Geometry of the Legendre transformation

Transcription

1 1. Introdction. Geometry of the Legendre transformation Paeł Urbański Katedra etod atematycznych Fizyki, Uniersytet Warszaski Hoża 74, Warszaa Lagrangian systems Hamiltonian systems offer different representations of the same object - the dynamics of a mechanical system. The Legendre transformation is the passage from one of these representations to the other. To get rogh idea hat is the story abot, let s consider the case of a hyperreglar lagrangian. Lagrangian is a fnction ith the vertical derivative L:, d v L: T. Lagrangian L is called hyperreglar if d v L is a diffeomorphism. If it is the case, then e can define Hamiltonian by H: T : p p, d 1 v L(p) L d 1 v L(p). We se this Hamiltonian to generate a vector field X H on T : dh, Y = ω(x H, Y ). This vector field can be obtained directly from the Lagrangian by the canonical diffeomorphism given by the eqivalence of fnctors TT T T We have for p = d v L(v) α : TT T. α (X H (p)) = dl(v). We can smmarize this example ith a pictre β T T TT T π T P Tπ P τ T π T ''' (1) [ [ π ') [ τ [^ The dynamics is in the center, Langrangian system is on the right, Hamiltonian system on the left. This diagram is knon as Tlczyje triple. It follos from this pictre that each of the top objects is a symplectic manifold also has to (compatible) vector bndle strctres, i.e. they are doble vector bndles. Ths the folloing strctres are involved in the Legendre transformation: (1) Lagrangian sbmanifolds generating objects, (2) Iterated tangent fnctors their eqivalence, (3) Doble vector bndles. α 1

2 2. Generating objects of Lagrangian sbmanifolds. In Physics, e have to accept more general dynamics then vector fields, more general lagrangian sbmanifolds in T then sections of π more general generating objects (Lagrangian, Hamiltonian systems) then fnctions on orse reglar families. Let be a differential fibration. We denote by V the bndle of vertical vectors defined by ρ X (2) V = {v T; Tρ(v) = 0} (3) We denote by V the polar of the vertical bndle defined by V = { s T ; v V τ (v) = π (s) s, v = 0 } (4) A fnction F : can be considered a family of fnctions defined on fibres of the fibration ρ. We ill represent a family of fnctions by a diagram F ρ X The critical set for a family of fnctions F : is the set } S(F, ρ) = {r ; df, v = 0 v Vr. (6) At each point r S(F, ρ) e define a bilinear mapping W (F, r): V r T r (5) : (v, ) D (1,1) (F χ)(0, 0), (7) here χ is a mapping from 2 to sch that v = tχ(, 0)(0) = tχ(0, )(0). The family F is said to be reglar if S(F, ρ) is a sbmanifold the rank of W (F, r) at each r S(F, ρ) is eqal to the codimension of S(F, ρ). The family F is called a orse family if the rank of W (F, r) is maximal at each r S(F, ρ). orse family is reglar. It is knon that if F is a reglar family, then the set { = p T X; r ρ(r) = q = π X (p) } v TqX Tρ() = v p, v = df, Tr (8) is an immersed Lagrangian sbmanifold of the symplectic space (T X, ϑ X ). This Lagrangian sbmanifold is said to be generated by the family (5). For each reglar family F e have the mapping κ: S(F, ρ) T X characterized by κ(r), v = df, (9) 2

3 for each v TX sch that τ X (v) = ρ(r) each T sch that Tρ() = v. The mapping κ is a sbimmersion. If F is a orse family, then κ is an immersion. The immersed Lagrangian sbmanifold (8) is the image im(κ) of the mapping κ. There is a srjective sbmersion λ : V T X characterized by λ(q), v = q, (10) for each v TX sch that τ X (v) = ρ(π (q)) each T sch that Tρ() = v. The intersection im(df ) V is clean if only if F is a reglar family. This intersection is transverse if only if F is a orse family. The immersed Lagrangian sbmanifold (8) is the set λ(im(df ) V ). If the fibration ρ is the identity morphism 1 X : X X, then the family of fnctions is a fnction the Lagrangian sbmanifold is the set F : X (11) = {p T } X; π X (p) = q, p, v = df, v v TqX = im(df ). (12) We have to consider even more general generating object. It is hen e generate a sbmanifold of T Q the manifold X of the family is not the hole Q, bt an immersed sbmanifold of Q only. T Q F π Q ρ (13) Q ι X X 2.2. edction of generating objects. Let F ρ X (14) be a family of fnctions generating a Lagrangian sbmanifold of (T X, dϑ x ). Proposition 1. If is generated by a family (14) sch that the fibration ρ is the composition ρ ρ of fibrations ρ : ρ: X the critical set S(F, ρ ) is the image of a section σ: of ρ, then is generated by the family F ρ X (15) ith F = F σ. 3

4 2.3. Lagrangian system. Let be the configration manifold of a mechanical system. A generating object TT Y L Tπ ι C η C (16) is called a Lagrangian system. The family of fnctions Y L η C (17) is called a Lagrangian family Hamiltonian system. Let (P, ω) be a symplectic manifold representing the phase space of a mechanical system. A generating object TP Z H τ P P ι K ζ K (18) is called a Hamiltonian system. The family of fnctions Z H ζ K (19) is called a Hamiltonian family. A Hamiltonian system T P π P P ι K K H (20) is called a Dirac system. A more general Hamiltonian system (18) is called a generalized Dirac system. 3. Doble vector bndles. Let K be a system (K r, K l, E, F) of vector bndles, here K r = (K, τ r, E), K l = (K, τ l, F ), E = (E, τ l, ) F = (F, τ r, ), sch that the diagram 4

5 y y K τ l τ r P F E τ r τ P l is commtative. By m r, m l, m r, m l e denote the operations of addition in K r, K l, E F respectively. A system K is called doble vector bndle if the diagram (21) is commtative the folloing conditions are satisfied: (1) pairs (τ l, τ l ), (τ r, τ r ) are vector bndle morphisms, (2) pairs of additions (m l, m l ) (m r, m r ) are vector bndle morphisms K r F K r K r K l E K l K l respectively, (3) zero sections 0 r : E K l, 0 l : F K r are vector bndle morphisms. Proposition 1. The vector bndle strctres of K r K l coincide on the intersection C of ker τ l ker τ r. Ths,C = (C, τ, ), here τ = τ l τ r = τ r τ l is a vector bndle is called the core of K. We have the diagram K τ l τ r P F C E τ r P τ τ l hich ill be sed instead of the diagram (21) to represent the doble vector bndle. Proposition 2. (1) ker τ r ith the vector bndle strctre indced from K r is canonically isomorphic to the Whitney sm F C. (2) ker τ r ith the vector bndle strctre indced from K l is canonically isomorphic to the vector bndle F C, i.e., to the pll-back of C by the projection τ r. (3) ker τ l ith the vector bndle strctre indced from K l is canonically isomorphic to the Whitney sm E C. (4) ker τ l ith the vector bndle strctre indced from K r is canonically isomorphic to the vector bndle E C, i.e., to the pll-back of C by the projection τ l Examples. Let (E, τ, ) be a vector bndle. The tangent manifold is in a canonical ay a doble vector bndle ith the diagram (21) (22) Tτ TE E τ P τ τ E P τ E. (23) 5

6 y Also its dal ith respect to the canonical projection, i.e the cotangent bndle T E is a doble vector bndle ith the diagram T E π T E τ P E 444 T E h τ 46 π πh h hk, (24) here T τ is dal to the vertical lift E E. This is an example of a general sitation Dality. Let K r be the vector bndle, dal to K r. We denote by K r the total fiber bndle space by π l the projection π l : K r E. Let a K r k C satisfy τ(k) = τ l (π l (a)). We can evalate a on a vector (π l (a), k) of ker τ l. We define a mapping π r : K r C by the formla k, π r (a) = (π l (a), k), a. (25) It follos directly from this constrction that Proposition 3. The mapping π r : K r C is a morphism of vector bndles We define a relation in the folloing ay: c P r (m l )(a, b) if π r : K r C. P r (m l ): K r K r K r (1) π l (c) = π l (a) + π l (b), (2), c = v, a + v, b for each, v, v K sch that τ r () = π l (c), τ r (v) = π l (a), τ r (v ) = π l (b) = m l (v, v ). Local coordinates. Let (x i, e a, f A, c α ) be an adapted coordinate system on K let (x i, e a, p A, q α ) be the adapted coordinate system on the dal bndle K r, i. e., the canonical evalation is given by the formla v, a = A p A (a)f A (v) + α q α (a)c α (v). (26) We se (x i, c α ) as a coordinate system in C (x i, q α ) as a coordinate system on C. In these coordinate sytems e have x i π r (a) = x i (a), q α π r (a) = q α (a) (27) x i (a + r b) = x i (a) = x i (b), e a (a + r b) = e a (a) + e a (b), p A (a + r b) = p A (a) + p A (b), q α (a + r b) = q α (a) = q α (b). (28) It follos that (K r, π r, C ) is a vector bndle. We denote it by K r r. The vector bndle (K r, π l, E) e denote by K r l. 6

7 y y y y y Theorem 4. The system K r diagram = (K r r, K r l, C, E) is a doble vector bndle ith the K r π π l r P E AA F C π A r AC π π r. (29) In the folloing, e sho that there is a canonical isomorphism of doble vector bndles K ((K r ) r ) r. Throgh this section e denote by τ r, π r, ξ r, ϑ r by τ l, π l, ξ l, ϑ l the right left projections in K, K r, (K r ) r, ((K r ) r ) r respectively. Identifying vector bndles ith their second dals, e have ξ r : (K r ) r F ξ l : (K r ) r C ϑ r : ((K r ) r ) r E ϑ l : ((K r ) r ) r F. The core of (K r ) r is E the core of ((K r ) r ) r is (C ) = C. We define a relation K K ((K r ) r ) r in the folloing ay: Let v K, φ ((K r ) r ) r be sch that τ l (τ r (v)) = ϑ l (ϑ r (φ)). We say that (v, φ) K if for each a K r, α (K r ) r sch that τ r (v) = π l (a), π r (a) = ξ l (α), ξ r (α) = ϑ l (φ) e have a, α = v, a + α, φ. (30) Theorem 5. The relation K is an isomorphism of doble vector bndles. Example. 2. Let K = T E: T τ T E E 4 T 44 π 46 π π E P τ h hk h E h. (31) Then the first, second third right dals can be identified ith doble vector bndles TE, TE T E, represented by the diagrams TE τ E Tτ P E E τ P τ τ Tπ AA τ TE A AC π E τ E P π E T E π T E π P E 4 T E 44 h π 46 π hτ h hk. (32) 7

8 In this case the formla (30) means that the relation K, interpreted as a sbmanifold of T (E E, is a lagrangian sbmanifold generated by the canonical pairing vieed as a fnction E E. If e replace the right-h side of (30) by a different combination of v, a α, φ, e obtain another isomorphism. The isomorphism corresponding to v, a α, φ e denote by ± K, the isomorphism corresponding to v, a + α, φ e denote by K the isomorphism corresponding to v, a α, φ e denote by = K. Ths the canonical isomorphisms K, ± K, K, = K define diffeomorphisms from T E to T E. For K, = K these diffeomorphisms are antisymplectomorphisms ith respect to the canonical symplectic strctre of the cotangent bndle for ± K, K e obtain symplectomorphisms. 4. Iterated tangent fnctors. The Lagrange formlation of the dynamics is based on the canonical diffeomorphism α : TT T. It can be obtained by by the composition of β : TT T T the canonical diffeomorphism (see previos sections) γ : T T T. It is knon that the canonical symplectic strctre on a tangent bndle is linear, i.e. the corresponding map β is a morphism of doble vector bndles. It follos that γ β is also a morphism of doble vector bndles. Hoever, to avoid discssion concerning proper choice of sign in (30), it is better to have α defined independently. Elements of the iterated bndle T are eqivalence classes of crves in a set of eqivalence classes of crves in. A simpler representation of these elements is needed. Let χ: 2 be a differentiable mapping. For each s e denote by t (0,1) χ(s, 0) the vector tχ(s, )(0). For each t e denote by t (1,0) χ(0, t) the vector tχ(, t)(0). We have crves t (0,1) χ(, 0): (33) t (1,0) χ(0, ):. (34) Vectors tt (0,1) χ(, 0)(0) T tt (1,0) χ(0, )(0) T ill be denoted by tt (0,1) χ(0, 0) tt (1,0) χ(0, 0) respectively. For each T there is a mapping χ: 2 sch that = tt (0,1) χ(0, 0). The mapping specified by coordinate relations (q κ χ)(s, t) = (q κ () + q κ ()t + q κ ()s + q κ ()st) (35) has the reqired property. We consider mappings χ: 2 χ : 2 eqivalent if These mappings are eqivalent if tt (0,1) χ (0, 0) = tt (0,1) χ(0, 0). (36) χ (0, 0) = χ(0, 0), (37) D (1,0) χ (0, 0) = D (1,0) χ(0, 0), (38) D (0,1) χ (0, 0) = D (0,1) χ(0, 0), (39) D (1,1) χ (0, 0) = D (1,1) χ(0, 0). (40) We have obtained an efficient representation of elements of T. In terms of this representation e define the canonical involtion κ : T T : tt (0,1) χ(0, 0) tt (1,0) χ(0, 0) = tt (0,1) χ(0, 0), (41) 8

9 ith χ: 2 : (s, t) χ(t, s). (42) The coordinate expression of this involtion is given by (q κ, q λ, q µ, q ν ) κ = (q κ, q λ, q µ, q ν ). (43) The commtative diagram T κ T Tτ τ (44) is a vector fibration isomorphism. The diagram τ T κ Tτ T (45) is the inverse isomorphism. It follos that κ is a morphism of doble vector bndles. T Tτ Q τ ' '') τ ' '* ' ' id τ κ id id τ Q τ T ' '') Tτ ' τ ' ' '*. (46) For a differentiable mapping α: e have TTα(tt {(0,1)} χ(0, 0)) = tt {(0,1)} (α χ)(0, 0) (47) i.e. κ is a natral eqivalence of fnctors. The morphism α is defined as the dal to (45). κ TTα = TTα κ, (48) Tπ Q TT τ ' '') τ T T id π α id id π Q τ T ' '') T τ T π. (49) 9

10 5. The Legendre transformation. Let be the configration manifold of a mechanical system. The phase space of the system is the symplectic manifold (T, dϑ ). The identity morphism 1 TT is a symplectic relation generated by the generating object TT TT τ T Tπ (50) T ι T T, by the generating object TT TT Tπ τ T (51) T ι T T, here, is the canonical pairing, : T : (v, p) p, v. (52) Starting ith a Lagrangian system TT Y L Tπ η (53) ι C C e constrct a Hamiltonian system TT Z H τ T ζ (54) T ι K K ith K = π 1 (τ (C)) = { p T ; v C π (p) = τ (v) }, (55) Z = {(p, v, y) T Y ; v C, η(y) = v}, (56) ζ: Z K : (p, v, y) p, (57) H: Z : (p, v, y) p, v L(y). (58) 10

11 This Hamiltonian system is obtained by composing the Lagrangian system ith the generating object (50). The to systems generate the same dynamics since the generating object (50) generates the identity relation. The passage from the Lagrangian system (53) to the Hamiltonian system (54) is called the Legendre transformation. Conversely, given a Hamiltonian system TT Z H τ T T e constrct a Lagrangian system ι K ζ K (59) (TT, d T ϑ ) Y L ith Tπ ι C η C (60) C = τ 1 (π (K)) { = v T } ; p K π (p) = τ (v), (61) Y = {(p, v, z) T Z; p K, ζ(z) = p}, (62) L: Y η: Y C : (p, v, z) v, (63) : (p, v, z) p, v H(z) (64) by composing the Hamiltonian system ith the generating object (51). This passage is called the inverse Legendre transformation. In addition to the Legendre transformations e have Legendre relations. Given a Lagrangian system (53) e have the first Legendre relation Λ 1 (L): Y T, (65) hose graph is the set { graph(λ 1 (L)) = (p, y) T Y ; y S(L, η), π (p) = q = τ (η(y)) } C Tq Tη(z) = χ z TyY (η(y), ) p, = dl, z the second Legendre relation (66) Λ 2 (L): T, (67) hose graph is the set { graph(λ 2 (L)) = (p, v) T } ; y Y η(y) = v, (p, y) graph(λ 1 (L)). (68) If D TT is the dynamics generated by the Lagrangian system, then graph(λ 2 (L)) = (τ T, Tπ )(D). (69) A Lagrangian system is said to be hyperreglar if the second Legendre relation Λ 2 (L) is a diffeomorphism. 11

12 6. Lie algebroids. A Lie algebroid strctre on a vector bndle τ: E is sally defined in terms of a Lie bracket on sections of τ. Hoever, it can be characterized ( defined) in terms of associated strctres (1) linear Poisson strctre on the dal bndle E, (2) a morphism ε: T E TE of doble vector bndle, (3) a relation κ: TE TE, dal to ε, (4) complete lift of mltisections of E to mltivector fields on E. For the Lie algebroid of a tangent bndle, the linear Poisson strctre is given by the canonical symplectic strctre on T, ε = α 1, κ = κ. The complete lift of a Lie algebroid is the ell knon complete lift d T of mltivector fields. Unlike for general Lie algebroid, it can be defined also on differential forms. In particlar, d T ω is a symplectic form on TT. With this form α ezb become symplectomorphisms. We see, that it is Lie algebroid strctre of the tangent bndle, hich makes possible Lagrangian formlation of the dynamics conseqently, the Legendre transformation. 7. Homogeneos systems. Here, e consider an important for physics class of singlar Lagrangian systems, hich appear in the calcls of variations for non-parameterized crves (space-time trajectories) Homogeneos Lagrangian systems. Let be a differential manifold let κ be the action κ: + : (k, v) kv. (70) Let = {v ; v 0} (71) be the tangent bndle of ith the image of the zero section removed. The set is an homogeneos open sbmanifold of. A Lagrangian system (TT, d T ϑ ) Y L Tπ η is said to be homogeneos ith respect to an action ι C C (72) + Y ρ Y 1 + η + C κ η C (73) if C is an homogeneos sbmanifold of, κ: + C C : (k, v) kv. (74) is the restriction of the action κ to C, L(ρ(k, y)) = kl(y) (75) 12

13 for each y Y each k +. The Lagrangian family Y L is said to be a homogeneos Lagrangian family. The action κ is lifted to the action η C (76) + TT κ TT 1 + Tπ Tπ (77) + κ ith the action defined by for each k +. The relation κ: + TT TT (78) κ(k, ) = α 1 κ(k, ) α (79) κ(k, ) = k (80) holds for each TT each k +. The dynamics D generated by a homogeneos Lagrangian system is homogeneos Homogeneos Hamiltonian systems. Let (P, ω) be a symplectic manifold representing the phase space of a mechanical system let (TP, i T ω) Z H be a Hamiltonian system. Let τ P P ι K ζ (81) K σ: + Z Z (82) be an action of the grop ( +, ) sch that ζ σ(k, ) = ζ for each k +. The Hamiltonian system (81) is said to be homogeneos ith respect to the grop action σ if for each z Z each k +. The Hamiltonian family H(σ(k, z)) = kh(z) (83) Z H is said to be a homogeneos Hamiltonian family. The same concepts of homogeneity apply to a Hamiltonian system ζ K (84) 13

14 (TT, i T dϑ ) Z H τ T ζ (85) T ι K K based on a phase space (T, dϑ ). The dynamics D generated by a homogeneos Hamiltonian system is homogeneos. 8. elativistic mechanics. Let (Q, V, σ, g) be the affine space-time of special relativity. The triple (Q, V, σ) is an affine space g: V V is the inkoski metric elativistic particle. The dynamics of a free particle of mass m is generated by the Lagrangian system (TT Q, d T ϑ Q ) Tπ Q (86) TQ ι C C L here C = Q { q V ; g( q), q > 0} (87) The dynamics is the set D = { L: C : (q, q) m g( q), q. (88) (q, p, q, ṗ) TT Q; g( q), q > 0, p = } mg( q), ṗ = 0. (89) g( q), q The set C is a homogeneos sbmanifold of TQ L(q, k q) = kl(q, q). It follos that the Lagrangian system (86) is homogeneos. The Legendre transformation applied to the Lagrangian system (86) leads to a Hamiltonian system (TT Q, i T dϑ Q ) Z H τ T Q T Q ζ T Q (90) ith Z = T Q {v V ; g(v), v > 0}, (91) ζ: Z T Q : (q, p, v) (q, p), (92) 14

15 H: Z : (q, p, v) p, v m g(v), v. (93) The Hamiltonian system (90) can be simplified. The fibration ζ can be represented as a composition ζ ζ of fibrations ζ : Z Z : (q, p, v) (q, p, v ), (94) ζ : Z T Q : (q, p, λ) (q, p), (95) here Z = T Q + (96) v = g(v), v. (97) Eqating to zero the derivative of H along the fibres of ζ e obtain the relation p = µg(v) (98) valid for some vales of the Lagrange mltiplier µ. It follos from this relation that the covector p is in the set { p V ; p, g 1 (p) > 0 } (99) that {(q, p, v) Z; v p = ± p g(v)} (100) is the critical set S(H, ζ ). We have denoted by p the norm p, g 1 (p) of p. The images K = ζ(s(h, ζ )) Z = ζ (S(H, ζ )) of the critical set S(H, ζ ) are the sets K = Q { p V ; p, g 1 (p) > 0 } (101) Z = K +. (102) The mapping ζ: Z K : (q, p, λ) (q, p). (103) is a differential fibration. The critical set S(H, ζ ) is the nion of images of to local sections ξ + : Z Z : (q, p, λ) ( ) λ q, p, p g 1 (p) (104) ξ : Z Z : (q, p, λ) ( q, p, λ ) p g 1 (p) (105) 15

16 of the fibration ζ. We have to families of fnctions Z H Z H + ζ K ζ K (106) The fnctions H : Z : (q, p, λ) λ ( p + m) (107) H + : Z : (q, p, λ) λ ( p m) (108) are the compositions H ξ H ξ + respectively. It is easily seen that the family (106) generates an empty set. The dynamics of the particle is generated by the redced Hamiltonian system (TT Q, i T dϑ Q ) Z H + τ T Q ζ (109) T Q ι K K The critical set S( H +, ζ) is the sbmanifold { (q, p, λ) Z; } p = m (110) the image ζ(s( H +, ζ)) is the mass shell The mapping K m = { (q, p) K; } p = m. (111) is a differential fibration ith connected fibres. ζ: S( H +, ζ) K m : (q, p, λ) (q, p) (112) 8.2. Space-time formlation of geometric optics. The dynamics of light rays in affine inkoski space-time (Q, V, σ, g) is generated by the Lagrangian system (TT Q, d T ϑ Q ) Y L Tπ Q η (113) TQ ι C C ith C = TQ, (114) 16

17 Y = C +, (115) η: Y C : (q, q, µ) (q, q), (116) L: Y : (q, q, µ) 1 g( q), q. (117) 2µ The dynamics is the set D = {(q, p, q, ṗ) TT Q; g( q), q = 0, µ + p = 1µ } g( q), ṗ = 0. (118) The Lagrangian system (113) is homogeneos. The reslt of the Legendre transformation applied to the Lagrangian system (113) is the Hamiltonian system (TT Q, i T dϑ Q ) Z H ith τ T Q ζ (119) T Q T Q Z = T Q V +, (120) H: Z ζ: Z T Q : (q, p, v, µ) (q, p), (121) : (q, p, v, µ) p, v 1 g(v), v. (122) 2µ The Hamiltonian system (119) can be simplified as in the case of a relativistic particle ith mass m > 0. The redced system is the Hamiltonian system (TT Q, i T dϑ Q ) Z H ith τ T Q ζ (123) T Q ι K K K = { (q, p) T Q; p 0 }, (124) ζ: Z K : (q, p, µ) (q, p), (125) H: Z o frther simplification is possible. : (q, p, µ) µ 2 p, g 1 (p). (126) 17

18 9. Dirac systems. In [1] Dirac proposed the folloing procedre of deriving Hamiltonian formlation of a system ith singlar Lagrangian. Let L: be the Lagrangian of a system. Sppose that the Legendre mapping d v L: C T is a fibration over a sbmanifold C ith connected fibres. It is easy task to verify that p, v L(v) is constant on fibres. Conseqently, e have a fnction H on C a Dirac system TT τ T T ι C K m H (127) The dynamics D of this system incldes the dynamics D of the original Lagrangian system. It may happen that they are not eqal. An example is given by the relativistic particle of previos section. The corresponding Dirac system is TT, τ T T ι Km K m 0 D D. 10. Legendre transformation convex analysis. ot yet ready. 11. eferences. [1] P. A.. Dirac, Generalized Hamiltonian Dynamics, Canad. J. ath. 2 (1950), [2] P. Libermann,.-C. arle, Symplectic Geometry Analytical echanics, D. eidel Pblishing Company, Dordrecht, [3] W.. Tlczyje, Hamiltonian Systems, Lagrangian Systems the Legendre Transformation, Symposia athematica 16 (1974), [4] W.. Tlczyje, The Legendre transformation, Ann. Inst. Henri Poincaré, 27 (1977), [5] G. Pidello W.. Tlczyje, Derivations of differential forms on jet bndles, Ann. at. Pra Appl., 147 (1987). [6] W.. Tlczyje, P. Urbanski, Homogeneos Lagrangian systems, in Gravitation, Electromagnetism Geometric Strctres, Pitagora Editrice, [7] W.. Tlczyje, P. Urbanski, A slo carefl Legendre transformation for singlar Lagrangians, Acta Phys. Pol. B 30 (1999). 18