Point symmetries of the Euler Lagrange equations

Transcription

1 RESEARCH Revsta Mexcana de Físca MARCH-APRIL 2014 Pont symmetres of the Euler Lagrange equatons G.F. Torres del Castllo Departamento de Físca Matemátca, Insttuto de Cencas, Unversdad Autónoma de Puebla, Puebla, Pue., Méxco. Receved 6 August 2013; accepted 7 January 2014 We gve an elementary dervaton of the equatons for the pont symmetres of the Euler Lagrange equatons for a Lagrangan of a system wth a fnte number of degrees of freedom. We show that gven a dvergence symmetry of a Lagrangan, there exsts an equvalent Lagrangan that s strctly nvarant under that transformaton. The correspondng descrpton n the Hamltonan formalsm s also nvestgated. Keywords: Lagrangans; symmetres; equvalent Lagrangans; constants of moton; Hamltonan formalsm. Damos una dervacón elemental de las ecuacones para las smetrías puntuales de las ecuacones de Euler Lagrange para una lagrangana de un sstema con un número fnto de grados de lbertad. Mostramos que dada una smetría hasta una dvergenca de una lagrangana, exste una lagrangana equvalente que es estrctamente nvarante bajo esa transformacón. Tambén se nvestga la descrpcón correspondente en el formalsmo hamltonano. Descrptores: Lagranganas; smetrías; lagranganas equvalentes; constantes de movmento; formalsmo hamltonano. PACS: Jj; Hq; Sv 1. Introducton One of the advantages of the Lagrangan formalsm n classcal mechancs s that, roughly speakng, each contnuous symmetry of the Lagrangan functon of a system can be related to the exstence of a constant of moton. However, usually, ths relatonshp s not fully exploted, and s employed only n connecton wth smple geometrcal transformatons, such as translatons, rotatons, and tme dsplacements. In the case of a system wth one degree of freedom, a constant of moton amounts to a frst-order ordnary dfferental equaton ODE so that, wth the ad of a constant of moton, nstead of havng to solve a second-order ODE, one only has to solve a frst-order ODE. When the number of degrees of freedom s greater than 1, any constant of moton also helps to smplfy the equatons of moton. The strct varatonal symmetres of a Lagrangan Lq, q, t = 1, 2,..., n, are the pont transformatons, q = q q 1,..., q n, t, t = t q 1,..., q n, t, that leave the acton ntegral t 1 nvarant, that s, t 0 L Lq, dq, t = Lq, dq, t, 1 wth / = t / t+ q t / here and henceforth there s summaton over repeated ndces. The one-parameter groups of strct varatonal symmetres are determned by the frst-order lnear partal dfferental equaton PDE η + dη q dξ + t ξ + L dξ = 0, 2 where η q j, t and ξq, t are n + 1 unknown functons see, e.g., Refs. [1 6] note that, e.g., dη/ = η/ t + q η/. A nontrval soluton of ths equaton yelds the constant of moton ϕq, q, t = η ξ q L. 3 q A wder class of varatonal symmetres, also related to constants of moton, s formed by the one-parameter famles of pont transformatons, q = q q 1,..., q n, t, s, t = t q 1,..., q n, t, s, such that Lq, dq, t = Lq, dq, t + d F q, t, s, 4 for all values of the parameter s for whch the transformaton s defned, where F s some functon of q, t, and s only. These transformatons are sometmes called Noether symmetres [2], or dvergence symmetres [3], but t seems more approprate to call them Noether Bessel-Hagen symmetres [7]. From Eq. 4 t follows that a set of functons ξq, t, η q j, t generates a one-parameter group of varatonal symmetres of L f there exsts a functon Gq, t defned up to an addtve trval constant such that η + dη q dξ + t ξ + L dξ = dg, 5 [cf. Eq. 2]. The functon G s equal to the partal dervatve of F wth respect to s, at s = 0, assumng that at s = 0 the transformaton reduces to the dentty. In ths case, n addton to ξ and the η, one has to fnd G. The constant of moton assocated wth a soluton of Eq. 5 s ϕq, q, t = η ξ q L q G. 6 Even though t s more complcated to solve Eq. 5 than Eq. 2, for some Lagrangans Eq. 5 leads to many more

2 130 G.F. TORRES DEL CASTILLO constants of moton than Eq. 2 see, e.g., the example gven n Sec of Ref. [6] and the examples below. The method usually employed to solve Eqs. 2 and 5 reles on the fact that ξ and η are functons of q and t only and, n many cases, the left-hand sdes of Eqs. 2 and 5 are polynomals n the q, wth coeffcents that depend on q and t. Snce Eqs. 2 and 5 must hold for all values of q, q, and t, wthout mposng the equatons of moton, by equatng the coeffcents of the products of the q on each sde of the equaton, one obtans a system of equatons that only nvolve the varables q and t see, e.g., Refs. [5, 6, 8]. It may be notced that the functon G cannot depend on the q because one would have terms proportonal to q on the rght-hand sde of Eq. 5, whle the left-hand sde of ths equaton s a functon of q, q, and t, only. Cf., Eq of Ref. [9]. In the case of Eq. 5, one obtans n ths manner some expressons for the partal dervatves, G/ t and G/, of the unknown functon G n terms of L, ξ, η, and ther frst partal dervatves. From the equalty of the mxed second partal dervatves of G wth respect to q and t, one fnds nn + 1/2 equatons, that do not contan G. Once ξ and η are determned from the set of equatons thus obtaned, the functon G can be fnally calculated see, e.g., Ref. [6]. As we shall show below, these calculatons can be smplfed f, nstead of startng from Eq. 5, one looks for the symmetres of the Euler Lagrange equatons correspondng to the gven Lagrangan see also Refs. [8, 10], because these are n PDEs for ξ and η only Eqs. 26 below. In Sec. 2 we prove that gven a varatonal symmetry of a Lagrangan, there exst Lagrangans equvalent to t for whch the transformaton s a strct varatonal symmetry. We derve the transformaton rules for the Euler Lagrange equatons under pont transformatons, whch lead to the equatons for the generators of the one-parameter groups of symmetres of the Euler Lagrange equatons. In Sec. 3 we show that, f the Lagrangan s regular.e., det 2 L/ q j 0, the varatonal symmetres are canoncal transformatons. 2. Symmetres of the Lagrangans and of the Euler Lagrange equatons As s well known, two Lagrangans, L 1 q, q, t and L 2 q, q, t, lead to the same Euler Lagrange equatons, that s 1 d 1 = 2 d 2, f and only f there exsts a functon F q, t such that L 2 = L 1 + F t + q F. 7 In such a case, t s sad that L 1 and L 2 are equvalent see, e.g., Ref. [8]. Note that ths ndeed defnes an equvalence relaton. In the lterature, ths equvalence s also called gauge equvalence and the functon F s called gauge functon. Thus, a varatonal symmetry of L s a pont transformaton that leaves L nvarant, or leads to a Lagrangan equvalent to L [see Eq. 4]. A straghtforward computaton shows that f the functons ξ, η generate a one-parameter group of varatonal symmetres of L 1.e., Eq. 5 holds for some functon G 1, then ξ, η also generate a one-parameter group of varatonal symmetres of L 2 wth G 2 = G 1 + ξ t + η F, 8 up to an addtve trval constant. In other words, each soluton, ξ, η of Eq. 5 represents a varatonal symmetry of a whole class of Lagrangans or, equvalently, a symmetry of a set of Euler Lagrange equatons, whch are common to all the Lagrangans of a class. Makng use of Eq. 8 we can readly show that f a pont transformaton s a varatonal symmetry of a gven Lagrangan L 1, then we can always fnd another Lagrangan, L 2, equvalent to L 1, for whch the pont transformaton s a strct varatonal symmetry, that s, G 2 = 0. Ths concluson follows from the fact that, for any functon G 1, t s always possble to fnd a functon F such that G 1 + ξ t + η F = 0. 9 In fact, the soluton s determned up to an addtve functon arbtrary functon of n varables Transformaton of the Euler Lagrange equatons In order to fnd the equatons for the symmetres of the Euler Lagrange equatons, we shall study the effect of a pont transformaton on the Euler Lagrange equatons. In the case of a coordnate transformaton of the form q = q q 1,..., q n, t, 10 where the new coordnates may depend explctly on t, but the tme tself s not changed, the nverse relatons, q = q q 1,..., q n, t, must exst, and makng use repeatedly of the chan rule one fnds that d = q j q d q j q j. 11 Equaton 11 explctly demonstrates the covarance of the Euler Lagrange equatons under the coordnate transformatons 10, whch means that dfferent choces of the generalzed coordnates lead to equvalent equatons of moton. Here t s assumed that the functon L appearng n both sdes of Eq. 11 s the same functon, expressed n terms of two dfferent coordnate systems, but, when t s also transformed, the Lagrangan Lq, q, t must be replaced by a new Lagrangan L accordng to L q, q, t = Lq, q, t, 12 Rev. Mex. Fs

3 POINT SYMMETRIES OF THE EULER LAGRANGE EQUATIONS 131 wth q dq /, so that the acton ntegral remans nvarant t 1 t 0 L = t 1 t 0 L. In order to fnd a relaton analogous to Eq. 11, applcable to an arbtrary pont transformaton q = q q 1,..., q n, t, t = t q 1,..., q n, t, 13 nstead of attemptng a drect computaton, t s convenent to defne q 0 t note that ths s not related to Relatvty, t s just a useful notaton, so that Eqs. 13 are equvalent to the sngle equaton q α = q αq 1,..., q n, q 0, α = 0, 1,..., n, 14 and we ntroduce an auxlary varable u n terms of whch the coordnates q and t wll be expressed. Then, snce t s now a functon of u, accordng to the elementary rules for a change of varable n an ntegral, t 1 t 0 Lq, dq /, t = u 1 u 0 Lq, dq /du, t /du du du. Hence, the use of the varable u must be accompaned by the use of the Lagrangan Lq α, dq α /du Lq, dq /du, t /du du. In fact, a straghtforward computaton usng agan the chan rule shows that, for = 1, 2,..., n, L d L du dq /du = d, 15 du provng that the orgnal Euler Lagrange equatons are ndeed reproduced wth L, and L d q 0 du L dq 0 /du = du d L dq du du = du = du q t 1 /du [ t d ] L q d, 16 whch s trvally equal to zero when the other n Euler Lagrange equatons for L are satsfed. That s, we only get n equatons of moton from L, as n the case of L. Applyng now the relaton 11 to the auxlary Lagrangan L, we fnd that under a pont transformaton 13, for = 1, 2,..., n, L d L du dq /du = q α q q L d q α du L dq α /du the lower case Greek ndces run over 0, 1, 2,..., n. Wth the ad of Eqs. 15 and 16 we see that ths last relaton amounts to du d = q j d du q j t du q j whch means e.g., takng u = t that q d qj = q t dq j q j d q j q j d q j q j. 17 Ths relaton reduces to Eq. 11 n the case where t = t, and demonstrates the covarance of the Euler Lagrange equatons under the pont transformatons Symmetres of the Euler Lagrange equatons We shall say that the pont transformaton 13 s a symmetry of the Euler Lagrange equatons correspondng to the Lagrangan Lq, q, t f Eq. 17 holds wth L = L. Accordng to the defntons gven n Sec. 1, f a pont transformaton 13 s a varatonal symmetry of a Lagrangan L, the Lagrangan L appearng n Eq. 17 s equal to L or s equvalent to L; n ether case, we can replace L by L on the lefthand sde of Eq. 17 and therefore, any varatonal symmetry of L s also a symmetry of ts Euler Lagrange equatons. In what follows t wll be convenent to use the abbrevaton [8] E d. 18 By contrast wth the Lagrangan L, the functons E depend on q j, q j, q j, and t. Thus, the pont transformaton 13 s a symmetry of the Euler Lagrange equatons f E q k, q k, q k, t qj = q t dq j E j q k, q k, q k, t, 19 or, equvalently, nterchangng the roles of q α and q α, E q k, q k, q k, t = q j t dq j E j q k, q k, q k, t. 20 Rev. Mex. Fs

4 132 G.F. TORRES DEL CASTILLO Example The Euler Lagrange equatons correspondng to the Lagrangan L = m 2 ẋ2 + ẏ 2 mgy, 21 where m and g are constants, and q 1, q 2 = x, y, possess a one-parameter group of symmetres gven by x = xe s/2, y = ye s/2 1 2 gt2 e 2s e s/2, t = te s. 22 In fact, for ths famly of pont transformatons treatng s as an ndependent parameter, and, therefore, dx dy = es/2 dx e s = e s/2 ẋ, = es/2 dy gte 2s e s/2 e s = e s/2 ẏ gte s e s/2, d 2 x 2 = ddx / = e s/2 dẋ e s = e 3s/2 ẍ, d 2 y 2 = ddy / = e s/2 dẏ ge s e s/2 e s = e 3s/2 ÿ g1 e 3s/2. On the other hand [see Eq. 18], E 1 = mẍ, E 2 = mg mÿ, 23 hence, for nstance, the rght-hand sde of Eq. 20 wth = 2 gves x t + dx mẍ t dy mg mÿ = e s/2 e s{ mg m [ e 3s/2 ÿ g1 e 3s/2 ]} = mg mÿ, whch concdes wth E 2, and, n a smlar manner, one fnds that the equaton wth = 1 also holds. It can be readly verfed that the pont transformatons 22 are not strct varatonal symmetres of the Lagrangan 21. In fact, they satsfy Eq. 4 wth [ F = e 3s/2 1mgt y + 1 ] 6 gt2 2e 3s/2 1. The one-parameter groups of symmetres of the Euler Lagrange equatons are more easly obtaned by fndng frstly ther generators. Proceedng as n Ref. [6], we now consder a one-parameter famly of symmetres of the Euler Lagrange equatons q = q q 1,..., q n, t, s, t = t q 1,..., q n, t, s, 24 and we shall assume that q q 1,..., q n, t, 0 = q and t q 1,..., q n, t, 0 = t. Takng the partal dervatve of both sdes of Eq. 20 wth respect to s, at s = 0, usng the chan rule, we obtan Ej q k 0 = δ j q k s + E j q k q k s + E j q k q k s + E j t q j t s + s + δ j s E j, or, equvalently, wth the ad of the standard defntons η q j, t q q j, t, s s, ξq, t t q, t, s s, 25 we have see, e.g., Refs. [4, 6] E η k + E dηk q k q k q dξ k + E d 2 η k q k 2 2 q dξ k q d 2 ξ k 2 + E t ξ + E η j dξ j + E = Equaton 26 s derved n Ref. [8], Sec. 8.3, makng use of the language of fbred manfolds and jet prolongatons. As n the case of Eqs. 2 and 5, Eqs. 26 are PDEs for the n + 1 functons η and ξ. Any lnear combnaton wth constant coeffcents of solutons of Eqs. 26 s also a soluton of these equatons, and the fact that Eqs. 26 have to hold for all values of q k and q k wthout mposng the equatons of moton, leads to several condtons that n some cases are readly solved see the example below. The man dfferences between Eqs. 26 and Eqs. 2 and 5 are that Eqs. 26 consttute a system of n PDEs, not a sngle equaton when n > 1. Equatons 26 determne the varatonal symmetres of a Lagrangan and all other Lagrangans equvalent to t, and t does not contan the unknown functon G. However, n order to fnd the constant of moton assocated wth a gven soluton ξ, η of Eqs. 26, we have to make use of Eq. 5 to obtan G and then of Eq. 6. Alternatvely, the dfferental of ths constant of moton s equal to the contracton of the vector feld ξ t + η dη + q dξ wth the dfferental of the Cartan 1-form, defned below [Eq. 37]. Rev. Mex. Fs

5 POINT SYMMETRIES OF THE EULER LAGRANGE EQUATIONS Example. Partcle n a unform gravtatonal feld We wll determne the generators of the pont symmetres of the Euler Lagrange equatons 23, correspondng to the standard Lagrangan for a partcle of mass m n a unform gravtatonal feld 21. Substtutng Eqs. 23 nto Eqs. 26 we obtan [wth x, y = q 1, q 2 ] m d 2 η 1 2 2ẍdξ ξ ẋd2 2 mẍ η 1 x mg + mÿ η 2 x mẍdξ = 0, d 2 η 2 dξ m 2 2ÿ ẏ d2 ξ 2 mẍ η 1 mg + mÿ η 2 mg + mÿdξ = 0. Cancelng the common factor m and wrtng more explctly the dervatves d 2 η k / 2, we have 2 η 1 t 2 2 η 2 t q 2 η 1 j q j t + q η 1 2 η 1 j + q j q k 2ẍ dξ q j q j q k ẋ d2 ξ 2 + ẍ η 1 x + g + ÿ η 2 x + ẍdξ = 0, + 2 q 2 η 2 j q j t + q η 2 2 η 2 j + q j q k 2ÿ dξ q j q j q k ẏ d2 ξ 2 + ẍ η 1 + g + ÿ η 2 + g + ÿdξ = Makng use of the fact that the coeffcents of ẍ and ÿ must be equal to zero we fnd that 2 η 1 x dξ = 0, η 1 + η 2 x = 0, 2 η 2 dξ = 0. Snce dξ/ = ξ/ t + q j ξ/ q j, from the precedng equatons t follows that ξ/ q j = 0, hence ξ = At, η 1 = x da 2 + B 1y, t, η 2 = y da 2 + B 2x, t, 28 where At, B 1 y, t, and B 2 x, t are some functons, wth B 1 + B 2 = x By consderng the coeffcents of the quadratc terms n the veloctes n Eqs. 27, one fnds that η 1 and η 2 must be polynomals of frst degree n x and y: B 1 y, t = D 1 ty + D 2 t, B 2 x, t = D 1 tx + D 3 t, where D 1, D 2, and D 3 are functons of a sngle varable, and we have taken nto account the condton 29. The vanshng of the lnear terms n the veloctes n Eqs. 27 mples that D 1 s some constant. Fnally, from the terms that do not contan q k or q k one obtans the condtons x d 3 A d2 D 2 2 gd 1 = 0, y d 3 A d2 D g 2 da = 0, whch mply that A s a polynomal of degree not greater than 2, and the soluton of Eqs. 27 s gven by ξ = c 3 + c 7 t + c 8 t 2, 1 η 1 = c 1 + c 4 t + c 6 2 gt2 + y + c 7 x 2 + c 8xt, η 2 = c 2 + c 5 t c 6 x + c 7 y gt2 + c 8 yt 1 2 gt3, 30 where c 1,..., c 8 are arbtrary real constants, whch agrees wth the soluton obtaned from Eq. 5 [6]. Indeed, substtutng these functons ξ and η nto Eq. 5 one readly fnds that, up to a trval constant, the correspondng functon G s gven by c 2 mgt + c 4 mx + c 5 m y 12 gt2 + c 6 mgxt + c 7 m 32 gyt + 14 g2 t 3 + c 8 m 32 gt2 y + 12 x2 + y g2 t 4. Then, makng use of Eq. 6, one can calculate the assocated constant of moton. The fact that c 1 and c 3 do not appear n G means that f only the constants c 1 and c 3 are dfferent from zero, the transformatons generated are strct varatonal symmetres of L, whch corresponds to the fact that x and t do not appear n L. When only c 1 and c 3 are dfferent from zero, the vector feld ξ / t + η / reduces to c 1 / x + c 3 / t; / x generates translatons along the x-axs, and / t generates tme dsplacements. The one-parameter group of pont transformatons 22 s generated by the vector feld ξ / t + η / wth c 7 = 1 and all the other constants c k equal to zero. The vector feld / obtaned settng c 2 = 1, and all the other constants c k equal to zero generates translatons along the y-axs, whch are not strct varatonal symmetres of the Lagrangan 21, snce n ths case G = mgt. Wth the ad of 8 we can readly obtan a Lagrangan, equvalent to 21, for whch / generates strct varatonal symmetres. In vew of Eq. 9, we need a functon F such that G + ξ t + η F = mgt + F = 0. Choosng F = mgty, from Eq. 7 we have L 2 = m 2 ẋ2 + ẏ 2 mgy + d mgty = m 2 ẋ2 + ẏ 2 + mgtẏ. Now y s gnorable, but L 2 depends explctly on t. Rev. Mex. Fs

6 134 G.F. TORRES DEL CASTILLO Example. A sngular Lagrangan As a second example, we shall consder the sngular Lagrangan L = q 1 q 3 q 2 q 3 + q 1 q 3, 31 already studed n Ref. [10] and the references cted theren. In ths case E 1 = q 3 q 3, E 2 = q 3, E 3 = q 1 q 1 + q 2. The soluton to Eqs. 26 for these functons E, gven n Ref. [10], s ξ = c 1, η 1 = c 2 q 1 + c 3 e t + c 4 e t, η 2 = c 2 q 2 + bq 3, η 3 = c 2 q 3, 32 where c 1,..., c 4 are arbtrary real constants, and b s an arbtrary real-valued functon of one varable. We omt an unnecessary addtve constant, denoted by C 3 n Ref. [10]. In fact, substtutng Eqs. 31 and 32 nto Eq. 5 we fnd that the left-hand sde of Eq. 5 amounts to d q 3 c 3 e t q 3 c 4 e t q 3 budu. 33 Hence, only c 1 and c 2 are related to strct varatonal symmetres of the Lagrangan 31; c 1 s related wth the obvous nvarance of L under tme dsplacements, whle c 2 s related to a scalng symmetry q 1 = q 1 e c 2s, q 2 = q 2 e c 2s, q 3 = q 3 e c2s, t = t. Even though the Lagrangan 31 s sngular, the general expressons gven n the precedng sectons are also applcable n ths case. For nstance, makng use of Eqs. 6, 32, and 33 we fnd the constant of moton ϕ = c 1 q 1 q 3 q 1 q 3 + c 2 q 3 q 1 q 1 q 3 q 2 q 3 q 3 + c 3 e t q 3 q 3 + c 4 e t q 3 + q 3 + budu, whch dffers from the expressons reported n Ref. [10]. The expressons presented n Ref. [10] are the ones gven by Eq. 3, whch are vald only n the case of a strct varatonal symmetry. If we look for a Lagrangan equvalent to 31, for whch bq 3 / q 2 represents a strct varatonal symmetry, makng use of Eq. 9, wth G = q 3 budu [see Eq. 33], we need a functon F such that q 3 budu + bq 3 F = 0 q 2 note that ths PDE s smpler than the system of second-order PDEs consdered n Ref. [10]. Thus, we can choose, e.g., F = q q 3 2 budu, bq 3 whch dffers from the result found n Ref. [10] the error n Ref. [10] was produced by the mplct assumpton that the new Lagrangan s also ndependent of q 2. As a consequence of the fact that L s sngular, ts varatonal symmetres contan an arbtrary functon, and the correspondng constants of moton are not useful because, by vrtue of the equatons of moton, q 3 = Hamltonan formulaton As s well known, for a gven system wth Hamltonan functon Hq, p, t, where q, p are local coordnates n the phase space, the coordnate transformaton Q = Q q j, p j, t, P = P q j, p j, t s canoncal f there exsts a functon F such that P dq K p dq H = df. The functon K s the Hamltonan that determnes the tme evoluton of the new coordnates Q, P see, e.g., Refs. [11, 12]. One can readly verfy that f T = T q j, p j, t s a new varable replacng t, then the exstence of a functon F such that P dq KdT p dq H = df 34 assures that the Hamlton equatons are equvalent to dq = H p, dq dt = K P, dp = H, 35 dp dt = K. 36 Q For nstance, ths follows from the fact that the Hamlton equatons determne the extremals of the ntegral p dq H. However, the Posson brackets are preserved only f T s a functon of t exclusvely ths happens n the case of the famly of transformatons 22 and n the examples gven above, snce ξ s a functon of t only. As n the case of the usual canoncal transformatons, the converse s not true, the equvalence of the Hamlton equatons 35 and 36 does not mply the exstence of a functon F such that Eq. 34 holds [12]. Accordng to the usual defntons of the canoncal momenta and the Hamltonan, p dq H = dq The lnear dfferental form q L = L + dq q. θ L L + dq q, 37 Rev. Mex. Fs

7 POINT SYMMETRIES OF THE EULER LAGRANGE EQUATIONS 135 s known as the Cartan 1-form see, e.g., Ref. [8] and the references cted theren. Makng use of ths defnton we see that, gven two equvalent Lagrangans L 1 and L 2 = L 1 + df/, θ L2 = L 1 + F F and the condton 4 s equvalent to t + q F dq q = θ L1 + df Lq, dq, t + dq q = Lq, dq, t + dq q + df. Thus, f the Lagrangan s regular, a varatonal symmetry corresponds to a canoncal transformaton [n the sense of Eq. 34]. In the standard Hamltonan formulaton one only consders transformatons of the coordnates of the phase space, mantanng t unchanged and for many purposes ths s enough. For nstance, any constant of moton can be assocated wth a one-parameter group of canoncal transformatons, wth t unchanged; n the case of the Hamlton Jacob method, one looks for a transformaton to new coordnates whch are constant of moton, but f a functon s a constant of moton, ts dervatve wth respect to any tme coordnate wll be equal to zero. Nevertheless, t seems nterestng to explore the applcatons of more general transformatons. By contrast, n the Lagrangan formalsm, t s very useful to consder pont transformatons n whch t s also transformed, as we can see n the examples gven above and n Refs. [6, 8]. 4. Concludng remarks As a by-product of the dervatons n ths paper, we have found the transformaton rules of the Euler Lagrange equatons under pont transformatons [Eq. 17]. As ponted out above, Eqs. 26 consttute a convenent way to obtan all the varatonal symmetres of a gven Lagrangan, because they do not contan the functon G, present n Eq. 5. As we have shown, when one consders pont transformatons n the phase space, there are two nonequvalent ways of defnng a canoncal transformaton; the transformatons nduced by the varatonal symmetres of a Lagrangan obvously preserve the form of the Hamlton equatons, but the Posson brackets may not be preserved. Acknowledgment The author s grateful to Dr. José Lus López Bonlla for brngng Ref. [10] to hs attenton. 1. H. Rund, The Hamlton Jacob Theory n the Calculus of Varatons Van Nostrand, London, Chap H. Stephan, Dfferental Equatons: Ther Soluton Usng Symmetres Cambrdge Unversty Press, Cambrdge, P.J. Olver, Applcatons of Le Groups to Dfferental Equatons, 2nd ed. Sprnger-Verlag, New York, P.E. Hydon, Symmetry Methods for Dfferental Equatons: A Begnner s Gude Cambrdge Unversty Press, Cambrdge, B. van Brunt, The Calculus of Varatons Sprnger-Verlag, New York, G.F. Torres del Castllo, C. Andrade Mrón, and R.I. Bravo Rojas, Rev. Mex. Fís. E Y. Kosmann-Schwarzbach, The Noether Theorems: Invarance and Conservaton Laws n the Twenteth Century Sprnger, New York, Chap O. Krupková, The Geometry of Ordnary Varatonal Equatons Sprnger-Verlag, Berln, S. Wenberg, Lectures on Quantum Mechancs Cambrdge Unversty Press, Cambrdge, M. Havelková, Communcatons n Mathematcs M.G. Calkn, Lagrangan and Hamltonan Mechancs World Scentfc, Sngapore, G.F. Torres del Castllo, Rev. Mex. Fs. E Rev. Mex. Fs