The Equations of Motion of a Charged Particle in the Five-Dimensional Model of the General Relativity Theory with the Four-Dimensional Nonholonomic Velocity Space

The Equations of Motion of a Charged P article in the Fiv e-Dimensional Mo del of the General Relativit y Theory with the F our-Dimensional Nonholonomic V elo cit y Space Victor R. Krym 1 , Nickolai N. Petr ov St.-P etersburg State Univ ersit y , Departmen t of Mathematics and Mechanics, St.-P etersburg, 198504 Russia Abstract W e consider the four-dimensional nonholonomic distribution defined b y the 4- p oten tial of the electromagnetic field on the manifold. This distribution has a metric tensor with the Loren tzian signature (+ , − , − , − ), therefore, the causal structure ap- p ears as in the general relativit y theory . By means of the Pon tryagin’s maxim um principle w e prov ed that the equations of the horizon tal geo desics for this distribution are the same as the equations of motion of a c harged particle in the general relativit y theory . This is a Kaluza – Klein problem of classical and quan tum physics solv ed by metho ds of sub-Lorentzian geometry . W e study the geodesics sphere which app ears in a constant magnetic field and its singular points. Sufficien tly long geodesics are not optimal solutions of the v ariational problem and define the nonholonomic wa vefron t. This w av efron t is limited by a conv ex elliptic cone. W e also study v ariational princi- ple approach to the problem. The Euler – Lagrange equations are the same as those obtained b y the P on tryagin’s maximum principle if the restriction of the metric tensor on the distribution is the same. Keyw ords: general relativit y , electromagnetic field, nonholonomic differential geometry AMS Sub j. Class. (MSC): 37J60, 37J55, 53B30, 53B50, 53D50, 58A30, 70F25, 81S05 P A CS: 02.40.Ky , 04.20.Cv, 04.20.Fy , 04.50.+h, 11.25.Mj, 41.90.+e Journal-ref: V estnik Sankt-P eterburgskogo Univ ersiteta, Ser. 1. Matematik a, Mekhanik a, Astronomiya, 2007, N1, pp. 62–70. 1 Nonholonomic mo del in sub-Loren tzian geometry Distribution on a smo oth manifold M is a family of subspaces A ( x ) ⊂ T x M (of the tangen t bundle of the manifold) with the same dimension, smoothly parametrized by the p oin ts of manifold [1, 2, 3, 4]. F or each x ∈ M on A ( x ) the bilinear form h u, u i x is defined whic h can b e equally describ ed by the metric tensor g ij ( x ). The metric tensor of a distribution smo othly dep ends of the p oin t of the manifold. The absolutely con tinous path x ( t ) is called horizontal , iff x 0 ( t ) ∈ A ( x ( t )) for almost an y t . In sub-Riemannian geometry the metric tensor g ij ( x ) of the distribution A is p ositiv ely defined. In this geometry we study the shortest horizon tal curv es connecting tw o giv en p oin ts. If the distribution is completely nonholonomic and the Riemannian manifold is full, then any tw o p oin ts of the manifold can b e connected b y the shortest horizon tal geo desic. On manifolds and also on distributions 1 E-mail: vkrym2007@rambler.ru 1 NONHOLONOMIC MODEL IN SUB-LORENTZIAN GEOMETR Y 2 with metric tensor the geo desic is the Euler – Lagrange solution for the length functional J ( x ( · ) , u ( · )) = R T t 0 h u ( t ) , u ( t ) i 1 / 2 x ( t ) dt . In sub-Loren tzian geometry the metric tensor g ij ( x ) of the distribution A has the signa- ture (+ , − , . . ., − ). In this geometry we study the longest horizontal curv es connecting tw o giv en p oin ts [5]. The equations of motion of a charged particle in the electromagnetic and gra vitational fields can b e obtained as the solutions of the v ariational problem for the dis- tribution [6]. Consider the 5-dimensional manifold M 5 with the 4-dimensional distribution A defined b y means of the differential form ω ( x ) = 3 P i =0 A i ( x ) dx i + dx 4 , where the co vector ( A i ) i =0 ,... , 3 is the 4-p oten tial of the electromagnetic field. Consider the length functional J ( x ( · ) , u ( · )) = Z T t 0 L ( x ( t ) , u ( t )) dt, (1) where L ( x, u ) = m h u, u i 1 / 2 x is the pseudonorm of the v ector u , m is the mass of the particle. Consider the problem of maximizing the length functional on the set of horizontal curv es connecting tw o giv en p oin ts. Solution of this problem as w e proov e in this paper leads to equations d dt ∂ L ∂ u k − ∂ L ∂ x k + p 4 3 X j =0 F j k u j = 0 , k = 0 , . . ., 3 , (2) where F j k are defined b y (5). These equations are the same as the equations of motion of a particle with the c harge p 4 in the electromagnetic and gra vitational fields of the general relativit y theory . This w ording of the problem is different from the classical problem of ph ysics [7, 8] of minimizing the action functional on a 4-dimensional manifold: S ( x ( · ) , u ( · )) = − mc Z t 1 0 h u ( t ) , u ( t ) i 1 / 2 x ( t ) dt − e c Z t 1 0 3 X i =0 A i ( x ( t )) u i ( t ) dt, (3) where e is the charge of the particle, c is the sp eed of ligh t. This pap er is dev oted to the problem of unification of electromagnetic, gravitational and other fundamental in teractions [9, 10, 11]. In this mo del the particles are considered on the 5-dimensional manifold M 5 with the sp ecial smo oth structure. Allow ed are only smo oth co ordinate transformations for which ∂ y i ∂ x 4 = 0 , i = 0 , . . ., 3 , ∂ y 4 ∂ x 4 = 1 . (4) Comp onen ts ∂ y 4 ∂ x i , i = 0 , . . ., 3, can b e in terpreted as gauge transformations (9). Since the co ordinate transformations are smo oth, ∂ ∂ x 4 ∂ y i ∂ x j = ∂ ∂ x j ∂ y i ∂ x 4 = 0, i, j = 0 , . . ., 4. There- fore the cylindricit y condition ∂ v ∂ x 4 = 0 for any tensor field v is inv ariant. In the tangen t bundle T M the subspace Lin { ∂ 4 } is inv ariant. In the cotangent bundle T ∗ M the subspace Lin { dx 0 , dx 1 , dx 2 , dx 3 } is in v ariant. Assume that the metric tensor and the 4-p oten tial of the electromagnetic field do not dep end of x 4 . This condition is inv ariant at the co ordinate 1 NONHOLONOMIC MODEL IN SUB-LORENTZIAN GEOMETR Y 3 transformations with the prop ert y (4). The 4-dimensional distribution A is the v elo cit y space, i.e. the set of all possible v elo cit y v ectors of the particle. The metric tensor of the distribution A defines 4-dimensional con vex elliptic cone, completely similar to the cone of future of the general relativit y theory . Causal structure in the prop osed theory is simi- lar to the causal structure of the general relativit y theory [12, 13]. In the presence of the electromagnetic field the manifold A is nonholonomic. The electromagnetic field is defined in ph ysics b y the antisymmetric tensor ( F j k ) j,k =0 ,... , 3 ( F j k ) j,k =0 ,... , 3 =     0 E x E y E z − E x 0 − H z H y − E y H z 0 − H x − E z − H y H x 0     . (5) Here E is the 3-dimensional vector of the electric field tension, H is the 3-dimensional vector of the magnetic field tension. There is the 4-p oten tial ( A i ) i =0 ,... , 3 whic h defines the tensor of the electromagnetic field: F j k = ∂ A k ∂ x j − ∂ A j ∂ x k , j, k = 0 , . . ., 3 . (6) 4-P otential is not defined unambiguously . It is defined with an arbitrary gauge transforma- tion A i 7→ A i + ∂ f ∂ x i , where f is an y smo oth function. Let U b e some co ordinate neighbourho o d on the smo oth manifold M 5 . The 4-dimensional distribution A on U can b e defined by the differential form ω x = 3 X i =0 A i ( x ) dx i + dx 4 . (7) This definition is correct since at the co ordinate transformation with the prop erty (4) w e ha ve 3 X i =0 A i dx i + dx 4 = 3 X j =0 3 X i =0 A i ∂ x i ∂ y j + ∂ x 4 ∂ y j ! dy j + dy 4 = 3 X j =0 e A j dy j + dy 4 . (8) Therefore the co ordinate transformations on the 5-manifold lead to transformations of the 4-p oten tial e A j = 3 X i =0 A i ∂ x i ∂ y j + ∂ x 4 ∂ y j , (9) that include 4-dimensional coordinate transformations of the general relativity theory and gauge transformations of the 4-p oten tial of the electromagnetic field [6] as in [7]. Lo cally the distribution A can b e defined by the basis vector fields e i = ∂ ∂ x i − A i ∂ ∂ x 4 , i = 0 , . . ., 3. V ector fields on a manifold can b e considered as op erators of differen tiation. During the co ordinate transformation with the prop ert y (4) we hav e e e j = 3 X i =0 ∂ x i ∂ y j e i , (10) 2 THE EQUA TIONS OF MOTION OF A CHAR GED P AR TICLE 4 where e e j = ∂ ∂ y j − e A j ∂ ∂ y 4 , j = 0 , . . ., 3. Therefore the 4-dimensional basis of the distribution A is transformed (10) exactly the same wa y as the co ordinate vector fields  ∂ ∂ x i  i =0 ,... , 3 in the general relativity theory . The commutators of vector fields e i , i = 0 , . . ., 3, generate the tensor of the electromagnetic field: [ e i , e j ] = − F ij ∂ ∂ x 4 . If F ij 6 = 0, then the distribution A is completely nonholonomic. Allo wed curves on the distribution are absolutely contin uous and satisfy almost ev erywhere the horizontalit y condition ω γ ( t ) ( γ 0 ( t )) = 0 . (11) In this pap er w e consider the optimization problem for the length functional 2 on the 5- dimensional manifold with the 4-dimensional distribution A . W e prov e that the solutions of the v ariational problem for this distribution satisfy the equations of motion for the charged particle of the general relativity theory . W e consider also the geo desic sphere for the distri- bution A . The ge o desic spher e with the radius r and cen ter x 0 is the set of p oin ts at distance r from the p oin t x 0 . T o obtain the geo desic sphere w e consider the set of solutions of the v ariational problem starting at p oin t x 0 . W e constructed the 4-dimensional pro jection of the geo desic sphere for the particle mo ving in the constan t magnetic field. 2 The equations of motion of a c harged particle The optimization problem for the length functional on the set of the horizontal curv es con- necting t w o giv en p oin ts is one of the main problems of the theory of optimal con trol [15, 16]. The v elo cit y v ector x 0 ( t ) of horizon tal curv es b elong to the distribution A , hence w e consider the metric tensor of the distribution A . If the vector fields e i ( x ), i = 0 , . . ., 3 form the basis of the distribution A ( x ), then its metric tensor is g ij ( x ) = h e i ( x ) , e j ( x ) i x , i, j = 0 , . . ., 3. Consider the length functional J ( x ( · ) , u ( · )) = Z T t 0 L ( x ( t ) , u ( t )) dt, (12) where L ( x, u ) = m 3 X i,j =0 g ij ( x ) u i u j ! 1 / 2 (13) and m is the mass of the particle. Since the metric tensor has the signature (+ , − , − , − ), the velocity v ector dx dt b elong to the cone of future V ( x ) = { v ∈ A ( x ) | h v , v i x > 0 , v 0 > 0 } . This set dep ends of the p oin t x . Hence let us consider another problem. All spaces A ( x ) are linearly isomorphic. Therefore w e can consider the cone V ( x ) = { u ∈ R 4 | 3 P i,j =0 g ij ( x ) u i u j > 0 , u 0 > 0 } . F or some neighbourho o d of the p oin t x 0 the inersection of all sets V ( x ) is nonempty and has a nonempty interior Q . In this neigh b ourho o d of the p oin t x 0 consider another problem for which u ( t ) ∈ Q . If ( x ( t ) , u ( t )) is an optimal pro cess 2 Note that nonholonomic v ariational problem is different from the nonholonomic mechanical problem. 2 THE EQUA TIONS OF MOTION OF A CHAR GED P AR TICLE 5 for the problem (12), for whic h u ( t ) ∈ V ( x ( t )), then this pair is lo cally an optimal pro cess in the case u ( t ) ∈ Q . F rom the horizontalit y condition (11) follows that for the v elo cit y vector of the particle          dx k dt = u k , k = 0 , . . ., 3 dx 4 dt = − 3 X j =0 A j ( x ) u j . (14) Let us consider the maximization problem for the length functional on the set of horizon tal curv es connecting tw o giv en p oints x i ( t 0 ) = x i 0 and x i ( T ) = x i T , i = 0 , . . ., 4. The Hamilton – Pon trjagin function for the functional (12), (14) has the form H ( x, u, p, a 0 ) = − a 0 L ( x, u ) + 3 X j =0 p j u j − p 4 3 X j =0 A j ( x ) u j . (15) Due to Pon trjagin’s maxim um principle [17, 18] there is a 0 ≥ 0 and v ector-function p k ( t ), k = 0 , . . ., 4, t 0 ≤ t ≤ T , satisfying the conjugate system of linear differential equations dp k dt = − ∂ H ∂ x k     ( x ( t ) ,u ( t ) ,p ( t ) ,a 0 ) = a 0 ∂ L ∂ x k + p 4 3 X j =0 ∂ A j ∂ x k u j ( t ) , k = 0 , . . ., 4 , (16) where ( x ( t ) , u ( t )) is the optimal pro cess for the problem (12). Since the metric tensor and 4-p oten tial of the electromagnetic field do not dep end of the co ordinate x 4 , the appropriate momen tum comp onen t is the integral of motion (is conserved): p 4 ( t ) = const. F rom the equation (2) follo ws that p 4 is the length of the particle. F or almost an y t ∈ [ t 0 , T ] the function u 7→ H ( x ( t ) , u, p ( t ) , a 0 ) reaches its upp er b ound on the set Q at u = u ( t ). If this maxim um is reached at an inner p oin t, then ∂ H ∂ u k = 0, i.e. p k − p 4 A k = a 0 ∂ L ∂ u k , k = 0 , . . ., 3 . (17) Since the length functional do es not dep end on the parametrization of the path x ( · ), we can assume that the optimal control b elong to the pseudosphere L ( x, u ) = 1. The normal co vector for this pseudosphere has the co ordinates ∂ L ∂ u k , k = 0 , . . ., 3. The pro jection of the impulse vector p k on the subset ( u 0 , . . ., u 3 ) has the co ordinates p k − p 4 A k . The Hamilton function reaches its maxim um when the pro jection of the vector p k and the normal for the unit pseudosphere are collinear. The equation (17) is exactly the condition for these t w o v ector to b e collinear. T o exclude the vector p k from the equations (16) and (17) w e use the iden tity A k ( x ( t )) − A k ( x ( t 0 )) = Z t t 0 3 X j =0 ∂ A k ∂ x j u j dt 0 , k = 0 , . . ., 3 . (18) Then a 0  ∂ L ∂ u k − Z t t 0 ∂ L ∂ x k dt 0  + p 4 Z t t 0 3 X j =0 F j k u j dt 0 + p 4 A k ( x ( t 0 )) − p k ( t 0 ) = 0 . (19) 3 THE GEODESIC SPHERE FOR THE P AR TICLE IN THE MA GNETIC FIELD 6 P arameters a 0 and p k , k = 0 , . . ., 4, can b e multiplied by any p ositiv e constant. With a 0 = 1 w e obtain the equations of horizon tal geo desics on the distribution A . If we additionnaly assume contin uity of functions u ( t ), then d dt ∂ L ∂ u k − ∂ L ∂ x k + p 4 3 X j =0 F j k u j = 0 , k = 0 , . . ., 3 . (20) These equations are identical with the equations of motion of a particle with the c harge p 4 of the general relativity theory . F or a 0 = 0 w e obtain the equations of abnormal geo desics. Then at least one of the parameters p k , k = 0 , . . ., 4, should b e non-zero. Since in this case (17) p k = p 4 A k , k = 0 , . . ., 3, then p 4 6 = 0. The solutions of the v ariational problem for a 0 = 0 has the form 3 X j =0 F j k u j = 0 , k = 0 , . . ., 3 . (21) The solutions of the v ariational problem (12) – (14) which do not satisfy Euler – Lagrange equations are called abnormal geo desics [19, 20]. Mon tgomery constructed a distribution in 3-dimensional space for which the commutator of the basis vector fields is 2 x 2 ∂ ∂ x 3 [21, 22]. In his example the abnormal geo desics are streigh t lines parallel with the x 2 axis. In our mo del the equations of abnormal geo desics on the distribution A has the form (21). If det F = 0, then this equation can p osess a non-trivial solution. 3 The geo desic sphere for the particle in the magnetic field The geo desic sphere has a compicated structure [2, 23, 24]. In the follo wing part of the pap er w e assume that M is the linear space R 5 and the metric tensor of the distribution A is diagonal, g = Diag(1 , − 1 , − 1 , − 1). The equations of motion of a c harged particle in the electromagnetic field (20) in the space with the diagonal metric tensor has the form m du k dt − q 3 X j =0 F k j u j = 0 , k = 0 , . . ., 3 , (22) where m is the mass of the particle, q is the charge of the particle, F is the tensor of the electromagnetic field, u k are comp onen ts of the 4- velocit y v ector of the particle, dx k dt = u k , k = 0 , . . ., 3. The fiveth comp onen t of the v elo cit y v ector dx 4 dt is determined b y the horizon tality condition. Let us consider the motion of the particle in the magnetic field. Assume that the tension of the magnetic field H x 6 = 0, and the tension of the electric field is zero. Then F 23 = 3 THE GEODESIC SPHERE FOR THE P AR TICLE IN THE MA GNETIC FIELD 7 Figure 1: Pro jection of the geo desic sphere on the co ordinates ( x 2 , x 3 , x 4 ). − F 32 6 = 0, and other components of the tensor of the electromagnetic field F are zero. In this co ordinates the equations of motion and the horizontalit y condition (11) has the form                  m du 2 dt − q H x u 3 = 0 m du 3 dt + q H x u 2 = 0 dx 4 dt + 3 X k =0 A k u k = 0 . (23) Other comp onen ts of the 4-velocity vector are constan t. Assume that H x = const. Then from the equations (23) follo ws that in the magnetic field the c harged particle mov es along the helicoid line. Consider the pro jection of the geo desic sphere on the co ordinates ( x 2 , x 3 , x 4 ). In the plane ( x 2 , x 3 ) the particle mov es along a circle. Cho ose the following comp onen ts of the 4-p oten tial: A 0 = 0, A 1 = 0, A 2 = ϕx 3 , A 3 = 0. Then F 23 = − ϕ , H x = ϕ . Designate p = q m ϕ . Then the equations of motion of a particle in the subspace ( x 2 , x 3 , x 4 ) has the form                du 2 dt − pu 3 = 0 du 3 dt + pu 2 = 0 dx 4 dt + ϕx 3 u 2 = 0 . (24) The norm of the v elo cit y v ector along the geo desic is constan t. Assume that ( u 2 ) 2 +( u 3 ) 2 = 1. 3 THE GEODESIC SPHERE FOR THE P AR TICLE IN THE MA GNETIC FIELD 8 Then                x 2 ( t ) = − 1 p cos( α + pt ) + b 2 x 3 ( t ) = 1 p sin( α + pt ) + b 3 x 4 ( t ) = ϕ 4 p 2 ( − 2 pt + sin(2 α + 2 pt )) + ϕ p b 3 cos( α + pt ) + b 4 . (25) In the plane ( x 2 , x 3 ) the charged particle is moving along the circle with the center ( b 2 , b 3 ) and the radius 1 | p | . Let b 2 = (cos α ) /p and b 3 = − (sin α ) /p . Cho ose the constant b 4 so that x 4 (0) = 0. Then                x 2 ( t ) = 1 p (cos α − cos( α + pt )) x 3 ( t ) = 1 p (sin( α + pt ) − sin α ) x 4 ( t ) = ϕ 4 p 2 ( − 2 pt + sin(2 α + 2 pt ) − sin 2 α ) + ϕ p 2 sin α (cos α − cos( α + pt )) . (26) The length of the geo desic γ : [0 , t ] → R 5 is equal to t . The equations (26) define mapping ( t, α, p ) 7→ γ α,p ( t ). In the plane ( x 2 , x 3 ) the p oin t mak es one turn around the cen ter at | pt | = 2 π . More long geo desics ( | pt | > 2 π ) are not the shortest paths. Here one should consider the shortest paths, since the restriction of the metric tensor on the tangent bundle for the co ordinate plane ( x 2 , x 3 , x 4 ) is positively defined. The geo desic sphere of the radius s can b e defined by the follo wing form ula: B ( s ) = { γ α,p ( t )   α ∈ [ − π , π ] , t = s, | pt | ≤ 2 π } . (27) Fig. 1 shows the pro jection of the geo desic sphere of the radius s = 1 for the co ordinates ( x 2 , x 3 , x 4 ) at ϕ = 1, with p ∈ [ − 2 π , 0] on the left and p ∈ [0 , 2 π ] on the righ t. If | pt | ≤ 2 π , the geo desics are the shortest paths. The nonholonomic geodesic sphere of any radius has a pair of p oin ts such that if we contin ue the geo desic after this p oin t we enter the geo desic sphere with some lesser radius. If | pt | ≥ 2 π , then the geo desic of length t ceases to b e optimal path. All p ossible endp oin ts γ α,p ( t ) of the geodesics of length t reach the x 4 axis at pt = 2 π k , k ∈ Z , this p oin t b eing the intersection p oin t for the considered surface. The densit y of these p oin ts increases when we approach the origin. This phenomena is called the nonholonomic wa ve front (fig. 2). Fig. 2 shows the pro jection of the surface defined b y the geo desics of length s = 1 for the co ordinates ( x 2 , x 3 , x 4 ) at ϕ = 1 with p ∈ [ − 8 π , − π ] on the left and p ∈ [ π , 8 π ] on the righ t. These geo desics cease to b e optimal paths for | p | > 2 π . F or large enough p this surface b elongs to some cone with the ap ex at the origin. This cone has the axis of symmetry x 4 . Indeed for p large enough from the equations (26) follows that x 4 ( t ) = − ϕ 2 p s + O ( 1 p 2 ) ( p → ∞ ) . (28) In the plane ( x 2 , x 3 ) the particle mov es along the circle of the radius 1 | p | . The distance from the starting p oin t to the ending p oin t is changing from 0 to 2 | p | . F or p large enough the 4 DIST ANCE ALONG THE X 4 COORDINA TE 9 Figure 2: The non-holonomic wa v e fron t defined by non-optimal geo desics. geo desics of the length s b elong to the interior of the cone with the angle of inclination | ϕ | 4 s and the ap ex at the origin. 4 Distance along the x 4 co ordinate In the subspace ( x 2 , x 3 ) the particle mo v es along the circle. Let us find the v alues of t for whic h the end of the geo desic is ov er the starting p oin t in the subspace ( x 2 , x 3 , x 4 ). The end of the geo desic is ov er the starting p oin t for pt = 2 π k , k ∈ Z . Use the T aylor expansion for the equation (26) at the p oin t p = 2 π k t + θ , θ → 0: x 4 ( t ) = − ϕ t 2 4 π k + ϕ t 3 4 π 2 k 2 θ + O ( θ 2 ) ( θ → 0) . (29) Since t = s , the distance for whic h the particle mov ed along the x 4 co ordinate is of the order ϕ s 2 4 π k . F or distribution with the p ositiv ely defined metric tensor there is the theorem about the ball shap e [25, 26]. In the co ordinate neigh b ourho od of an y p oin t one can c ho ose the v ector fields e i whic h form the basis of the distribution A . The field of planes that span comm utators of all p ossible v ector fields of the distribution A is designated A 2 = [ A , A ]. One also consider A k +1 = [ A k , A ]. This sequence is stabilizing: there is the minim um m suc h that A m = A m +1 . The n umber m whic h can dep end of the point of the manifold is called the degree of nonholonomity of the distribution A . Designate n k = dim A k and define the function ϕ ( i ) = j , if n j − 1 < i ≤ n j . The function ϕ is defined on the set of n umbers { 1 , . . ., dim A m } with v alues at { 1 , . . ., m } . F or completely nonholonomic distribution on the Riemannian manifold there are the co ordinates x i suc h that the ball of accessibilit y is b ounded from ab o ve and from b elo w b y the set | x i | ≤ ε ϕ ( i ) . F or the tw o-dimensional distribution on the 3-dimensional manifold n 0 = 0, n 1 = 2, n 2 = 3. Therefore the function ϕ (1) = 1, ϕ (2) = 1, ϕ (3) = 2. 5 LA GRANGE F ORMULA TION F OR THE EQUA TIONS OF MOTION 10 5 Lagrange form ulation for the equations of motion 5.1 General case Let us consider the classical problem of the calculus of v ariations: ho w to find an absolutely con tinuous vector-function whic h maximizes the functional [2, 17] J ( x ( · ) , u ( · )) = Z T 0 L ( t, x ( t ) , u ( t )) dt, (30) where u ( t ) = dx dt . Assume that x (0) = x 0 , x ( T ) = x 1 , and allow ed paths satisfy the conditions ϕ i ( t, x, dx dt ) = 0, i = 1 , . . .k . Then there are k measurable and limited functions λ i ( t ) called Lagrange m ultipliers and a constan t a 0 ≥ 0, at least one non-zero and suc h that the function L λ ( t, x ( t ) , . x ( t )) = a 0 L ( t, x ( t ) , . x ( t )) + k X i =1 λ i ( t ) ϕ i ( t, x ( t ) , . x ( t )) (31) almost anywhere on [0 , T ] satisfies the Euler – Lagrange equations in the integral form ∂ L λ ( t, x ( t ) , . x ( t )) ∂ . x i ( t ) = Z T 0 ∂ L λ ( τ , x ( τ ) , . x ( τ )) ∂ x i dτ + c i , i = 1 , . . .n, (32) where c i are constants [17, p. 279]. The parameters a 0 , λ i can b e multiplied b y an y p ositiv e constan t. With a 0 = 1 we obtain regular geo desics. With a 0 = 0 we obtain abnormal geo desics. No w assume (for the general case) that the distribution B is defined by the family of differen tial forms ω j , j = m +1 , . . .n . The horizontalit y conditions hav e the form ω j ( u ) = 0 , j = m +1 , . . .n, . x = u. (33) Then the Lagrange function for the considered problem is L λ ( x, u ) = a 0 h u, u i 1 / 2 + n X j = m +1 λ j ω j ( u ) , (34) where h· , ·i is some non-degenerate 3 bilinear form on T M n . The equations of the horizontal geo desics are a 0 h D u dt , · i + n X j = m +1 . λ j ω j + n X j = m +1 λ j dω j ( u, · ) = 0 , (35) where D u dt is the cov ariant deriv ative of the v elosit y v ector of the particle along the path. The velocity vector can b e decomp osed by the basis v ector fields of the distribution: u ( t ) = m X k =1 v k ( t ) ξ k ( x ( t )) . (36) 3 This assumption for our mo del is not necessary , as shown b elo w. It is enough for the r estriction of the metric tensor on the distribution to b e non-degenerate. 5 LA GRANGE F ORMULA TION F OR THE EQUA TIONS OF MOTION 11 Then the co v ariant deriv ativ e is D u dt = m X k =1 dv k dt ξ k + m X i,j =1 v i v j ∇ ξ i ξ j , (37) where ∇ is the symmetric connection defined by the bilinear form. Use the Maurer – Cartan form ula dω ( ξ , η ) = ξ ω ( η ) − η ω ( ξ ) − ω ([ ξ , η ]) . (38) Since ω j ( ξ i ) = 0, i = 1 , . . .m , j = m +1 , . . .n , then for the velocity v ector and the basis of the distribution dω j ( u, ξ i ) = − ω j ([ u, ξ i ]). This comm utator can b e rewritten using the structural constants of the distribution B : [ u, ξ i ] = m X k =1 v k [ ξ k , ξ i ] = m X k =1 v k n X s =1 c s ki ξ s , (39) where [ ξ k , ξ i ] = n P s =1 c s ki ξ s . This basis and the differen tial forms can b e selected suc h that ω j ( ξ i ) = δ j i , i, j = m +1 , . . .n . Then for the distribution B a 0 h D u dt , ξ l i + n X j = m +1 λ j m X k =1 c j lk v k = 0 , l = 1 , . . .m. (40) Since ω j ( ξ i ) = δ j i , i, j = m +1 , . . .n , then for the v elo cit y v ector and all basis vector fields of the tangen t space dω j ( u, ξ i ) = − ω j ([ u, ξ i ]). Hence w e can write the pro jection of the equation (35) at the basis vectors whic h do not b elong to the distribution: a 0 h D u dt , ξ l i + . λ l + n X j = m +1 λ j m X k =1 c j lk v k = 0 , l = m +1 , . . .n. (41) 5.2 Lagrange form ulation for the distribution A in our mo del The distribution A is defined by the differential form ω x = 3 P i =0 A i ( x ) dx i + dx 4 . The Euler – Lagrange equations for the length functional (12) with the condition ω γ ( t ) ( γ 0 ( t )) = 0 are a 0 h D . γ dt , · i + . λω + λdω ( . γ , · ) = 0 , (42) where the co v ariant deriv ativ e D . γ dt = 3 X k =0 dv k dt e k + 3 X i,j =0 v i v j ∇ e j e i , (43) ∇ is the symmetric connection defined by the metric tensor. Please note that in the Lagrange metho d w e ha ve to use the metric tensor defined on the whole tangen t space T M whereas in the maximum principle we used the restriction of the metric tensor on the distribution only . REFERENCES 12 This is no problem b ecause in the result we use the comp onen ts of the metric tensor of the distribution only . W e can also use the connection (Christoffel symbols) with all lo wer indexes and in this case the bilinear form do es not ha ve to b e non-degenerate. The velocity vector . γ ( t ) = 3 P k =0 v k ( t ) e k ( γ ( t )). The Euler – Lagrange metho d also has the constant a 0 ≥ 0 [17, p. 279]. The parameters a 0 , λ can b e multiplied by any p ositiv e constan t. With a 0 = 1 w e obtain the equations of regular geo desics. With a 0 = 0 we obtain the equations of abnormal geo desics. Pro jecting the equations of motion on the basis of the distribution A w e obtain a 0 h D . γ dt , e i i + λ 3 X j =0 c 4 ij v j = 0 , i = 0 , . . ., 3 . (44) The structural constan ts are defined b y [ ξ i , ξ j ] = 4 P l =0 c l ij ξ l , where ξ j = e j , j = 0 , . . ., 3, ξ 4 = ∂ 4 . F or the distribution A [ e i , e j ] = − F ij ∂ 4 , [ e j , ∂ 4 ] = 0. Therefore c 4 ij = F j i , and other structural constan ts are zero. Since ω x ( ∂ 4 ) = 1, then a 0 h D . γ dt , ∂ 4 i + . λ = 0 . (45) W e can assume that h· , ∂ 4 i = 0. Then λ = const, and λ can be interpreted as the c harge of the particle (or charge to mass ratio dep ending on the Lagrangian). Note that applying the maximum principle in section 2 w e used the restriction of the metric tensor on the distribution only . Since this restriction is non-degenerate, w e can rise appropriate indexes of the Christoffel syb ols and get ordinary differen tial equations. The result do es not dep end on the extension of the metric tensor on the whole tangen t space. All figures published in this paper were pro duced using our o wn 3D graphics program c  V.R. Krym. References [1] Gromov M. Carnot-Car athe o dory Sp ac es Se en F r om Within. Preprint IHES/M/94/6 (Institut des Hautes Etudes Scien tifiques, 1994) [2] V ershik A.M., Gershko vich V.Y a. The Nonholonomic Dynamic al Systems. 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