α j, p j, p i αj) = (p i αidu α + p i αdu α i ) η

21 Construction of the mapping β Ω β Ω : J 1 (π ν) Λ m+1 2 J 1 π + β Ω(j 1 x λ) = i h Ω (m + 1) Ω where Ω is the canonical multisymplectic structure on J 1 π + and J 1 (π ν) h : T λ J 1 π + T λ J 1 π + β Ω(x i, u α, p, p i α, u α j, p j, p i αj) = (p j αj duα 1dp u α j dp j α) η

22 Tulczyjew s triple for Classical Field Theory S L = S H m+1 2 J 1 π A π J 1 (ν π) b Ω m+1 2 J 1 π + j 1 (Leg L j 1 σ) dl J 1 π π 1,0 Leg L leg L J 1 π + π 10 d(f h (π ν) (η)) µ h J 1 π j 1 (h τ) j 1 σ E τ σ π M

23 Affine degenerate Lagrangians Let γ : E J 1 π + be a local section of the fibration ν : J 1 π + E. ˆγ : J 1 π R for all j 1 x φ J 1 π, ˆγ(j 1 x φ) = (γ(y))(j 1 x φ), where y = π 10 (j 1 x φ). Then, we can consider S L = A 1 π (dl(j 1 π)) where the Lagrangian density L = ˆγη. γ = γ (x, u)d m x + γ i αdu α d m 1 x i, ˆγ(x i, u α, u α i ) = γ (x, u) + γ i α(x, u)u α i. S L = {(x i, u α, p, pα, i uj α, p j, pαj) i : pα i = γi α ; pαi i = γ u α + γi β u α uβ i }. Examples: Metric-affine gravity, Dirac fermion fields.

24 Quadratic degenerate Lagrangians Let b : J 1 π J 1 π + be a morphism of affine bundles over the identity of E such that µ b : J 1 π J 1 π, is an affine bundle isomorphim. We define the quadratic Lagrangian L(z) = 1 2 b(z)(z). b(x i, u α, u α i ) = (x i, u α, b (x, u)+b i α(x, u)u α i, b i α(x, u)+b ij αβ (x, u)uβ j ), (µ b)(x i, u α, u α i ) = (x i, u α, b i α(x, u) + b ij αβ (x, u)uβ j ). The condition that µ b be an affine isomorphism is equivalent to (b ij αβ ) is regular. L(x i, u α, u α i ) = 1 2 (b + (b i α + b i α)u α i + b ij αβ uα i u β j ). Then, let S L = A 1 π (dl(j 1 π)). Example: Electromagnetic fields.

25 References: W Tulczyjew, C.R. Acad. Sci. Paris 283 (1976), M de León, D Martín de Diego and A Santamaría-Merino, in Applied Differential Geometry and Mechanics, Editors W Sarlet and F Cantrijn, Univ. of Gent, Academia Press, (2003), F Cantrijn, A Ibort and M de León, J. Austral. Math. Soc. (Series A) 66 (1999), E Echeverría-Enríquez, M de León, MC. Muñóz-Lecanda and N Román-Roy, Journal of Mathematical Physics 48(2007) I Kolár and M Modugno, Annalti di Matematica pura ed applicata (IV). CLVIII(1991), DJ Saunders, London Math. Soc., Lecture Note Series, 142 Cambridge Univ. Press, (1989). W Tulczyjew, C.R. Acad. Sci. Paris 283 (1976),

26 THANK YOU!