On singular lagrangians and Dirac s method

Transcription

1 INVESTIGACIÓN Revista Mexicana de Física 58 ( FEBRERO 01 On singular lagrangians and Dirac s method J.U. Cisneros-Parra Facultad de Ciencias, Universidad Autonoma de San Luis Potosi, Zona Uniiversitaria, San Luis Potosi 7890, Mexico. Recibido el de septiembre de 011; aceptado el 9 de diciembre de 011 We show some instances of singular lagrangians from the classical mechanics of particles and apply Dirac s method for building the canonical equations. We find then the reason for the singularity, and therefore, we get the Hamilton equations with the familiar procedure, that is without the need of Dirac s procedure. Known cases of singular lagrangians in special relativity are also presented, and their non-singular alternatives. Keywords: Singular lagrangian; classical mechanics; special relativity. Se presentan algunos lagrangianos singulares del ámbito de la mecánica clásica de partículas, y se les aplica el método de Dirac para construir las ecuaciones canónicas. Se halla la razón de la singularidad, y, con ello, se obtienen las ecuaciones de Hamilton por el camino acostumbrado, esto es, sin necesidad del método de Dirac. Se presentan también casos conocidos de lagrangianos singulares en la relatividad especial y sus alternativas no singulares. Descriptores: Lagrangiano singular; mecánica clásica; relatividad especial. PACS: 45.0.Jj; Ef 1. Introduction The transition from the lagrangian to the hamiltonian formalism is carried out by expressing the generalized velocities q i (i=1,..., n in terms of the momenta p j (q, q, t= L/ q j, and eliminating them in the function H = p q L. This is possible if the mathematical condition p i q j = L q i q j 0, (1 is satisfied. This signifies that they build a set of independent variables. But if the determinant vanishes there exists one or more relations between the p s: φ k (p, q, t = 0, k = 1,..., α, ( where n α is the rank of the matrix ( L/ q i q j. Thus not all p s are independent. In such situation one says that the lagrangian is degenerated or singular, and the Hamilton equations of motion cannot be obtained by the familiar procedure. In an attempt to generalize the hamiltonian dynamics, Dirac [1-3] developed a method for building the canonical equations starting from the complete hamiltonian H = H 0 + v k φ k, (3 where H 0 = q l p l L depends on the coordinates and the independent p s, and v k are new independent variables. This comes from taking a virtual variation of H 0 ([7]: or δh 0 = ( q i δp i L δq i = ( q i δp i ṗ i δq i, (3a ( q i H ( 0 δp i ṗ i + H 0 δq i = 0, p i (3b for all δp i, δq i, consistent with the restrictions: n ( φk δq i + φ k δp i = 0, k = 1,..., α, (3c p i i=1 that is, α δ s of all δp i, δq i depend on the remaining ones. Eliminating them from Eq. (3b by the well-known multiplier s procedure, one has or briefly with q i = H 0 p i ṗ i = H 0 q i = H p i, + v k φ k p i, v k φ k, i = 1,..., n, (3d ṗ i = H, i = 1,..., n, (3e H = H 0 + v k φ k. Dirac then imposes to the primary restrictions φ k the consistency conditions φ k = 0, from which one can obtain additional restrictions. Some of these can be identities (0 = 0, others of the form f m (q, p = 0 (like functions (, and others of type g l (q, p + v l h l (p, q = 0, that can be used to fix some of the unknown variables v k. The second possibility is treated in a similar way as conditions φ k = 0.. Cases of singular lagrangians It is remarkable that most classical mechanics textbooks do not treat the topic on singular lagrangians, or if they do they lack on a discussion of some specific cases (not even artificial examples, although the aim of generally building the appropriate canonical equations have had a considerable

2 6 J.U. CISNEROS-PARRA development, since the beginning of the way initiated by Dirac [1]. But self in specialized papers one seldom finds examples of systems with such behavior. Here, we write down a set of particular lagrangians of a special type: L= 1 m( q 1 + q + l q 3 + l q 1 q 3 cos q 3 + l q q 3 sin q 3 + V (q1, q, q 3, (4 where l and m are constants, the Mittelstaedt s lagrangian [4] L = 1 m ( q 1 + q + 1 µ q 3 + V (q 1, q, q 3, (5 that of Cawley ([5] L = q 1 q + V (q 1, q, q 3 the lagrangian of Deriglazov [6] ( V = 1 q q3, (6 L = q q 1 + q 1 q + q 1 q q 1 q + V (q 1, q (7 (V = q 1 + q. They all have in common that the potential energy V depends only on the coordinates of the system, and that they are singular. Certainly, it is not difficult to see that there exists one relation between the p s in each instance: φ 1 = p 3 l p 1 cos q 3 l p sin q 3 = 0, φ 1 = p p 1 = 0, φ 1 = p 3 = 0, φ 1 = q 1 p 1 q p = 0. (4a (5a (6a (7a Thus in these cases one cannot arrive at the canonical equations of motion using the well-known procedure, and we are forced to use Dirac s method (see however, Sec Dirac s method for lagrangians (5 and (7 Actually, we will only give the details for lagrangian (5 because the results for the other are found in the reference [6]. We start obtaining the momenta of the system, using Eq. (5: p 1 = 1 m ( q 1 + q, p = 1 m ( q 1 + q, p 3 = 1 µ q 3, (8 so, p depends on p 1 and only p 1 (or p and p 3 are independent. The primary restriction is then Eq. (5a p p 1 = 0. For getting H 0 (Eq. (3, we eliminate the velocities from qi p i L in favor of the independent p s, resulting The complete hamiltonian is then H 0 = m p 1 + µ p 3 V. (9 H = m p 1 + µ p 3 V + v (p p 1. (10 The consistency condition φ 1 = ṗ ṗ 1 = 0 leads to the secondary restriction φ = V q 1 V q = 0. (11 This is a relation between q 1, q and q 3 which we briefly write as φ = q F (q 1, q 3 = 0. (1 We then build the consistency condition φ = 0, or from which we find φ = [φ, H] = 0, v (1 + F,1 m p 1 F,1 µ p 3 F,3 = 0 ( F,i F. (13 [φ, H] is the Poisson bracket of φ and H. Eq. (13 allows fixing variable v: v = m p 1 F,1 + µ p 3 F,3 1 + F,1. (14 With the additional relations (1 and (14, we can now write the canonical equations of motion: with q 1 = mp 1 v, q = v, q 3 = µp 3, ṗ 1 = V,1, ṗ = V,, ṗ 3 = V,3, (15 q = F (q 1, q 3, v = m p 1 F,1 + µ p 3 F,3 1 + F,1. Thus, the independent equations of motion are q 1 = m p 1 µ p 3 F,3 1 + F,1, q 3 = µp 3, ṗ 1 = (V,1 q =F, ṗ 3 = (V,3 q =F. (16 Eqs. (16 can easily be written in newtonian form: 1 + F,1 q 1 + F,3 q 3 + F,11 q 1 + F,13 q 1 q 3 + F,33 q 3 = m (V,1 q =F, (17 q 3 = µ (V,3 q=f. (18 On the other hand, for Deriglazov s lagrangian (7 it is found that and H = p 1 4 q V (q 1, q + v (q 1 p 1 q p, (19 φ = q 1 V,1 q V, = 0, or φ = q F (q 1 = 0, p 1 v = F (F + q 1 F,1 F,1. Rev. Mex. Fis. 58 (

3 ON SINGULAR LAGRANGIANS AND DIRAC S METHOD 63 Therefore, the independent canonical equations are p 1 q 1 = F, + q 1 F F,1 ṗ 1 = p 1 F (F + q 1 F,1 F,1 + (V,1 q =F (q 1, (0 and from here one also gets the Newton s equation of motion (Deriglazov uses V (x, y = x + y and F (x = ±x F (F + q 1 F,1 q 1 + F (F,1 + q 1 F,11 q 1 (V,1 q=f (q 1 = 0, (1 4. An alternative procedure to arrive to the equations of motion The particular cases here considered, imply the relations φ 1 = 0 (4a to (7a between the momenta. Without regarding the hamiltonian formalism, we can deduce the consequence of such relations. For lagrangian (4 the p s are given by p 1 = m q 1 + m l q 3 cos q 3, p = m q + m l q 3 sin q 3, ( p 3 = m l q 3 + m l ( q 1 cos q 3 + q sin q 3, from which, we know, follows Eq. (4a. If we now take the time derivative of Eq. (4a, substitute there p 1 and p from Eqs. ( and take into account Lagrange s equations, we write V,1 l cos q 3 + V, l sin q 3 V,3 = 0. (3a In a similar way, the implication of φ 1 = 0 for the remaining cases is V,1 = V, or q = F (q 1, q 3, (3b V, = 0, (3c q 1 V,1 = q V, or q = F (q 1, (3d Equations (3a-(3d are relations between the coordinates of each system, thus one coordinate cannot be independent. In these cases, the reason for the lagrangian to be singular is that the coordinates are not independent (see Appendix, and so the canonical equations cannot be obtained by the familiar procedure, in which it is necessary that the coordinates be generalized (independent. Therefore, eliminating one of the coordinates from the corresponding lagrangian, it would be possible to build straightforwardly the Hamilton s equations. Let us do it for lagrangians (5 and (7. with Substituting Eq. (3b into (5 we get L = 1 m (1 + F,1 q ( 1 m F,3 + 1 µ + 1 m (1 + F,1 F,3 q 1 q 3 + V, (4 V (q 1, q 3 = V ( q 1, F (q 1, q 3, q 3. Likewise, the substitution of Eq. (3d into (7 leads to L = (q 1 F,1 + F q 1 + V (q 1, V (q 1 = V (q 1, F (q 1, H = p 1 4 (F + q 1 F,1 V (q 1. (5 Let us write the equation of motion for Deriglazov s lagrangian (5, dp 1 /dt = L/ q 1 : (F + q 1 F,1 q 1 + (F + q 1 F,1 ( F,1 + q 1 F,11 q 1 V,1 = 0. (6 This equation is exactly the same as Eq. (1. This can be seen from Eq. (3d that we write at q = F (q 1 : so that q 1 (V,1 q=f (q 1 = (q V, q=f (q 1, (7 (V,1 q =F (q 1 = 1 q 1 F (V, q =F (q 1, (8 V,1 = 1 q 1 (F + q 1 F,1 (V, q=f (q 1, (9 and thus factor F cancels out from Eq. (1, and F + q 1 F,1 from Eq. (6. Regarding (5 we get, after substituting (3b into (5, L = 1 m A q 1+ ( B m + 1 µ where we have done the abbreviations q 3+ A B m A = 1 + F,1, B = F,3, q 3 q 1q 3 +V, (30 V (q 1, q 3 = V ( q 1, q = F (q 1, q 3, q 3. (31 The two momenta and the generalized velocities are then given by p 1 = L = A q 1 m q 1 + A B m q 3, p 3 = A B ( B m q 1 + m + 1 q 3, (3 µ q 1 = m + µ B A p 1 µ B A p 3, q 3 = µ B A p 1+µ p 3. (33 Thus the hamiltonian is H = m A p 1 + µ A (B p 1 A p 3 V (q 1, q 3. (34 Rev. Mex. Fis. 58 (

4 64 J.U. CISNEROS-PARRA We can now easily write the canonical equations of motion: q 1 = m + µb A p 1 µb A p 3, q 3 = µ B A p 1 + µ p 3, ṗ 1 = V,1 + m A 3 A,1 p 1 µ A (B p 1 A p 3 ṗ 3 = V,3 + m A 3 A,3 p 1 µ A (B p 1 A p 3 (B,1 p 1 BA A,1 p 3, (35 (B,3 p 1 BA A,3 p 3. From here we come to the equations of motion for q 1 and q 3 by eliminating p 1 and p 3 in the two last equations (35. For this purpose, we derive Eqs. (3 with respect to t and substitute the result in (35. We get, after solving for q 1 and q 3 and taking into account that the equations A,3 = B,1 = F,13, (36 A q 1 + A,1 q 1 + B,3 q 3 + A,3 q 1 q 3 By Eq. (3b, regarding that + µ B V,3 m + µ B V,1 = 0, (37 A q 3 µv,3 + µ B A V,1 = 0. (38 (V,1 q=f = (V, q=f, we can see that Eqs. (37 and (38 are fully equivalent to Eqs. (17 and (18. For the cases presented here we then see that even though there exists a relation between the momenta, it is not necessary to apply Dirac s method for building the canonical equations. We can continue using the conventional procedure, without the need of invoking any generalization of the dynamics. These lagrangians are of the type i L = L 0 (Q m, Q m + V (Q, Q m which is known to be singular. Of course, this does not change the fact that they are so because one uses more coordinates than there are degrees of freedom. Lowering the number of coordinates accordingly, the problems reduce to ordinary ones (see Appendix. Moreover, restrictions (3b and (3d are not set on the fly, rather they are a consequence of the way we build the lagrangian. We can summarize the results in other terms. If we interpret the velocity dependent part of (4 to (7 as the kinetic energy of the system ii, T = 1 ( ds m = 1 dt mg ij q i q j, where g ij are the components of the metric tensor, and sum over repeated indices is understood, then the volume element in the space of the system can be written as dτ = g mn dq 1 dq dq 3, (39 where g mn is the determinant of the metric tensor. But if, as it is here the case, Eq. (1 is violated, then the volume element vanishes, and thus the system is restricted to a space of lower dimension (e.g. a surface. 5. Relativistic lagrangians There are several possibilities to build the free particle relativistic lagrangian that reduce to the classical expression in the limit c. For instance, for the one dimensional motion is L = m c c q, (40 L m c + 1 m q when the velocity of the particle is much smaller than c. The corresponding hamiltonian is, therefore H = p q L = c p + m c p + m c = c p + m c, (41 where p = m c q c q, q = c p p + m c. One tries to come in another way to the hamiltonian by using the proper time τ of the particle: c dτ = c dt dq, (4 instead of the coordinate time t. q and t are then functions of the parameter τ: q(τ, t(τ, so that the lagrangian now is where L = m c c t q, (43 t = dt dτ, q = dq dτ. For the lagrangian (43 we can construct two momenta p 0 and p, given by p 0 = L m c 3 t t = c t q, p = L q = m c q c t q. (44 It is not difficult to see that there exists a relation between them: p 0 = c p + m c 4, (45 Rev. Mex. Fis. 58 (

5 ON SINGULAR LAGRANGIANS AND DIRAC S METHOD 65 so that (43 is singular. This lagrangian is peculiar in a certain sense. For all lagrangians of the form L(t, q = F (c t q, (46 where F is an arbitrary function of the invariant c t q, the only function F that violates Eq. (1 is just the square root. Indeed, the determinant (1 for the function (46 is L q i q j = 4c F ( F + ( c t q F, (47 where F is the derivative of F with respect of its argument, so that the determinant is zero for a function F satisfying the equation F + ( c t q F = 0, (48 that is F (x = a x + b, (49 where a and b are constants. (43 is in this sense the worst choice one can take, much in the same manner as the construction of lagrangian (A3 in the Appendix. If one should have started with the relativistic covariant Newton s second law m d q i dτ = m dq i dτ = 0, (50 where q 0 = c t, q 1 = q, and the line element is given by ds = (dq 0, dq, and the metric tensor g ij has components g 00 = 1, g 11 = 1, g 10 = g 01 = 0, (51 one would have arrived at L = 1 m (c t q, (5 that is certainly not singular. With lagrangian (5 one can directly get the hamiltonian by the familiar procedure: H = p 0 m c p m. (54a The equations of motion are according to (43 c 3 mq (q t t q (c t q 3/ = 0, (53 c 3 mt (q t t q (c t q 3/ = 0, (54 and they clearly reduce to only one equation, from which it follows q = c 1 t + c, (55 a relation between q and t. On the contrary, from (5 one get the equations t = 0, q = 0, (56 or t = a 1 τ + b 1, q = a τ + b. (57 Lagrangian (40 describes a relativistic particle if we demand it to be real, so that v < c. In the case represented by Eq. (54, one can add the condition v < c for completeness, or demand that the proper time τ (appearing in Eq. (59, for example must be real. We are not diminishing the interesting properties of lagrangian (43, like invariance, parametrization independence iii, rather we are only showing here the consequences for the existence of a relation between momenta, and how can one overcome it without the necessity of generalize the classical dynamics. There is another example of a (relativistic singular lagrangian, namely ([6], we write it here for a one dimensional motion L = 1 q ( q 0 q m q, (58 where q 0 = q 0 (τ, q 1 = q 1 (τ, q = q(τ are the unknowns and m is a constant. L is singular because L q = 0, or p = 0 (59 and this is a relation between p s. On the other hand, the equations of motion are ( ( d q0 d q1 = 0, = 0, dτ q dτ q 1 q ( q 0 q 1 m = 0, (60 from which the third, that is a consequence of Eq. (59, can be solved for q(τ: q = ± 1 q 0 m q 1. (61 The first two Eqs. (60 can thus be expressed in the form ( ( d d = 0, dτ dτ q 0 q 0 q 1 q 1 q 0 q 1 = 0, (6 and they are equivalent to the equations of motion resulting from (43. According to Deriglazov, Dirac s method applied to (60, leads to the hamiltonian H = q (p 0 p 1 m + v p, (63 and hence the canonical equations are q i = q p i, ṗ i = 0, q = v, ṗ = 0, i = 0, 1, (64 with the conditions (primary and secondary p = 0, p 0 p 1 m = 0. (65 The secondary condition is similar to the primary one (45 for lagrangian (43. Rev. Mex. Fis. 58 (

6 66 J.U. CISNEROS-PARRA The canonical equations of motion (64 contain an undetermined variable v, that equals q. One can intend to fix it employing the Eqs. (64. From Eqs. (64 one sees that p 0 and p 1 are constant, so that or q 1 = p 1 p 0 q 0 = A q 0, A = constant, (66 q 1 = A q 0 + B, (67 where B is an arbitrary constant. On the other side, variable q can be written as from which it follows and hence q = 1 A m q 0, (68 q = ± v = ± 1 A m q 0, (69 1 A m q 0, (70 Of course, from the canonical equations of motion q 0 and q 1 cannot be determined as functions of τ, so that v, like q, remains undetermined. For lagrangians (43 and (58, one cannot avoid the use of Dirac s method for constructing the hamiltonian, not even by employing the restrictions as was done in Sec. 4. In the case of (43 the alternative is to take a different lagrangian, for instance that given by Eq. (5. FIGURE 1. Two ends of the springs are fixed at x = l 1 and x = l. The other ends are joined at a point, whose coordinates are (x, y at time t. The pendulum is inclined ϑ at this time. The position of the mass is given by (x 1, y 1. Appendix A. Singular lagrangian of a physical system The system in the plane shown in Fig. 1 consists of two massless springs of lengths l 1 and l and constants k 1 and k with ends fixed at x = l 1, y = 0 and x = l, y = 0, the other ends being joined at the free point (x, y where a pendulum of length l and mass m hangs. In setting the newtonian equations of motion for the mass m one takes into account that at the point (x, y 6. Conclusions F 1 + F + τ = 0, (A1 In the classical mechanics of particles, there is no case reported of a singular lagrangian for a real system; all instances that we know are of artificially built systems. Even that of the system described in the Appendix is really not singular. Thus, it seems that the lagrangians of classical mechanics are basically non degenerate. Singularities appears first when we treat to generalize to cases where there is not a previously given rule for building L, like in the special relativity. There one has the freedom to choose the lagrangian among several possibilities, some of which are regular and others singular. One does not care too much about this because one have a method, Dirac s method, for working out the problem, even at the expense of frequently introducing undetermined variables like the v s. Perhaps it would be more natural to set the condition on new lagrangians to be regular. One can argue against this that the additional variables v that appear in the theory can reveal symmetries of the system, like gauges. However, if two lagrangians, one regular and the other singular, lead to the same set of equations (for example, field equations, they must share comparable symmetries. where F 1 and F are the forces exerted by each spring and τ is the tension of the string. This implies that the resultant of F 1 and F must have the same inclination ϑ as the string, that is tan ϑ = F x, (A F y where F x = F 1x + F x and F y = F 1y + F y. The equilibrium condition (A can also be deduced directly from the Lagrange s equations of motion, for which the lagrangian is given by L = 1 (ẋ m + ẏ + l ϑ + l ẋ ϑ cos ϑ + l ẏ ϑ sin ϑ V (x, y m g (l cos ϑ + y. V (x, y stands for the potential energy of both springs: V (x, y = 1 k 1 (r 1 l k (r l, (A3 (A4 r 1 = (l 1 + x + y, r = (l x + y. Rev. Mex. Fis. 58 (

7 ON SINGULAR LAGRANGIANS AND DIRAC S METHOD 67 It is not difficult to show that the momenta p x = m ẋ + m l ϑ cos ϑ, p y = m ẏ + m l ϑ sin ϑ, p ϑ = m l ϑ + m l (ẋ cos ϑ + ẏ sin ϑ, are not independent, but satisfy the relation p x l cos ϑ + p y l sin ϑ p ϑ = 0. (A5 (A6 The consequence of restriction (A6 on the p s can be found by deriving it by t (and taking into account that ṗ = L/ q: m g l sin ϑ = ṗ ϑ = F x l cos ϑ + (F y + m g l sin ϑ + l ϑ ( p x sin ϑ + p y cos ϑ, (A7 where F x and F y are the components of the net force of the springs. Substituting now here p x and p y as given by Eqs. (A5, one gets that is F x cos ϑ + (F y + m g sin ϑ = m g sin ϑ, tan ϑ = F x F y, (A8 and it fully agrees with the equilibrium condition (A. Since F x = V (x, y V (x, y, F y =, x y are certain functions of (x, y, expression (A8 is a relation between x, y and ϑ, so that alone two of the three coordinates are independent. Our system has only (obviously two degrees of freedom. In other words, the existence of a relation of p s in the present case implies the presence of a restriction in the coordinates. Of course, because the system has only two degrees of freedom, we might as well have used the two obvious coordinates x 1 and y 1 for characterizing the position of the mass point m. The potential energy V (x, y depends on the point (x, y and needs to be expressed in terms of (x 1, y 1 by the relations (see Fig. 1 x 1 = x + l sin ϑ, y 1 = y + l cos ϑ, (A9 where ϑ is a given function of (x, y (Eq. A, what can be done by inverting the relation (A9. Perhaps it is simpler to take x and y as independent variables and x 1 and y 1 depending on them trough Eqs. (A9. Then, lagrangian is transformed into L 1 = L 1 (x 1, y 1, ẋ 1, ẏ 1, L(x, y, ẋ, ẏ = L 1 ( x1 (x, y, y 1 (x, y, ẋ 1 ((x, y, ẋ, ẏ, ẏ 1 (x, y, ẋ, ẏ. (A10 (A11 By the well-known property, the coordinate transformation leaves invariant Lagrange s equations. Thus, variables x and y are well suited for building the equations of motion (lagrangian (A9 as x 1 and y 1. On the other side, the function H 0 that does not contain the dependent momentum p ϑ is, taking into account Eq. (A6, H 0 = p x + p y m + V (x, y + m g (l cos ϑ + y, (A1 and so, the complete hamiltonian is given by (Eq. (3 H = p x + p y m + V (x, y + m g (l cos ϑ + y + v (p x l cos ϑ + p y l sin ϑ p ϑ. (A13 Thus, specially, the velocities are ẋ = p x + lv cos θ m ẏ = p y + lv sin θ, ϑ m = v. (A14 The one-dimensional version of the system shown in Fig. 1 is a horizontal spring of length l 1 and constant k 1 in series with another of length l and constant k, with a mass m attached at its end. The mass has the coordinate x relative to the joint of the springs, which has the coordinate x 1 with respect to the fixed end of the first spring. Thus, the lagrangian is given by L = 1 m (ẋ 1 + ẋ V (x 1, x, where the potential energy V is expressed as V (x 1, x = 1 k 1 (x 1 l k (x l. (A15 (A16 (A15 is singular (because the additional condition at the union point, with the relation between the momenta φ = p p 1 = 0. (A17 i. For lagrangian (5 Q 1 = q 1 + q, Q 3 = q 3, Q = q, whereas for lagrangian (7 Q 1 = q 1 q, Q = q. ii. As seen in the Appendix, for lagrangian (4 this is really the case, for (5, (6 and (7 we cannot assure that, because we do not know the physical system, they refer to. iii. Generally, parameters are not observable quantities, so that one prefers parameter independent theories. However, there are interesting procedures in mechanics, which contain unobservable variables, for example, Lagrange s treatment of a constrained system through Lagrange multipliers, that are not observable, Rev. Mex. Fis. 58 (

8 68 J.U. CISNEROS-PARRA the mechanics of Hertz, the Kaluza-Klein theory; even Dirac s method introduces variables v (Eq. (3 that are not observable. 1. P.A.M. Dirac, Canadian Journal of Math. ( P.A.M. Dirac, Proc. Roy. Soc. London A 46 ( P.A.M. Dirac, Lectures on Quantum Mechanics (Belfer Graduate School of Science, Yeshiva University, P. Mittelstaedt, Klassische Mechanik (Bibliographishes Institut, Hochschultaschenbücher Verlag, R. Cawley, Phys. Rev. Letters 4 ( A. Deriglazov, Classical Mechanics Hamiltonian and Lagrangian Formalism (Springer Verlag, M. Henneaux, C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Rev. Mex. Fis. 58 (