aa r X i v : . [ m a t h - ph ] M a y Relativistic mechanics in a general setting
G. Sardanashvily
Department of Theoretical Physics, Moscow State University, Moscow, Russia
Abstract . Relativistic mechanics on an arbitrary manifold is formulated in the terms of jets ofits one-dimensional submanifolds. A generic relativistic Lagrangian is constructed. Relativisticmechanics on a pseudo-Riemannian manifold is particularly considered.
Classical non-relativistic mechanics is adequately formulated as Lagrangian and Hamil-tonian theory on a fibre bundle Q → R over the time axis R [1, 3, 6, 8, 9, 11].If a configuration space Q of a mechanical system has no preferable fibration Q → R ,we obtain a general formulation of relativistic mechanics, including Special Relativity onthe Minkowski space Q = R [3, 9, 11]. A velocity space of relativistic mechanics is thefirst order jet manifold J Q of one-dimensional submanifolds of the configuration space Q . The notion of jets of submanifolds [2, 4, 5, 7] generalizes that of jets of sections offibre bundles which are utilized in field theory and non-relativistic mechanics (Section 2).The jet bundle J Q → Q is projective, and one can think of its fibres as being spaces ofthe three-velocities of a relativistic system (Section 3). The four-velocities of a relativisticsystem are represented by elements of the tangent bundle T Q of the configuration space Q , while the cotangent bundle T ∗ Q , endowed with the canonical symplectic form, playsa role of the phase space of relativistic theory (Section 6).We develop Lagrangian formalism on the jet bundle J Q → Q (Section 4). We showthat, in the framework of this formalism, Lagrangians possess a certain gauge symmetry(27) and, consequently, the corresponding Lagrange operators obey the rather restrictiveNoether identity (28). Solving this Noether identity, we obtain the generic Lagrangian (31)and the equation of motion (51) of relativistic mechanics on a manifold Q . In particular,if Q is the Minkowski space, we are in the case of Special Relativity (Example 4).Generalizing this example, we consider relativistic mechanics on an arbitrary pseudo-Riemannian manifold. Its equation of motion is the relativistic geodesic equation (55).Hamiltonian relativistic mechanics on a pseudo-Riemannian manifold is developed in Sec-tion 6. Its generic Hamiltonian takes the form (69).1 Jets of submanifolds
Jets of sections of fibre bundles are particular jets of submanifolds of a manifold [2, 4, 5, 7].Given an m -dimensional smooth real manifold Z , a k -order jet of n -dimensional sub-manifolds of Z at a point z ∈ Z is defined as an equivalence class j kz S of n -dimensionalimbedded submanifolds of Z through z which are tangent to each other at z with order k ≥
0. Namely, two submanifolds i S : S → Z, i S ′ : S ′ → Z through a point z ∈ Z belong to the same equivalence class j kz S if and only if the imagesof the k -tangent morphisms T k i S : T k S → T k Z, T k i S ′ : T k S ′ → T k Z coincide with each other. The set J kn Z = [ z ∈ Z j kz S of k -order jets of submanifolds is a finite-dimensional real smooth manifold, called the k -order jet manifold of submanifolds. For the sake of convenience, we put J n Z = Z .If k >
0, let Y → X be an m -dimensional fibre bundle over an n -dimensional base X and J k Y the k -order jet manifold of sections of Y → X . Given an imbedding Φ : Y → Z ,there is the natural injection J k Φ : J k Y → J kn Z, j kx s → [Φ ◦ s ] k Φ( s ( x )) , (1)where s are sections of Y → X . This injection defines a chart on J kn Z . These chartsprovide a manifold atlas of J kn Z .Let us restrict our consideration to first order jets of submanifolds. There is obviousone-to-one correspondence λ (1) : j z S → V j z S ⊂ T z Z (2)between the jets j z S at a point z ∈ Z and the n -dimensional vector subspaces of thetangent space T z Z of Z at z . It follows that J n Z is a fibre bundle ρ : J n Z → Z (3)with the structure group GL ( n, m − n ; R ) of linear transformations of the vector space R m which preserve its subspace R n . The typical fibre of the fibre bundle (3) is the Grassmannmanifold G ( n, m − n ; R ) = GL ( m ; R ) /GL ( n, m − n ; R ) . { ( U ; z A ) } be a coordinate atlas of Z . Though J n Z = Z , let us provide J n Z withan atlas where every chart ( U ; z A ) on a domain U ⊂ Z is replaced with the mn ! = m ! n !( m − n )!charts on the same domain U which correspond to different partitions of the collection( z · · · z A ) in the collections of n and m − n coordinates( U ; x λ , y i ) , λ = 1 , . . . , n, i = 1 , . . . , m − n. (4)The transition functions between the coordinate charts (4) of J n Z associated with acoordinate chart ( U, z A ) of Z are reduced to exchange between coordinates x λ and y i .Transition functions between arbitrary coordinate charts of the manifold J n Z take theform x ′ λ = x ′ λ ( x µ , y k ) , y ′ i = y ′ i ( x µ , y k ) . (5)Given the coordinate atlas (4) – (5) of a manifold J n Z , the first order jet manifold J n Z is endowed with an atlas of adapted coordinates( ρ − ( U ) = U × R ( m − n ) n ; x λ , y i , y iλ ) , (6)possessing transition functions y ′ iλ = ∂y ′ i ∂y j y jα + ∂y ′ i ∂x α ! ∂x α ∂y ′ k y ′ kλ + ∂x α ∂x ′ λ ! . (7) As was mentioned above, a velocity space of relativistic mechanics is the first order jetmanifold J Q of one-dimensional submanifolds of a configuration space Q [3, 9, 11].Given an m -dimensional manifold Q coordinated by ( q λ ), let us consider the jet ma-nifold J Q of its one-dimensional submanifolds. Let us provide Q = J Q with the coor-dinates (4):( U ; x = q , y i = q i ) = ( U ; q λ ) . (8)Then the jet manifold ρ : J Q → Q is endowed with coordinates (6):( ρ − ( U ); q , q i , q i ) (9)3ossessing transition functions (5), (7) which read q ′ = q ′ ( q , q k ) , q ′ = q ′ ( q , q k ) , (10) q ′ i = ( ∂q ′ i ∂q j q j + ∂q ′ i ∂q )( ∂q ′ ∂q j q j + ∂q ′ ∂q ) − . (11)A glance at the transformation law (11) shows that J Q → Q is a fibre bundle in projectivespaces. Example 1.
Let Q = M = R be a Minkowski space whose Cartesian coordinates( q λ ), λ = 0 , , , , are subject to the Lorentz transformations (10): q ′ = q ch α − q sh α, q ′ = − q sh α + q ch α, q ′ , = q , . (12)Then q ′ i (11) are exactly the Lorentz transformations q ′ = q ch α − sh α − q sh α + ch α q ′ , = q , − q sh α + ch α of three-velocities in relativistic mechanics [9, 11].In view of Example 1, one can think of the velocity space J Q of relativistic mechanicsas being a space of three-velocities. For the sake of convenience, we agree to call J Q the three-velocity space and its coordinate transformations (10) – (11) the relativistictransformations, though a dimension of Q need not equal 3 + 1. Given the coordinate chart (9) of J Q , one can regard ρ − ( U ) ⊂ J Q as the first orderjet manifold J U of sections of the fibre bundle π : U ∋ ( q , q i ) → ( q ) ∈ π ( U ) ⊂ R . (13)Then three-velocities ( q i ) ∈ ρ − ( U ) of a relativistic system on U can be treated as absolutevelocities of a local non-relativistic system on the configuration space U (13). However,this treatment is broken under the relativistic transformations q i → q ′ i (10) since theyare not affine. One can develop first order Lagrangian formalism with a Lagrangian L = L dq ∈ O , ( ρ − ( U ))on a coordinate chart ρ − ( U ), but this Lagrangian fails to be globally defined on J Q (seeRemark 3 below). The graded differential algebra O ∗ ( ρ − ( U )) of exterior forms on ρ − ( U )is generated by horizontal forms dq and contact forms dq i − q i dq . Coordinate transfor-mations (10) preserve the ideal of contact forms, but horizontal forms are not transformedinto horizontal forms, unless coordinate transition functions q (10) are independent ofcoordinates q ′ i . 4n order to overcome this difficulty, let us consider a trivial fibre bundle Q R = R × Q → R , (14)whose base R is endowed with a Cartesian coordinate τ [5]. This fibre bundle is providedwith an atlas of coordinate charts( R × U ; τ, q λ ) , (15)where ( U ; q , q i ) are the coordinate charts (8) of the manifold J Q . The coordinate charts(15) possess transition functions (10). Let J Q R be the first order jet manifold of thefibre bundle (14). Since the trivialization (14) is fixed, there is the canonical isomorphismof J Q R to the vertical tangent bundle J Q R = V Q R = R × T Q (16)of Q R → R [5, 6].Given the coordinate atlas (15) of Q R , the jet manifold J Q R is endowed with thecoordinate charts(( π ) − ( R × U ) = R × U × R m ; τ, q λ , q λτ ) , (17)possessing transition functions q ′ λτ = ∂q ′ λ ∂q µ q µτ . (18)Relative to the coordinates (17), the isomorphism (16) takes the form( τ, q µ , q µτ ) → ( τ, q µ , ˙ q µ = q µτ ) . (19) Example 2.
Let Q = M be a Minkowski space in Example 1 whose Cartesian coor-dinates ( q , q i ) are subject to the Lorentz transformations (12). Then the correspondingtransformations (18) take the form q ′ τ = q τ ch α − q τ sh α, q ′ τ = − q τ sh α + q τ ch α, q ′ , τ = q , τ of transformations of four-velocities in relativistic mechanics.In view of Example 2, we agree to call fibre elements of J Q R → Q R the four-velocitiesthough the dimension of Q need not equal 4. Due to the canonical isomorphism q λτ → ˙ q λ (16), by four-velocities also are meant the elements of the tangent bundle T Q , which iscalled the space of four-velocities.Obviously, the non-zero jet (19) of sections of the fibre bundle (14) defines some jetof one-dimensional subbundles of the manifold { τ } × Q through a point ( q , q i ) ∈ Q , butthis is not one-to-one correspondence. 5ince non-zero elements of J Q R characterize jets of one-dimensional submanifolds of Q , one hopes to describe the dynamics of one-dimensional submanifolds of a manifold Q as that of sections of the fibre bundle (14). For this purpose, let us refine the relationbetween elements of the jet manifolds J Q and J Q R .Let us consider the manifold product R × J Q . It is a fibre bundle over Q R . Given acoordinate atlas (15) of Q R , this product is endowed with the coordinate charts( U R × ρ − ( U ) = U R × U × R m − ; τ, q , q i , q i ) , (20)possessing transition functions (10) – (11). Let us assign to an element ( τ, q , q i , q i ) ofthe chart (20) the elements ( τ, q , q i , q τ , q iτ ) of the chart (17) whose coordinates obey therelations q i q τ = q iτ . (21)These elements make up a one-dimensional vector space. The relations (21) are main-tained under coordinate transformations (11) and (18) [4, 5]. Thus, one can associate:( τ, q , q i , q i ) → { ( τ, q , q i , q τ , q iτ ) | q i q τ = q iτ } , (22)to each element of the manifold R × J Q a one-dimensional vector space in the jet manifold J Q R . This is a subspace of elements q τ ( ∂ + q i ∂ i )of a fibre of the vertical tangent bundle (16) at a point ( τ, q , q i ). Conversely, given anon-zero element (19) of J Q R , there is a coordinate chart (17) such that this elementdefines a unique element of R × J Q by the relations q i = q iτ q τ . (23)Thus, we have shown the following. Let ( τ, q λ ) further be arbitrary coordinates on theproduct Q R (14) and ( τ, q λ , q λτ ) the corresponding coordinates on the jet manifold J Q R . Theorem 1 . (i) Any jet of submanifolds through a point q ∈ Q defines some (butnot unique) jet of sections of the fibre bundle Q R (14) through a point τ × q for any τ ∈ R in accordance with the relations (21).(ii) Any non-zero element of J Q R defines a unique element of the jet manifold J Q by means of the relations (23). However, non-zero elements of J Q R can correspond todifferent jets of submanifolds.(iii) Two elements ( τ, q λ , q λτ ) and ( τ, q λ , q ′ λτ ) of J Q R correspond to the same jet ofsubmanifolds if q ′ λτ = rq λτ , r ∈ R \ { } . ✷ In the case of a Minkowski space Q = M in Examples 1 and 2, the equalities (21)and (23) are the familiar relations between three- and four-velocities.6ased on Theorem 1, we can develop Lagrangian theory of one-dimensional submani-folds of a manifold Q as that of sections of the fibre bundle Q R (14). Let L = L ( τ, q λ , q λτ ) dτ, (24)be a first order Lagrangian on the jet manifold J Q R . The corresponding Lagrangeoperator reads δL = E λ dq λ ∧ dτ, E λ = ∂ λ L − d τ ∂ τλ L . (25)It yields the Lagrange equation E λ = ∂ λ L − d τ ∂ τλ L = 0 . (26)In accordance with Theorem 1, it seems reasonable to require that, in order to describejets of one-dimensional submanifolds of Q , the Lagrangian L (24) on J Q R possesses agauge symmetry given by vector fields u = χ ( τ ) ∂ τ on Q R or, equivalently, their verticalpart u V = − χq λτ ∂ λ , (27)which are generalized vector fields on Q R [5, 6]. Then the variational derivatives of thisLagrangian obey the Noether identity: q λτ E λ = 0 . (28)We call such a Lagrangian the relativistic Lagrangian.In order to obtain a generic form of a relativistic Lagrangian L , let us regard theNoether identity (28) as an equation for L . It admits the following solution. Let12 N ! G α ...α N ( q ν ) dq α ∨ · · · ∨ dq α N be a symmetric tensor field on Q such that the function G = G α ...α N ( q ν ) ˙ q α · · · ˙ q α N (29)is positive: G > , (30)everywhere on T Q \ b Q ). Let A = A µ ( q ν ) dq µ be a one-form on Q . Given the pull-backof G and A onto J Q R due to the canonical isomorphism (16), we define a Lagrangian L = ( G / N + q µτ A µ ) dτ, G = G α ...α N q α τ · · · q α N τ , (31)7n J Q R \ ( R × b Q )) where b T Q → Q . The correspondingLagrange equation reads E λ = ∂ λ G N G − / N − d τ ∂ τλ G N G − / N ! + F λµ q µτ = (32) E β [ δ βλ − q βτ G λν ...ν N q ν τ · · · q ν N τ G − ] G / N − = 0 ,E β = ∂ β G µα ...α N N − ∂ µ G βα ...α N ! q µτ q α τ · · · q α N τ − (33)(2 N − G βµα ...α N q µττ q α τ · · · q α N τ + G − / N F βµ q µτ ,F λµ = ∂ λ A µ − ∂ µ A λ . It is readily observed that the variational derivatives E λ (32) satisfy the Noether identity(28). Moreover, any relativistic Lagrangian obeying the Noether identity (28) is of type(31).A glance at the Lagrange equation (32) shows that it holds if E β = Φ G βν ...ν N q ν τ · · · q ν N τ G − , (34)where Φ is some function on J Q R . In particular, we consider the equation E β = 0 . (35)Because of the Noether identity (28), the system of equations (32) is underdetermined.To overcome this difficulty, one can complete it with some additional equation. Given thefunction G (31), let us choose the condition G = 1 . (36)Owing to the property (30), the function G (31) possesses a nowhere vanishing differential.Therefore, its level surface W G defined by the condition (36) is a submanifold of J Q R .Our choice of the equation (35) and the condition (36) is motivated by the followingfacts. Lemma 2 . Any solution of the Lagrange equation (32) living in the submanifold W G is a solution of the equation (35). ✷ Proof.
A solution of the Lagrange equation (32) living in the submanifold W G obeys thesystem of equations E λ = 0 , G = 1 . (37)Therefore, it satisfies the equality d τ G = 0 . (38)8hen a glance at the expression (32) shows that the equations (37) are equivalent to theequations E λ = ∂ λ G µα ...α N N − ∂ µ G λα ...α N ! q µτ q α τ · · · q α N τ − (2 N − G βµα ...α N q µττ q α τ · · · q α N τ + F βµ q µτ = 0 , (39) G = G α ...α N q α τ · · · q α N τ = 1 . QED
Lemma 3 . Solutions of the equation (35) do not leave the submanifold W G (36). ✷ Proof.
Since d τ G = − N N − q βτ E β , any solution of the equation (35) intersecting the submanifold W G (36) obeys the equality(38) and, consequently, lives in W G . QED
The system of equations (39) is called the relativistic equation. Its components E λ (33) are not independent, but obeys the relation q βτ E β = − N − N d τ G = 0 , G = 1 , similar to the Noether identity (28). The condition (36) is called the relativistic constraint.Though the equation (32) for sections of a fibre bundle Q R → R is underdetermined,it is determined if, given a coordinate chart ( U ; q , q i ) (8) of Q and the correspondingcoordinate chart (15) of Q R , we rewrite it in the terms of three-velocities q i (23) as anequation for sections of a fibre bundle U → π ( U ) (13).Let us denote G ( q λ , q i ) = ( q τ ) − N G ( q λ , q λτ ) , q τ = 0 . (40)Then we have E i = q τ " ∂ i G N G − / N − ( q τ ) − d τ ∂ i G N G − / N ! + F ij q j + F i . Let us consider a solution { s λ ( τ ) } of the equation (32) such that ∂ τ s does not vanish andthere exists an inverse function τ ( q ). Then this solution can be represented by sections s i ( τ ) = ( s i ◦ s )( τ ) (41)of the composite bundle R × U → R × π ( U ) → R s i ( q ) = s i ( τ ( q )) are sections of U → π ( U ) and s ( τ ) are sections of R × π ( U ) → R .Restricted to such solutions, the equation (32) is equivalent to the equation E i = ∂ i G N G − / N − d ∂ i G N G − / N ! + (42) F ij q j + F i = 0 , E = − q i E i . for sections s i ( q ) of a fibre bundle U → π ( U ).It is readily observed that the equation (42) is the Lagrange equation of the Lagrangian L = ( G / N + q i A i + A ) dq (43)on the jet manifold J U of a fibre bundle U → π ( U ). Remark 3.
Both the equation (42) and the Lagrangian (43) are defined only on acoordinate chart (8) of Q since they are not maintained by transition functions (10) –(11).A solution s i ( q ) of the equation (42) defines a solution s λ ( τ ) (41) of the equation (32)up to an arbitrary function s ( τ ). The relativistic constraint (36) enables one to overcomethis ambiguity as follows.Let us assume that, restricted to the coordinate chart ( U ; q , q i ) (8) of Q , the rela-tivistic constraint (36) has no solution q τ = 0. Then it is brought into the form( q τ ) N G ( q λ , q i ) = 1 , (44)where G is the function (40). With the condition (44), every three-velocity ( q i ) defines aunique pair of four-velocities q τ = ± ( G ( q λ , q i )) / N , q iτ = q τ q i . (45)Accordingly, any solution s i ( q ) of the equation (42) leads to solutions τ ( q ) = ± Z ( G ( q , s i ( q ) , ∂ s i ( q )) − / N dq , s i ( τ ) = s ( τ )( ∂ i s i )( s ( τ ))of the equation (37) and, equivalently, the relativistic equation (39). Example 4.
Let Q = M be a Minkowski space provided with the Minkowski metric η µν of signature (+ , −−− ). This is the case of Special Relativity. Let A λ dq λ be a one-formon Q . Then L = [ m ( η µν q µτ q ντ ) / + e A µ q µτ ] dτ, m, e ∈ R , (46)is a relativistic Lagrangian on J Q R which satisfies the Noether identity (28). The corre-sponding relativistic equation (39) reads mη µν q νττ − eF µν q ντ = 0 , (47) η µν q µτ q ντ = 1 . (48)10his describes a relativistic massive charge in the presence of an electromagnetic field A . It follows from the relativistic constraint (48) that ( q τ ) ≥
1. Therefore, passing tothree-velocities, we obtain the Lagrangian (43): L = " m (1 − X i ( q i ) ) / + e ( A i q i + A ) dq , and the Lagrange equation (42): d mq i (1 − P i ( q i ) ) / + e ( F ij q j + F i ) = 0 . Example 5.
Let Q = R be an Euclidean space provided with the Euclidean metric ǫ . This is the case of Euclidean Special Relativity. Let A λ dq λ be a one-form on Q . Then L = [( ǫ µν q µτ q ντ ) / + A µ q µτ ] dτ is a relativistic Lagrangian on J Q R which satisfies the Noether identity (28). The corre-sponding relativistic equation (39) reads mǫ µν q νττ − eF µν q ντ = 0 , (49) ǫ µν q µτ q ντ = 1 . (50)It follows from the relativistic constraint (50) that 0 ≤ ( q τ ) ≤
1. Passing to three-velocities, one therefore meets a problem.
A glance at the relativistic Lagrangian (31) shows that, because of the gauge symmetry(27), this Lagrangian is independent of τ and, therefore, it describes an autonomousmechanical system. Accordingly, the relativistic equation (39) on Q R is conservative and,therefore, it is equivalent to an autonomous second order equation on Q whose solutionsare parameterized by the coordinate τ on a base R of Q R . Given holonomic coordinates( q λ , ˙ q λ , ¨ q λ ) of the second tangent bundle T Q , this autonomous second order equation(called the autonomous relativistic equation) reads ∂ λ G µα ...α N N − ∂ µ G λα ...α N ! ˙ q µ ˙ q α · · · ˙ q α N − (2 N − G βµα ...α N ¨ q µ ˙ q α · · · ˙ q α N + F βµ ˙ q µ = 0 , (51) G = G α ...α N ˙ q α · · · ˙ q α N = 1 . Due to the canonical isomorphism q λτ → ˙ q λ (16), the tangent bundle T Q is regarded as aspace of four-velocities. 11eneralizing Example 4, let us investigate relativistic mechanics on a four-dimensionalpseudo-Riemannian manifold Q = X , coordinated by ( x λ ) and provided with a pseudo-Riemannian metric g of signature (+ , − − − ). We agree to call X a world manifold. Let A = A λ dx λ be a one-form on X . Let us consider the relativistic Lagrangian (31): L = [( g αβ x ατ x βτ ) / + A µ x µτ ] dτ, and the relativistic constraint (36): g αβ x ατ x βτ = 1 . The corresponding autonomous relativistic equation (39) on X takes the form¨ x λ − { µλν } ˙ x µ ˙ x ν − g λβ F βν ˙ x ν = 0 , (52) g = g αβ ˙ x α ˙ x β = 1 , (53)where { µλν } is the Levi–Civita connection. A glance at the equality (52) shows that it isa geodesic equation on T X with respect to an affine connection K λµ = { µλν } ˙ x ν + g λν F νµ . (54)on T X .A particular form of this connection follows from the fact that the geodesic equation(52) is derived from a Lagrange equation, i.e., we are in the case of Lagrangian relativisticmechanics. In a general setting, relativistic mechanics on a pseudo-Riemannian manifold(
X, g ) can be formulated as follows.The geodesic equation¨ x µ = K µλ ( x ν , ˙ x ν ) ˙ x λ , (55)on the tangent bundle T X with respect to a connection K = dx λ ⊗ ( ∂ λ + K µλ ˙ ∂ µ ) (56)on T X → X is called a relativistic geodesic equation if a geodesic vector field of K livesin the subbundle of hyperboloids W g = { ˙ x λ ∈ T X | g λµ ˙ x λ ˙ x µ = 1 } ⊂ T X (57)defined by the relativistic constraint (53).One can show that the equation (55) is a relativistic geodesic equation if the condition( ∂ λ g µν ˙ x µ + 2 g µν K µλ ) ˙ x λ ˙ x ν = 0 (58)holds. 12bviously, the connection (54) fulfils the condition (58). Any metric connection, e.g.,the Levi–Civita connection { λµν } on T X satisfies the condition (58).Given a Levi–Civita connection { λµν } , any connection K on T X → X can be writtenas K µλ = { λµν } ˙ x ν + σ µλ ( x λ , ˙ x λ ) , (59)where σ = σ µλ dx λ ⊗ ˙ ∂ λ (60)is some soldering form on T X . Then the condition (58) takes the form g µν σ µλ ˙ x λ ˙ x ν = 0 . (61)With the decomposition (59), one can think of the relativistic geodesic equation (55):¨ x µ = { λµν } ˙ x ν ˙ x λ + σ µλ ( x λ , ˙ x λ ) ˙ x λ , (62)as describing a relativistic particle in the presence of a gravitational field g and a non-gravitational external force σ . We are in the case of relativistic mechanics on a pseudo-Riemmanian world manifold(
X, g ). Given the coordinate chart (13) of its configuration space X , the homogeneousLegendre bundle corresponding to the local non-relativistic system on U is the cotangentbundle T ∗ U of U . This fact motivate us to think of the cotangent bundle T ∗ X as being thephase space of relativistic mechanics on X . It is provided with the canonical symplecticform Ω = dp λ ∧ dx λ (63)and the corresponding Poisson bracket { , } .A relativistic Hamiltonian is defined as follows [9, 10, 11]. Let H be a smooth realfunction on T ∗ X such that the morphism f H : T ∗ X → T X, ˙ x µ ◦ f H = ∂ µ H, (64)is a bundle isomorphism. Then the inverse image N = f H − ( W g )of the subbundle of hyperboloids W g (57) is a one-codimensional (consequently, coisotropic)closed imbedded subbundle N of T ∗ X given by the condition H T = g µν ∂ µ H∂ ν H − . (65)13e say that H is a relativistic Hamiltonian if the Poisson bracket { H, H T } vanishes on N . This means that the Hamiltonian vector field γ = ∂ λ H∂ λ − ∂ λ H∂ λ (66)of H preserves the constraint N and, restricted to N , it obeys the equation γ ⌋ Ω N + i ∗ N dH = 0 , (67)which is the Hamilton equation of a Dirac constrained system on N with a Hamiltonian H [3].The morphism (64) sends the vector field γ (66) onto the vector field γ T = ˙ x λ ∂ λ + ( ∂ µ H∂ λ ∂ µ H − ∂ µ H∂ λ ∂ µ H ) ˙ ∂ λ on T X . This vector field defines the autonomous second order dynamic equation¨ x λ = ∂ µ H∂ λ ∂ µ H − ∂ µ H∂ λ ∂ µ H (68)on X which preserves the subbundle of hyperboloids (57), i.e., it is the autonomousrelativistic equation (51). Example 6.
The following is a basic example of relativistic Hamiltonian mechanics.Given a one-form A = A µ dq µ on X , let us put H = g µν ( p µ − A µ )( p ν − A ν ) . (69)Then H T = 2 H − { H, H T } = 0. The constraint H T = 0 (65) defines aone-codimensional closed imbedded subbundle N of T ∗ X . The Hamilton equation (67)takes the form γ ⌋ Ω N = 0. Its solution (66) reads˙ x α = g αν ( p ν − A ν ) , ˙ p α = − ∂ α g µν ( p µ − A µ )( p ν − A ν ) + g µν ( p µ − A µ ) ∂ α A ν . The corresponding autonomous second order dynamic equation (68) on X is¨ x λ − { µλν } ˙ x µ ˙ x ν − g λν F νµ ˙ x µ = 0 , (70) { µλν } = − g λβ ( ∂ µ g βν + ∂ ν g βµ − ∂ β g µν ) ,F µν = ∂ µ A ν − ∂ ν A µ . It is a relativistic geodesic equation with respect to the affine connection (54).Since the equation (70) coincides with the generic Lagrange equation (52) on a worldmanifold X , one can think of H (69) as being a generic Hamiltonian of relativistic me-chanics on X . 14 eferences [1] Echeverr´ıa Enr´ıquez, A., Mu˜noz Lecanda, M. and Rom´an Roy, N. (1991). Geomet-rical setting of time-dependent regular systems. Alternative models, Rev. Math.Phys. , 301.[2] Giachetta, G., Mangiarotti, L. and Sardanashvily, G. (1997). New Lagrangian andHamiltonian Methods in Field Theory (World Scientific, Singapore).[3] Giachetta, G., Mangiarotti, L. and Sardanashvily, G. (2005).
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