Tulczyjew triples: from statics to field theory

Transcription

1 Tulczyjew triples: from statics to field theory Janusz Grabowski Polish Academy of Sciences 3rd Iberoamerican Meetings on Geometry, Mechanics and Control Salamanca, Sept. 3, 2012 J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

2 Contents Variational Calculus in Statics Mechanics for finite time interval Mechanics: infinitesimal version Classical Tulczyjew triple Classical fields for bounded domains Classical fields: infinitesimal version Tulczyjew triple: Lagrangian side for Field Theory Tulczyjew triple: Hamiltonian side for Field Theory The Tulczyjew triple for Field Theory Tulczyjew triple for time-dependent systems Scalar fields References J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

3 Some affine geometry Notation: τ : A N an affine bundle, v(τ) : v(a) N its model vector bundle, (x i, y a ) (x i ). Definition An AV-bundle is an affine bundle θ : Z M modeled on a trivial one-dimensional vector bundle M Y. Example Consider the affine dual bundle τ : A = Aff(A, Y ) N, (x i, p a, r) (x i ) of fiberwise affine maps from an affine bundle A over N into a one-dimensional vector space Y (or N Y ). The bundle A is simultaneously an AV-bundle over Hom(v(A), Y ) = v(a) Y modeled on v(a) Y Y, θ : A v(a) Y, (x i, p a, r) (x i, p a ). J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

4 The affine phase bundle Given an AV-bundle θ : Z M and F 1, F 2 Sec(Z), we have F 1 F 2 : M Y, and d(f 1 F 2 )(m) Y, so d(f 1 F 2 ) = 0 is well defined. Definition The phase bundle PZ of the AV-bundle Z is the affine bundle of cosets df (m) ( affine differentials ) of the equivalence relation (m 1, F 1 ) (m 2, F 2 ) m 1 = m 2, d(f 1 F 2 ) = 0. Fixing a section F 0 : M Z and a basic vector y Y we get a diffeomorphism ψ : PZ T M, df (m) d(y (F F 0 )). The pullback of the canonical symplectic form on T M does not depend on the choice of F 0 and y and makes PZ into a symplectic manifold. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

5 Variational calculus in statics δq Q - manifold of configurations Γ - admissible processes, i.e., one-dimensional oriented submanifolds with boundary (sometimes, however, we use a parametrization) W : Γ R - the cost function W(γ) = γ W, for W being a positively homogeneous function on the set TQ of vectors δq tangent to admissible processes. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

6 Variational calculus in statics Definition Point q Q is an equilibrium point of the system if for all processes starting in q the cost function is non-negative, at least initially. First-order condition: W (δ) 0. Interactions between systems are described by composite systems 1 system (1) and (2) have the same configurations Q = W = W 1 + W 2 The interaction with an external system is usually described in terms of forces ϕ T Q: W 2 (δq) = ϕ, δq. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

7 Variational Calculus The subset C T Q of all external forces in equilibrium with our system is called the constitutive set. We will consider only potential systems without constraints, where = TQ and W (δq) = du, δq for a function U : Q R, so that the constitutive set is C = du(q). In general, also for other theories, e.g. statics of an elastic rod, mechanics, different field theories, etc., we need Configurations Q, Processes (or at least infinitesimal processes), Functions on Q (to define regular systems), Covectors T Q (to define constitutive sets). J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

8 Mechanics for finite time interval Let M be a manifold of positions of mechanical system. We will use smooth paths in M and first-order Lagrangians L : TM R. Configurations: Q = {q : [t 0, t 1 ] M}. Functions: S(q) = t1 t 0 L( q)dt. Processes in Q come from homotopies q s (t) = χ(s, t), χ : R 2 [0, 1] [t 0, t 1 ] M. Tangent vectors are equivalence classes of curves. Cotangent vectors are equivalence classes of functions. s t 0 t 1 t J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

9 Mechanics for finite time interval We need convenient representations of vectors and covectors: d t1 ds S q s = EL( q), δq dt + PL( q), δq s=0 t 0 t 1, t 0 where EL : T 2 M T M and PL = d v L : TM T M are bundle maps. Tangent vectors are in one-to-one correspondence with paths δq in TM δq Covectors are in one-to-one correspondence with triples (f, p 0, p 1 ) p 0 f : [t 0, t 1 ] T M, p i T q(t i ) M. f p 1 J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

10 Mechanics for finite time interval We have found another representation of covectors (Liouville structure): α : PQ = {(f, p 0, p 1 )} T Q The (phase) dynamics is a subset D of PQ = {(f, p 0, p 1 )} i.e., D = α 1 (ds(q)), D = {(f, p 0, p 1 ) : f (t) = EL( q(t)), p a = PL( q(t a )), a = 0, 1}. Explicitly, writing q = (x i (t)), q = (x i (t), ẋ j (t)), f (t) = L x i ( q(t)) d ( ) L dt ẋ i ( q(t)), p a = L ẋ i ( q(t a)), a = 0, 1. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

11 Mechanics: infinitesimal version Let M be a manifold of positions of mechanical system. We will use smooth curves in M and first-order Lagrangians Configurations: Q = TM, q = (x, ẋ) Functions: S(q) = L(x, ẋ) Curves in Q come from homotopies: χ : R 2 M Tangent vectors: TQ = TTM, i.e, equivalence classes of curves in TM, δq = δẋ. Additionally, κ M : TTM TTM, κ(χ)(s, t) = χ(t, s). Covectors: T Q = T TM s ẋ(t) t J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

12 Mechanics: infinitesimal version Tangent vectors δẋ are in one-to-one correspondence with vectors tangent to curves t δx(t) in TM κ M : TTM δẋ (δx) TTM s t δq We get also the tangent evaluation between TT M and TTM defined on elements ṗ and (δx) with the same tangent projection δx on TM: ṗ, (δx) = d dt p(t), δx(t). t=0 The map dual to κ, α M : TT M T TM gives us an identification of covectors from T TM with elements of TT M. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

13 Double vector bundles The bundles TTM and T TM are examples of double vector bundles. Definition A double vector bundle is a manifold with two compatible vector bundle structures. Compatibility means that the Euler vector fields (generators of homothethies) associated with the two structures commute. TTM τ TM Tτ M TM TM τ M τ M M (x a, ẋ b, δx c, ẍ d ), 1 = δx a δx a + ẍ a ẍa, 2 = ẋ a ẋa + ẍ a ẍa. The diffeomorphism κ M : TTM TTM interchanges the two vector bundle structures, (x a, ẋ b, δx c, ẍ d ) (x a, δx c, ẋ b, ẍ d ). It contains the information about the bracket of vector fields. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

14 Examples: TE and T E τ E : TE E (x a, y i, ẋ b, ẏ j ) (x a, y i ) Tτ : TE TM (x a, y i, ẋ b, ẏ j ) (x a, ẋ b ) π E : T E E (x a, y i, p b, ξ j ) (x a, y i ) ζ : T E E (x a, y i, p b, ξ j ) (x a, ξ j ) E TE τ E Tτ TM τ τ M M E T E τ E ζ E τ π M J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

15 Canonical isomorphism: T E T E. T E π E ξ E E E π τ M (x a, η i, p b, y j ) T E ξ E π E E E π τ M (x a, y i, p b, η j ) R E : T E T E is a symplectic isomorphism of double v.b. structures, R E : (x a, y i, p b, η j ) (x a, η i, p b, y j ) In particular, α M : TT M T TM is the composition α = R T M β M, where β M : TT M T T M is induced by the symplectic structure. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

16 Classical Tulczyjew triple Lagrangian side of the triple M - positions, TT M T TM πtm TM - (kinematic) τ Tπ configurations, T M ξ M TM TM L : TM R - Lagrangian T τ M M - phase space τ T M M T M π M πm M M D = α 1 M (dl(tm))) PL : TM T M, PL = ξ dl, { D = (x, p, ẋ, ṗ) : whence the Euler-Lagrange equation: p = L ẋ, L x = d dt α M PL(x, ẋ) = (x, L ẋ ). ṗ = L }, x ( L ) ẋ. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

17 Classical Tulczyjew triple M - positions, D TT M TM - (kinematic) configurations, TM L : TM R - Lagrangian T M - phase space T M Lagrangian side of the triple M D = α 1 M (dl(tm))) PL : TM T M, PL = ξ dl, { D = (x, p, ẋ, ṗ) : whence the Euler-Lagrange equation: p = L ẋ, L x = d dt α M T TM ξ π TM PL T M M PL(x, ẋ) = (x, L ẋ ). ṗ = L }, x ( L ) ẋ. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35 dl TM

18 Classical Tulczyjew triple Hamiltonian side of the triple H : T M R β M T T M TT M TπM τ T M π ζ T M TM TM τ M τ T M M T M πm πm M M D = D = β 1 M (dh(t M)) { (x, p, ẋ, ṗ) : ṗ = H x, ẋ = H } p J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

19 Classical Tulczyjew triple Hamiltonian side of the triple H : T M R T T M ζ dh π T M TM T M β M TT M D TM T M M M D = D = β 1 M (dh(t M)) { (x, p, ẋ, ṗ) : ṗ = H x, ẋ = H } p J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

20 Tulczyjew triple in mechanics β M T T M TT M T TM dl dh TM TM TM T M T M T M M M M D α M J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

21 Classical fields for bounded domains Configurations in Q (fields) are smooth sections σ : D E of a locally trivial fibration ζ : E M over a manifold M of dimension m supported on compact discs D M with the smooth boundary D. We will use the coordinates (x i, y a ) in E. Processes are vertical homotopies χ : R M [0, 1] D E, so that the infinitesimal processes (tangent vectors) are vertical vector fields δσ over D, δσ : D VE. Functions are associated with first-order Lagrangians, i.e. bundle maps L : J 1 E Ω m, where we denote Ω k = k T M, S(σ) = L(j 1 σ). Covectors are equivalence classes of functions. D J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

22 Classical fields for bounded domains We have to find convenient representatives of covectors and momenta. According to the Stokes theorem, δs, δσ = EL j 2 σ, δσ + Here is the Euler-Lagrange operator and D D EL : J 2 E V E E Ω m PL j 1 σ, δσ. PL = ξ d v L : J 1 E V E E Ω m 1 is the Legendre map, where ξ is a certain canonical map ξ : V J 1 E E Ω m V E E Ω m 1. It follows that covectors are represented by pairs of sections (f, p), where f : D V + E = V E E Ω m, p : D PE = V E E Ω m 1. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

23 Classical fields: infinitesimal version Configurations: Q = J 1 E, q = (x i, y a, y c k ). Functions: L : J 1 E Ω m. Curves in Q come from homotopies: χ : R M E. Tangent vectors: TQ = VJ 1 E, i.e, equivalence classes of vertical curves in J 1 E; covectors: T Q = V J 1 E Additionally, κ : VJ 1 E J 1 VE is an isomorphism of double, affine-vector bundles: VJ 1 E J 1 VE J 1 E VE E Momenta: PE = V E E Ω m 1 with coordinates (q i, y a, p j b ). J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

24 Classical fields: infinitesimal version We get also a bilinear evaluation, : J 1 PE J 1 E J 1 VE Ω m defined on j 1 p(x 0 ) and j 1 δσ(x 0 ) by the formula j 1 p(x 0 ), j 1 δσ(x 0 ) = d p, δσ (x 0 ). The identification of Hom(VJ 1 E, Ω m ) with V + J 1 E defines the map α : J 1 PE V + J 1 E which is dual to κ. In the adapted coordinates, α(q i, y a, p j b, y c k, pl dm ) = (qi, y a, y c k, l p l dl, pj b ). J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

25 Tulczyjew triple: Lagrangian side for Field Theory PE J 1 PE E J 1 E α V + J 1 E J 1 E PE E D = α 1 (d v L(J 1 E)) PL : J 1 E PE, PL = ξ d v L, PL(q i, y a, yj b ) = (q i, y a, L yj b ). { } D = (q i, y a, p j b, y k c, pl dm ) : pj b = L y b, pdl l = L y d, whence the Euler-Lagrange equations: L y a j = x i l L yi a. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

26 Tulczyjew triple: Lagrangian side for Field Theory D J 1 PE J 1 E PE α V + J 1 E d v L ξ J 1 E PL PE E E D = α 1 (d v L(J 1 E)) PL : J 1 E PE, PL = ξ d v L, PL(q i, y a, yj b ) = (q i, y a, L yj b ). { } D = (q i, y a, p j b, y k c, pl dm ) : pj b = L y b, pdl l = L y d, whence the Euler-Lagrange equations: L y a j = x i l L yi a. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

27 Tulczyjew triple: Hamiltonian side for Field Theory To define the Hamiltonian bundle, take x M, the affine bundle A = J 1 xe over N = E x, and the vector space Y = Ω m x. Let J xe = A = Aff(A, Y ) be the affine-dual bundle which is an AV-bundle over T x M V xe Ω m x P x E, θ x : J xe P x E, (y a, p j b, r) (y a, p j b ). Let PJ xe be the corresponding affine phase bundle of affine differentials dh x of sections H x : P x E J xe. Collecting the affine phase bundles PJ xe point by point x M, we obtain the affine phase bundle PJ E which is the bundle of vertical affine differentials d v H of sections H of the bundle θ : J E PE, Pθ : PJ E PE, (x i, y a, p j b, p c, y d l ) (x i, y a, p j b ). J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

28 Tulczyjew triple: Hamiltonian side for Field Theory The bundle PJ E is actually a double, affine-vector bundle isomorphic canonically with V + J 1 E, V + J 1 R E ξ J 1 E id PE id E Composing α with R we get β = R α, In the adapted coordinates, β : J 1 PE PJ E. PJ E J 1 E Pθ PE E β(q i, y a, p j b, y c k, pl dm ) = (qi, y a, p j b, l p l dl, y c k ). J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

29 Tulczyjew triple: Hamiltonian side for Field Theory Hamiltonians are sections of the one-dimensional affine bundle θ : J E PE. PJ β E J 1 PE Pθ J 1 E PE E PE E J 1 E D = { D = β 1 M (dv H(PE)) (q i, y a, p j b, y c k, pl dm ) : l p l dl = H y d, } y k c = H pc k J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

30 Tulczyjew triple: Hamiltonian side for Field Theory Hamiltonians are sections of the one-dimensional affine bundle θ : J E PE. PJ β E J 1 PE D d v H Pθ PE J 1 E J 1 E PE E E D = { D = β 1 M (dv H(PE)) (q i, y a, p j b, y c k, pl dm ) : l p l dl = H y d, } y k c = H pc k J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

31 The Tulczyjew triple for Field Theory D PJ E J 1 PE β α V + J 1 E PE d v H PE PE J 1 E J 1 E J 1 E d v L E E E J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

32 Tulczyjew triple for time-dependent systems Put ζ : E = Q R R = M. We have the identifications: J 1 E TQ R, PE T Q R, V + J 1 E T TQ R, J 1 PE TT Q R, PJ E T T Q R. d v H β D T T Q R TT α Q R T TQ R TQ R TQ R TQ R T Q R T Q R T Q R id Q R Q R Q R d v L J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

33 Scalar fields The theory of a scalar field is based on the fibration E = M R M. On the Lagrangian side we get: J 1 E R T M (ϕ, f ) P V E E Ω m 1 R Ω m 1 (ϕ, p) V + J 1 E R T M M Ω m M Ω m 1 (ϕ, f, a, p) J 1 P R T M M J 1 Ω m 1 (ϕ, f, j 1 p) The map α : J 1 P V + J 1 E reads α(ϕ, f, j 1 p) = (ϕ, f, dp(x), p), where x p(x) is any representative of j 1 p. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

34 Scalar fields Let g be a metric tensor on M. Denote with G : TM T M the corresponding isomorphism and ω the volume form. For the Lagrangian we get L(ϕ, f ) = 1 2 f, G 1 (f ) ω d v L(ϕ, f ) = (ϕ, f, 0, G 1 (f ) ω) R T M M Ω m M Ω m 1. The Lagrange equations for a section M x (ϕ(x), p(x)) read dϕ(x) = f, p = G 1 (f ) ω, dp = 0, j 1 (ϕ, p)(x) d v L(ϕ(x), dϕ(x)). The corresponding Euler-Lagrange equation is therefore dg 1 (dϕ) = 0, i.e ϕ = 0. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

35 Scalar fields Since J 1 E = R T M M R = E is a vector bundle, the Hamiltonian side is simplified. An affine map A : R T M Ω m is determined by p Ω m 1 and a Ω m : We have therefore A(ϕ, f ) = f p + a. J E R Ω m 1 M Ω m (ϕ, p, a) PJ E R Ω m 1 M T M M Ω m (ϕ, p, f, a) The map β : J 1 P PJ E reads β(ϕ, f, j 1 p) = (ϕ, p, f, dp(x)), where x p(x) is any representative of j 1 p. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

36 Scalar fields Since the bundle θ : J E P is trivial, Hamiltonians are maps P Ω m. For the Hamiltonian H(ϕ, p) = 1 2 p ( p), where is the Hodge operator associated with g, we get d v H(ϕ, p) = (ϕ, p, p, 0). The Hamilton equations for a section M x (ϕ(x), p(x)) read dϕ(x) = p, dp = 0. The above equations lead to the following equation for x ϕ(x) d dϕ = 0, i.e ϕ = 0. J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

37 References K. Grabowska: The Tulczyjew triple for classical fields, J. Phys. A 45 (2012), (35pp). W. Tulczyjew: Hamiltonian systems, Lagrangian systems, and the Legendre transformation, Symposia Math. 14, (1974), W. M. Tulczyjew: Les sous-variétés lagrangiennes et la dynamique lagrangienne (French), C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 8, Av, A675 A678. W. M. Tulczyjew, P. Urbański: A slow and careful Legendre transformation for singular Lagrangians, The Infeld Centennial Meeting (Warsaw, 1998), Acta Phys. Polon. B, 30 (1999), K. Gawȩdzki: On the geometrization of the canonical formalism in the classical field theory, Rep. Math. Phys. 3 (1972), J. Kijowski and W. Szczyrba: A canonical structure for classical field theories, Commun. Math. Phys. 46 (1976), J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

38 References J. Kijowski and W. Tulczyjew [1979]: A symplectic framework for field theories, Lecture Notes in Phys. 107 (1979). J. F. Cariñena, M. Crampin, L. A. Ibort: On the multisymplectic formalism for first order theories, Differential Geom. Appl., 1 (1991), M. J. Gotay, J. Isenberg, J. E. Marsden: Momentum maps and classical relativistic fields, Part I: Covariant field theory, arxiv:physics/ v2. M. J. Gotay, J. Isenberg, J.E. Marsden, Momentum maps and classical relativistic fields, Part II: Canonical analysis of field theories, arxiv:math-ph/ v1 (2004). M. J. Gotay: A multisymplecitc framework for classical field theory and the calculus of variations: in Mechanics, Analysis, and Geometry: 200 Years After Lagrange, North Holland, Amsterdam, (1991) J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

39 References F. Cantrijn, L. A. Ibort, M. De Leon: Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), L. Vitagliano: The Hamilton-Jacobi Formalism for Higher Order Field Theories, Int. J. Geom. Meth. Mod. Phys, 07 (2010), C. M. Campos, E. Guzmán, J. C. Marrero: Classical field theories of first order and lagrangian submanifolds of premultisymplectic manifolds, arxiv: K. Grabowska, J. Grabowski, P. Urbański: Geometrical Mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), J. Grabowski, K. Grabowska: Dirac Algebroids in Lagrangian and Hamiltonian Mechanics J. Geom. Phys. 61 (2011), J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35

40 The final use of affine geometry THANK YOU FOR YOUR ATTENTION! J.Grabowski (IMPAN) Tulczyjew triples: from statics to field theory 03/09/ / 35