Lagrangian Dynamics of Submanifolds. Relativistic Mechanics

JOURNAL OF GEOMETRIC MECHANICS doi:10.3934/jgm.2012.4.99 c American Institute of Mathematical Sciences Volume 4, Number 1, March 2012 pp. 99–110

LAGRANGIAN DYNAMICS OF SUBMANIFOLDS. RELATIVISTIC MECHANICS

Gennadi Sardanashvily Department of Theoretical Physics Moscow State University Moscow, Russia

(Communicated by Manuel de Le´on)

Abstract. Geometric formulation of Lagrangian relativistic mechanics in the terms of jets of one-dimensional submanifolds is generalized to Lagrangian theory of submanifolds of arbitrary dimension.

1. Introduction. Classical non-relativistic mechanics is adequately formulated as Lagrangian and Hamiltonian theory on a fibre bundle Q → R over the time axis R, where R is provided with the Cartesian coordinate t possessing transition functions t0 = t+const. [1,5,7,8, 11]. A velocity space of non-relativistic mechanics is a first order jet manifold J 1Q of sections of Q → R. Lagrangians of non-relativistic mechanics are defined as densities on J 1Q. This formulation is extended to time- reparametrized non-relativistic mechanics subject to time-dependent transformations which are bundle automorphisms of Q → R [5,8]. Thus, one can think of non-relativistic mechanics as being particular classical field theory on fibre bundles over X = R. However, an essential difference between non-relativistic mechanics and field theory on fibre bundles Y → X, dim X > 1, lies in the fact that connections on Q → R always are flat. Therefore, they fail to be dynamic variables, but characterize non-relativistic reference frames. In comparison with non-relativistic mechanics, relativistic mechanics admits transformations of the time depending on other variables, e.g., Lorentz transformations in Special Relativity on a Minkowski space Q = R4. Therefore, a configuration space Q of relativistic mechanics has no preferable fibration Q → R, and its ve- 1 locity space is a first order jet manifold J1 Q of one-dimensional submanifolds of a 1 configuration space Q [5,8, 12]. Fibres of a jet bundle J1 Q → Q are projective spaces, and one can think of them as being spaces of three-velocities of a relativistic system. Four-velocities of a relativistic system are represented by elements of the tangent bundle TQ of a configuration space Q. This work is devoted to generalization of the above mentioned formulation of relativistic mechanics to the case of submanifolds of arbitrary dimension. Let us consider n-dimensional submanifolds of an m-dimensional smooth real manifold Z. The notion of jets of submanifolds [4,6,9] generalizes that of jets of sections of fibre bundles, which are particular jets of submanifolds (Section 2). Namely, a space of jets of submanifolds admits a cover by charts of jets of sections.

2000 Mathematics Subject Classiﬁcation. Primary: 58A20, 70H40; Secondary: 83A05. Key words and phrases. Lagrangian theory, jet manifold, relativistic mechanics.

99 100 GENNADI SARDANASHVILY

Just as in relativistic mechanics, we restrict our consideration to first order jets of 1 submanifolds which form a smooth manifold JnZ. One however meets a problem 1 how to develop Lagrangian formalism on a manifold JnZ because it is not a fibre bundle. For this purpose, we associate to n-dimensional submanifolds of Z the sections of a trivial fibre bundle π : ZΣ = Σ × Z → Σ, (1) where Σ is some n-dimensional manifold. We obtain a relation between the elements 1 of JnZ and the jets of sections of the fibre bundle (1) (Section 3). This relation fails to be one-to-one correspondence. The ambiguity contains, e.g., diffeomorphisms of Σ. Then Lagrangian formalism on a fibre bundle ZΣ → Σ is developed in a standard way, but a Lagrangian is required to possess the gauge symmetry (19) which leads to the rather restrictive Noether identities (20) (Section 4). If n = 2, this is the case, e.g., of the Nambu–Goto Lagrangian (21) of classical string theory (Example 4.1). If n = 1, solving these Noether identities, we obtain a generic Lagrangian (41) of relativistic mechanics (Section 5). These examples confirm the correctness of our description of Lagrangian dynamics of submanifolds of a manifold Z as that of sections of the fibre bundle ZΣ (1).

2. Jets of submanifolds. Given an m-dimensional smooth real manifold Z, a k-order jet of n-dimensional submanifolds of Z at a point z ∈ Z is defined as an k equivalence class jz S of n-dimensional imbedded submanifolds of Z through z which are tangent to each other at z with order k > 0. Namely, two submanifolds 0 iS : S → Z, iS0 : S → Z k through a point z ∈ Z belong to the same equivalence class jz S if and only if the images of k-tangent morphisms k k k k k 0 k T iS : Tz S → Tz Z,T iS0 : Tz S → Tz Z coincide with each other. The set k [ k Jn Z = jz S z∈Z of k-order jets of submanifolds is a finite-dimensional real smooth manifold, called the k-order jet manifold of submanifolds [4,6,9]. Let Y → X be an m-dimensional fibre bundle over an n-dimensional base X and J kY a k-order jet manifold of sections of Y → X. Given an imbedding Φ : Y → Z, there is a natural injection k k k k k J Φ: J Y → Jn Z, jx s → [Φ ◦ s]Φ(s(x)), k where s are sections of Y → X. This injection defines a chart on Jn Z. These charts k provide a manifold atlas of Jn Z. Let us restrict our consideration to first order jets of submanifolds. There is obvious one-to-one correspondence 1 ζ : j S → V 1 ⊂ T Z (2) z jz S z 1 between the jets jz S at a point z ∈ Z and the n-dimensional vector subspaces of 1 the tangent space TzZ of Z at z. It follows that JnZ is a fibre bundle 1 ρ : JnZ → Z (3) LAGRANGIAN DYNAMICS OF SUBMANIFOLDS 101 with a structure group GL(n, m − n; R) of linear transformations of a vector space Rm which preserve its subspace Rn. A typical fibre of the fibre bundle (3) is a Grassmann manifold GL(m; R)/GL(n, m − n; R). This fibre bundle is endowed with the following coordinate atlas. Let {(U; zA)} be a coordinate atlas of Z. Let us provide Z with an atlas obtained by replacing every chart (U, zA) of Z with m! n!(m − n)! charts on U which correspond to different partitions of (zA) in collections of n and m − n coordinates (U; xa, yi), a = 1, . . . , n, i = 1, . . . , m − n. (4) Transition functions between the coordinate charts (4) associated with a coordinate chart (U, zA) are reduced to exchange between coordinates xa and yi. Transition functions between arbitrary coordinate charts (4) take the form x0a = x0a(xb, yk), y0i = y0i(xb, yk). (5) 0 Let JnZ denote a manifold Z provided with the coordinate atlas (4)–(5). 0 1 Given this atlas of JnZ = Z, a first order jet manifold JnZ is endowed with coordinate charts −1 (m−n)n a i i (ρ (U) = U × R ; x , y , ya), (6) possessing the following transition functions. With respect to the coordinates (6) 1 a i on a jet manifold JnZ and the induced fibre coordinates (x ˙ , y˙ ) on the tangent bundle TZ, the above mentioned correspondence ζ (2) reads i a i 1 ζ :(ya) → x˙ (∂a + ya(jz S)∂i). It implies the relations ∂y0j ∂y0j ∂xb ∂xb y0j = yk + y0i + , a ∂yk b ∂xb ∂y0i a ∂x0a (7) ∂xb ∂xb ∂x0c ∂x0c y0i + yk + = δc, ∂y0i a ∂x0a ∂yk b ∂xb a i which jet coordinates ya must satisfy under coordinate transformations (5). Let us consider a non-degenerate n × n matrix M with entries ∂x0c ∂x0c M c = yk + . b ∂yk b ∂xb Then the relations (7) lead to equalities ∂xb ∂xb y0i + = (M −1)b . ∂y0i a ∂x0a a Hence, we obtain the transformation law of first order jet coordinates ∂y0j ∂y0j y0j = yk + (M −1)b . (8) a ∂yk b ∂xb a In particular, if coordinate transition functions x0a (5) are independent of coordinates yk, the transformation law (8) comes to the familiar transformations of jets of sections. 102 GENNADI SARDANASHVILY

1 3. The fibre bundle (1). Given a coordinate chart (6) of JnZ, one can regard −1 1 1 ρ (U) ⊂ JnZ as a first order jet manifold J U of sections of a fibre bundle χ : U 3 (xa, yi) → (xa) ∈ χ(U). A graded differential algebra of exterior forms on ρ−1(U) is generated by horizontal a i i a forms dx and contact forms dy − yadx . Coordinate transformations (5) and (8) preserve an ideal of contact forms, but horizontal forms are not transformed into horizontal forms, unless coordinate transition functions x0a (5) are independent of coordinates yk. Therefore, one can develop first order Lagrangian formalism with a Lagrangian L = Ldnx on a coordinate chart ρ−1(U), but this Lagrangian fails to 1 be globally defined on JnZ. In order to overcome this difficulty, let us consider the trivial fibre bundle ZΣ → Σ (1) whose trivialization throughout holds fixed. This fibre bundle is provided with an atlas of coordinate charts µ a i (UΣ × U; σ , x , y ), (9) a i 0 where (U; x , y ) are the above mentioned coordinate charts (4) of a manifold JnZ. The coordinate charts (9) possess transition functions σ0µ = σµ(σν ), x0a = x0a(xb, yk), y0i = y0i(xb, yk). (10) 1 Let J ZΣ be a first order jet manifold of the fibre bundle (1). Since the trivialization (1) is fixed, it is a vector bundle 1 1 π : J ZΣ → ZΣ isomorphic to a tensor product 1 ∗ J ZΣ = T Σ ⊗ TZ (11) Σ×Z ∗ of the cotangent bundle T Σ of Σ and the tangent bundle TZ of Z over ZΣ. 1 Given the coordinate atlas (9)-(10) of ZΣ, a jet manifold J ZΣ is endowed with the coordinate charts 1 −1 mn µ a i a i ((π ) (UΣ × U) = UΣ × U × R ; σ , x , y , xµ, yµ), (12) possessing transition functions ∂x0a ∂x0a ∂σν ∂y0i ∂y0i ∂σν x0a = yk + xb , y0i = yk + xb . (13) µ ∂yk ν ∂xb ν ∂σ0µ µ ∂yk ν ∂xb ν ∂σ0µ Relative to the coordinates (12), the bundle isomorphism (11) takes the form a i µ a i (xµ, yµ) → dσ ⊗ (xµ∂a + yµ∂i). µ a i a i Obviously, a jet (σ , x , y , xµ, yµ) of sections of the fibre bundle (1) defines some jet of n-dimensional submanifolds of a manifold {σ} × Z through a point a i a i (x , y ) ∈ Z if an m × n matrix with entries (xµ, yµ) is of maximal rank n. This property is preserved under the coordinate transformations (13). An element of 1 J ZΣ is called regular if it possesses this property. Regular elements constitute an 1 open subbundle of a jet bundle J ZΣ → ZΣ. 1 Since regular elements of J ZΣ characterize first jets of n-dimensional submanifolds of Z, one hopes to describe the dynamics of these submanifolds of a manifold Z as that of sections of the fibre bundle (1). For this purpose, let us refine the 1 1 relation between elements of jet manifolds JnZ and J ZΣ. LAGRANGIAN DYNAMICS OF SUBMANIFOLDS 103

1 Let us consider a manifold product Σ × JnZ. It is a fibre bundle over ZΣ. Given the coordinate atlas (9)-(10) of ZΣ, this product is endowed with coordinate charts −1 (m−n)n µ a i i (UΣ × ρ (U) = UΣ × U × R ; σ , x , y , ya), (14) µ a i i possessing the transition functions (8). Let us assign to an element (σ , x , y , ya) µ a i a i of the chart (14) the elements (σ , x , y , xµ, yµ) of the chart (12) whose coordinates obey the relations i a i yaxµ = yµ. (15) These elements make up an n2-dimensional vector space. The relations (15) are maintained under the coordinate transformations (10) and induced transformations of the charts (12) and (14) as follows: ∂y0i ∂y0i ∂x0a ∂x0a ∂σν y0ix0a = yk + (M −1)c yk + xb = a µ ∂yk c ∂xc a ∂yk ν ∂xb ν ∂σ0µ ∂y0i ∂y0i ∂x0a ∂x0a ∂σν yk + (M −1)c yk + xb = ∂yk c ∂xc a ∂yk b ∂xb ν ∂σ0µ ∂y0i ∂y0i ∂σν ∂y0i ∂y0i ∂σν yk + xb = yk + xb = y0i. ∂yk b ∂xb ν ∂σ0µ ∂yk ν ∂xb ν ∂σ0µ µ Thus, one can associate: 0 µ a i i µ a i a i i a i ζ :(σ , x , y , ya) → {(σ , x , y , xµ, yµ) | yaxµ = yµ}, 1 2 to each element of a manifold Σ × JnZ an n -dimensional vector space in a jet 1 manifold J ZΣ. This is a subspace of elements a µ i xµdσ ⊗ (∂a + ya∂i) of a fibre of the tensor bundle (11) at a point (σµ, xa, yi). This subspace always a contains regular elements, e.g., whose coordinates xµ form a non-degenerate n × n matrix. 1 1 Conversely, given a regular element jz s of J ZΣ, there is a coordinate chart (12) a 1 1 such that coordinates xµ of jz s constitute a non-degenerate matrix, and jz s defines 1 a unique element of Σ × JnZ by the relations i i −1 µ ya = yµ(x )a . (16) Thus, we have shown the following. Let (σµ, zA) further be arbitrary coordinates µ A A on the product ZΣ (1) and (σ , z , zµ ) the corresponding coordinates on a jet 1 1 manifold J ZΣ. In these coordinates, an element of J ZΣ is regular if an m × n A matrix with entries zµ is of maximal rank n. Theorem 3.1. (i) Any jet of submanifolds through a point z ∈ Z defines some (but not unique) jet of sections of the fibre bundle ZΣ (1) through a point σ × z for any σ ∈ Σ in accordance with the relations (15). 1 (ii) Any regular element of J ZΣ defines a unique element of a jet manifold 1 1 JnZ by means of the relations (16). However, non-regular elements of J ZΣ can correspond to different jets of submanifolds. µ A A µ A 0A 1 (iii) Two elements (σ , z , zµ ) and (σ , z , zµ ) of J ZΣ correspond to the same jet of submanifolds if 0A ν A zµ = Mµ zν , where M is some matrix, e.g., it comes from a diffeomorphism of Σ. 104 GENNADI SARDANASHVILY

4. Lagrangian formalism. Based on Theorem 3.1, we can describe the dynamics of n-dimensional submanifolds of a manifold Z as that of sections of the ﬁbre bundle ZΣ (1) for some n-dimensional manifold Σ. Let A A n L = L(z , zµ )d σ (17) 1 be a ﬁrst order Lagrangian on a jet manifold J ZΣ. The corresponding Euler– Lagrange operator reads

A n µ δL = EAdz ∧ d σ, EA = ∂AL − dµ∂AL. (18) It yields the Euler–Lagrange equations

µ EA = ∂AL − dµ∂AL = 0. In view of Theorem 3.1, it seems reasonable to require that, in order to describe 1 jets of n-dimensional submanifolds of Z, the Lagrangian L (17) on J ZΣ must be invariant under diffeomorphisms of a manifold Σ. To formulate this condition, it is sufficient to consider infinitesimal generators of one-parameter subgroups of these µ diffeomorphisms which are vector fields u = u ∂µ on Σ. Since ZΣ → Σ is a trivial µ bundle, such a vector field gives rise to a vector field u = u ∂µ on ZΣ. Its jet 1 prolongation onto J ZΣ reads

1 µ A ν µ µ ν A ν A µ J u = u ∂µ − zν ∂µu ∂ = u dµ + [−u zν ∂A − dµ(u zν )∂ ], A A (19) A A ν dµ = ∂µ + zµ ∂A + zµν ∂µ + ··· .

1 One can regard it as a generalized vector ﬁeld on J ZΣ depending on parameter functions uµ(σν ), i.e., it is a gauge transformation [3,4]. Let us require that J 1u (19) or, equivalently, its vertical part

ν A ν A µ uV = −u zν ∂A − dµ(u zν )∂A is a variational symmetry of the Lagrangian L (17). Then by virtue of the sec- ond Noether theorem, the Euler–Lagrange operator δL (18) obeys the irreducible Noether identities A zν EA = 0. (20) One can think of these identities as being a condition which a Lagrangian L on 1 J ZΣ must satisfy in order to be a Lagrangian of submanifolds of Z. It is readily observed that this condition is rather restrictive.

Example 4.1. Let Z be a locally aﬃne manifold, i.e., a toroidal cylinder Rm−k×T k. Its tangent bundle can be provided with a constant non-degenerate ﬁbre metric ηAB. Let Σ be a two-dimensional manifold. Let us consider a 2 × 2 matrix with entries

A B hµν = ηABzµ zν . Then its determinant provides a Lagrangian

1/2 2 A B A B A B 2 1/2 2 L = (det h) d σ = ([ηABz1 z1 ][ηABz2 z2 ] − [ηABz1 z2 ] ) d σ (21)

1 on the jet manifold J ZΣ (11). This is the well known Nambu–Goto Lagrangian of classical string theory [10]. It satisﬁes the Noether identities (20). LAGRANGIAN DYNAMICS OF SUBMANIFOLDS 105

5. Relativistic mechanics. As was mentioned above, if n = 1, we are in the case of relativistic mechanics. In this case, one can obtain a complete solution of the Noether identities (20) which provides a generic Lagrangian of relativistic mechanics. Given an m-dimensional manifold Q coordinated by (qλ), let us consider a jet 1 manifold J1 Q of its one-dimensional submanifolds. It is treated as a velocity space 0 of relativistic mechanics [5,8, 12]. Let us provide Q = J1 Q with the coordinates (4): (U; x0 = q0, yi = qi) = (U; qλ). (22) 1 Then a jet manifold ρ : J1 Q → Q is endowed with the coordinates (6): −1 0 i i (ρ (U); q , q , q0) (23) possessing transition functions (5), (8): q00 = q00(q0, qk), q00 = q00(q0, qk), ∂q0i ∂q0i ∂q00 ∂q00 −1 (24) q0i = qj + qj + . 0 ∂qj 0 ∂q0 ∂qj 0 ∂q0 1 A glance at the transformation law (24) shows that J1 Q → Q is a ﬁbre bundle in projective spaces.

Example 5.1. Let Q = M 4 = R4 be a Minkowski space whose Cartesian coordinates (qλ), λ = 0, 1, 2, 3, are subject to the Lorentz transformations (24): q00 = q0chα − q1shα, q01 = −q0shα + q1chα, q02,3 = q2,3. (25) Then q0i (24) are exactly the Lorentz transformations 1 2,3 01 q0chα − shα 02,3 q0 q0 = 1 q0 = 1 −q0shα + chα −q0shα + chα of three-velocities in Special Relativity.

1 In view of Example 5.1, let us call a velocity space J1 Q of relativistic mechanics the space of three-velocities, though a dimension of Q need not equal 3 + 1. In order to develop Lagrangian formalism of relativistic mechanics, let us consider the trivial fibre bundle (1): QR = R × Q → R, (26) whose base Σ = R is endowed with a global Cartesian coordinate τ. This fibre bundle is provided with an atlas of coordinate charts λ (R × U; τ, q ), (27) 0 i 0 where (U; q , q ) are the coordinate charts (22) of a manifold J1 Q. The coordinate 1 charts (27) possess the transition functions (24). Let J QR be a first order jet manifold of the fibre bundle (26). Since the trivialization (26) is fixed, there is the 1 canonical isomorphism (11) of J QR to the vertical tangent bundle 1 J QR = VQR = R × TQ (28) of QR → R. 1 Given the coordinate atlas (27) of QR, a jet manifold J QR is endowed with coordinate charts 1 −1 m λ λ ((π ) (R × U) = R × U × R ; τ, q , qτ ), (29) 106 GENNADI SARDANASHVILY possessing transition functions ∂q0λ q0λ = qµ. (30) τ ∂qµ τ Relative to the coordinates (29), the isomorphism (28) takes the form

µ µ µ µ µ (τ, q , qτ ) → (τ, q , q˙ = qτ ). (31) Example 5.2. Let Q = M 4 be a Minkowski space in Example 5.1 whose Carte- sian coordinates (q0, qi) are subject to the Lorentz transformations (25). Then the corresponding transformations (30) take the form

00 0 1 01 0 1 02,3 2,3 qτ = qτ chα − qτ shα, qτ = −qτ shα + qτ chα, qτ = qτ of transformations of four-velocities in Special Relativity.

1 In view of Example 5.2, we agree to call fibre elements of J QR → QR the four-velocities though the dimension of Q need not equal 4. Due to the canonical λ λ isomorphism qτ → q˙ (28), by four-velocities also are meant the elements of the tangent bundle TQ, which is called the space of four-velocities. In accordance with the terminology of Section 3, the non-zero jet (31) of sections of the fibre bundle (26) is regular, and it defines some jet of one-dimensional submanifolds of a manifold {τ} × Q through a point (q0, qi) ∈ Q. Although this is not one-to-one correspondence, just as in Section 4, one can describe the dynamics of one-dimensional submanifolds of a manifold Q as that of sections of the fibre bundle (26). 1 Let us consider a manifold product R × J1 Q. It is a fibre bundle over QR. Given the coordinate atlas (27) of QR, this product is endowed with the coordinate charts (14): −1 m−1 0 i i (UR × ρ (U) = UR × U × R ; τ, q , q , q0), (32) 0 i i possessing transition functions (24). Let us assign to an element (τ, q , q , q0) of the 0 i 0 i chart (32) the elements (τ, q , q , qτ , qτ ) of the chart (29) whose coordinates obey the relations (15): i 0 i q0qτ = qτ . (33) These elements make up a one-dimensional vector space. The relations (33) are maintained under coordinate transformations (24) and (30). Thus, one can associate 1 to each element of a manifold R × J1 Q the one-dimensional vector space 0 i i 0 i 0 i i 0 i (τ, q , q , q0) → {(τ, q , q , qτ , qτ ) | q0qτ = qτ } (34) 1 0 i in a jet manifold J QR. This is a subspace of elements qτ (∂0 + q0∂i) of a fibre of the vertical tangent bundle (28) at a point (τ, q0, qi). Conversely, given a non-zero 1 element (31) of J QR, there is a coordinate chart (29) such that this element defines 1 a unique element of R × J1 Q by the relations (16): i i qτ q0 = 0 . (35) qτ Thus, we come to Theorem 3.1 for the case of n = 1 as follows. Let (τ, qλ) further λ λ be arbitrary coordinates on the product QR (26) and (τ, q , qτ ) the corresponding 1 coordinates on a jet manifold J QR. LAGRANGIAN DYNAMICS OF SUBMANIFOLDS 107

Theorem 5.1. (i) Any jet of submanifolds through a point q ∈ Q defines some (but not unique) jet of sections of the fibre bundle QR (26) through a point τ × q for any τ ∈ R in accordance with the relations (33). 1 (ii) Any non-zero element of J QR defines a unique element of a jet manifold 1 1 J1 Q by means of the relations (35). However, non-zero elements of J QR can correspond to different jets of submanifolds. λ λ λ 0λ 1 (iii) Two elements (τ, q , qτ ) and (τ, q , qτ ) of J QR correspond to the same jet 0λ λ of submanifolds if qτ = rqτ , r ∈ R \{0}. In the case of a Minkowski space Q = M 4 in Examples 5.1 and 5.2, the equalities (33) and (35) are familiar relations between three- and four-velocities. Let λ λ L = L(τ, q , qτ )dτ (36) 1 be a first order Lagrangian on a jet manifold J QR. The corresponding Lagrange operator reads λ τ δL = Eλdq ∧ dτ, Eλ = ∂λL − dτ ∂λL. (37) Let us require that, in order to describe jets of one-dimensional submanifolds of Q, 1 the Lagrangian L (36) on J QR possesses a gauge symmetry given by vector fields u = u(τ)∂τ on QR or, equivalently, their vertical part λ uV = −u(τ)qτ ∂λ, (38) which are generalized vector fields on QR. Then variational derivatives of this Lagrangian obey the Noether identities (20): λ qτ Eλ = 0. (39) We call such a Lagrangian the relativistic Lagrangian. In order to obtain a generic form of a relativistic Lagrangian L, let us regard the Noether identities (39) as an equation for L. It admits the following solution. Let 1 G (qν )dqα1 ∨ · · · ∨ dqα2N 2N! α1...α2N be a symmetric tensor field on Q such that a function

ν α1 α2N G = Gα1...α2N (q )q ˙ ··· q˙ (40) is positive (G > 0) everywhere on TQ \ b0(Q), where b0(Q) is a global zero section of ν µ TQ → Q. Let A = Aµ(q )dq be a one-form on Q. Given the pull-back of G and 1 A onto J QR due to the canonical isomorphism (28), we deﬁne a Lagrangian

1/2N µ α1 α2N L = (G + qτ Aµ)dτ, G = Gα1...α2N qτ ··· qτ , (41) 1 on J QR \ (R × b0(Q)). The corresponding Lagrange equations read ∂ G ∂τ G E = λ − d λ + F qµ = λ 2NG1−1/2N τ 2NG1−1/2N λµ τ h i β β ν2 ν2N −1 1/2N−1 Eβ δλ − qτ Gλν2...ν2N qτ ··· qτ G G = 0, ∂βGµα ...α (42) E = 2 2N − ∂ G qµqα2 ··· qα2N − β 2N µ βα2...α2N τ τ τ

µ α3 α2N 1−1/2N µ (2N − 1)Gβµα3...α2N qττ qτ ··· qτ + G Fβµqτ ,

Fλµ = ∂λAµ − ∂µAλ. 108 GENNADI SARDANASHVILY

It is readily observed that the variational derivatives Eλ (42) satisfy the Noether identities (39). Moreover, any relativistic Lagrangian obeying the Noether identity (39) is of the type (41). A glance at the Lagrange equations (42) shows that they hold if

ν2 ν2N −1 Eβ = ΦGβν2...ν2N qτ ··· qτ G , (43) 1 where Φ is some function on J QR. In particular, we consider the equations

Eβ = 0. (44) Because of the Noether identities (39), the system of equations (42) is underdetermined. To overcome this difficulty, one can complete it with some additional equation. Given the function G (41), let us choose the condition G = 1. (45) Being positive, the function G (41) possesses a nowhere vanishing differential. Therefore, its level surface WG defined by the condition (45) is a submanifold of 1 J QR. Our choice of the equations (44) and the condition (45) is motivated by the following facts. Lemma 5.2. Any solution of the Lagrange equations (42) living in a submanifold WG is a solution of the equation (44).

Proof. A solution of the Lagrange equations (42) living in a submanifold WG obeys the system of equations Eλ = 0,G = 1. (46) Therefore, it satisﬁes the equality

dτ G = 0. (47) Then a glance at the expression (42) shows that the equations (46) are equivalent to the equations ∂λGµα ...α E = 2 2N − ∂ G qµqα2 ··· qα2N − λ 2N µ λα2...α2N τ τ τ

µ α3 α2N µ (48) (2N − 1)Gβµα3...α2N qττ qτ ··· qτ + Fβµqτ = 0, α1 α2N G = Gα1...α2N qτ ··· qτ = 1.

Lemma 5.3. Solutions of the equations (44) do not leave the submanifold WG (45). Proof. Since 2N d G = − qβE , τ 2N − 1 τ β any solution of the equations (44) intersecting the submanifold WG (45) obeys the equality (47) and, consequently, lives in WG. The system of equations (48) is called the relativistic equation. Its components Eλ (42) are not independent, but obeys the relation 2N − 1 qβE = − d G = 0,G = 1, τ β 2N τ similar to the Noether identities (39). The condition (45) is called the relativistic constraint. LAGRANGIAN DYNAMICS OF SUBMANIFOLDS 109

Though the system of equations (42) for sections of a fibre bundle QR → R is underdetermined, it is determined if, given a coordinate chart (U; q0, qi)(22) of Q and the corresponding coordinate chart (27) of QR, we rewrite it in the terms of i three-velocities q0 (35) as equations for sections of a fibre bundle U → χ(U) ⊂ R. Let us denote λ i 0 −2N λ λ 0 G(q , q0) = (qτ ) G(q , qτ ), qτ 6= 0. (49) Then we have ∂ G ∂0G E = q0 i − (q0)−1d i + F qj + F . i τ 1−1/2N τ τ 1−1/2N ij 0 i0 2NG 2NG λ 0 Let us consider a solution {s (τ)} of the equations (42) such that ∂τ s does not van- ish and there exists an inverse function τ(q0). Then this solution can be represented by sections si(τ) = (si ◦ s0)(τ) (50) of a composite bundle R × U → R × π(U) → R i where s (q0) = si(τ(q0)) are sections of U → χ(U) and s0(τ) are sections of R × π(U) → R. Restricted to such solutions, the equations (42) are equivalent to the equations ∂ G ∂0G E = i − d i + F qj + F = 0, i 1−1/2N 0 1−1/2N ij 0 i0 2NG 2NG (51) i E0 = −q0Ei. for sections si(q0) of a fibre bundle U → χ(U). It is readily observed that the equations (51) are a Lagrange equation of the Lagrangian 1/2N i 0 L = (G + q0Ai + A0)dq (52) on a jet manifold J 1U of a fibre bundle U → χ(U). It should be emphasized that, both the equations (51) and the Lagrangian (52) are defined only on a coordinate chart (22) of Q since they are not maintained under the transition functions (24). A solution si(q0) of the equations (51) defines a solution sλ(τ)(50) of the equations (42) up to an arbitrary function s0(τ). The relativistic constraint (45) enables one to overcome this ambiguity as follows. Let us assume that, restricted to the coordinate chart (U; q0, qi)(22) of Q, the 0 relativistic constraint (45) has no solution qτ = 0. Then it is brought into the form 0 2N λ i (qτ ) G(q , q0) = 1, (53) i where G is the function (49). With the condition (53), every three-velocity (q0) defines a unique pair of four-velocities 0 λ i 1/2N i 0 i qτ = ±(G(q , q0)) , qτ = qτ q0. (54) Accordingly, any solution si(q0) of the equations (51) leads to solutions Z 0 0 i 0 i −1/2N 0 i 0 i 0 τ(q ) = ± (G(q , s (q ), ∂0s (q0)) dq , s (τ) = s (τ)(∂is )(s (τ)) of the equations (46) and, equivalently, the relativistic equations (48). 110 GENNADI SARDANASHVILY

Example 5.3. Let Q = M 4 be a Minkowski space provided with a Minkowski metric ηµν of signature (+, − − −). This is the case of Special Relativity. Let λ A = Aλdq be a one-form on Q. Then µ ν 1/2 µ L = [m(ηµν qτ qτ ) + eAµqτ ]dτ, m, e ∈ R, (55) 1 is a relativistic Lagrangian on J QR which satisﬁes the Noether identity (39). The corresponding relativistic equation (48) reads ν ν mηµν qττ − eFµν qτ = 0, µ ν (56) ηµν qτ qτ = 1. This describes a relativistic massive charge in the presence of an electromagnetic 0 2 or gauge potential A. It follows from the relativistic constraint (56) that (qτ ) ≥ 1. Therefore, passing to three-velocities, we obtain a Lagrangian (52): X i 2 1/2 i 0 L = [m(1 − (q0) ) + e(Aiq0 + A0)]dq , i and the Lagrange equations (51):   mqi d 0 + e(F qj + F ) = 0. 0  P i 2 1/2  ij 0 i0 (1 − (q0) ) i

REFERENCES [1] A. Echeverr´ıa Enr´ıquez,M. MuñozLecanda and N. Román Roy, Geometrical setting of time- dependent regular systems. Alternative models, Reviews in Mathematical Physica, 3 (1991), 301–330. [2] G. Giachetta, L. Mangiarotti and G. Sardanashvily, “New Lagrangian and Hamiltonian Meth- ods in Field Theory,” World Scientific Publishing Co., Inc., River Edge, NJ, 1997. [3] G. Giachetta, L. Mangiarotti and G. Sardanashvily, On the notion of gauge symmetries of generic Lagrangian field theory, Journal of Mathematical Physics, 50 (2009), 012903, 19 pp. [4] G. Giachetta, L. Mangiarotti and G. Sardanashvily, “Advanced Classical Field Theory,” World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009. [5] G. Giachetta, L. Mangiarotti and G. Sardanashvily, “Geometric Formulation of Classical and Quantum Mechanics,” World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. [6] I. Krasil’shchik, V. Lychagin and A. Vinogradov, “Geometry of Jet Spaces and Nonlinear Partial Differential Equations,” Gordon and Breach, Glasgow, 1985. [7] M. De Leónand P. Rodrigues, “Methods of Differential Geometry in Analytical Mechanics,” North-Holland, Amsterdam, 1989. [8] L. Mangiarotti and G. Sardanashvily, “Gauge Mechanics,” World Scientific Publishing Co., Inc., River Edge, NJ, 1998. [9] M. Modugno and A. Vinogradov, Some variations on the notion of connections, Annali di Matematica Pura ed Applicata, CLXVII (1994), 33–71. [10] J. Polchinski, “String Theory,” Cambridge University Press, Cambridge, 1998. [11] G. Sardanashvily, Hamiltonian time-dependent mechanics, Journal of Mathematical Physics, 39 (1998), 2714–2729. [12] G. Sardanashvily, Relativistic mechanics in a general setting, International Journal of Geo- metric Methods in Modern Physics, 7 (2010), 1307–1319. Received September 2011; revised January 2012. E-mail address: [email protected]