Generation of solutions of the Hamilton Jacobi equation

Transcription

1 RESEARCH Revista Mexicana de Física 60 (2014) JANUARY-FEBRUARY 2014 Generation of solutions of the Hailton Jacobi equation G.F. Torres del Castillo Departaento de Física Mateática, Instituto de Ciencias Universidad Autónoa de Puebla, Puebla, Pue., México. Received 17 Septeber 2013; accepted 7 October 2013 It is shown that any function G(q i, p i, t), defined on the extended phase space, defines a one-paraeter roup of canonical transforations which act on any function f(q i, t), in such a way that if G is a constant of otion then fro a solution of the Hailton Jacobi (HJ) equation one obtains a one-paraeter faily of solutions of the sae HJ equation. It is also shown that any coplete solution of the HJ equation can be obtained in this anner by eans of the transforations enerated by n constants of otion in involution. Keywords: Hailton Jacobi equation; canonical transforations; constants of otion. Se uestra que cualquier función G(q i, p i, t), definida en el espacio fase extendido, define un rupo uniparaétrico de transforaciones canónicas las cuales actúan sobre cualquier función f(q i, t), de tal anera que si G es una constante de oviiento entonces a partir de cualquier solución de la ecuación de Hailton Jacobi (HJ) uno obtiene una failia uniparaétrica de soluciones de la isa ecuación de HJ. Se uestra tabién que cualquier solución copleta de la ecuación de HJ puede obtenerse de esta anera por edio de las transforaciones eneradas por n constantes de oviiento en involución. Descriptores: Ecuación de Hailton Jacobi; transforaciones canónicas; constantes de oviiento. PACS: Jj; Jr; Qs 1. Introduction In the Hailtonian forulation of classical echanics, the canonical transforations, iven locally by expressions of the for Q i = Q i (q j, p j, t), P i = P i (q j, p j, t), act on the extended phase space (the Cartesian product of the phase space and the tie axis) and, therefore, there is a naturally defined action of a canonical transforation on any function defined on the extended phase space (that is, on any function of q i, p i, and t). On the other hand, Hailton s principal function, usually denoted by an S, is a function defined on the extended confiuration space (that is, a function of q i and t), just like the wave function in the eleentary Schrödiner equation, but there is no obvious way to define the action of a canonical transforation on a function of q i and t. However, in Ref. 1] it was shown that, under certain conditions, one can define an action of a tie-independent canonical transforation that is, a canonical transforation that does not involve the tie, Q i = Q i (q j, p j ), P i = P i (q j, p j )] on functions defined on the confiuration space, in such a way that if a (tie-independent) Hailtonian is invariant under the canonical transforation, a solution of the correspondin tie-independent Hailton Jacobi (HJ) equation is apped into another solution. Furtherore, it was shown that the action of the one-paraeter roup of canonical transforations enerated by an arbitrary function G(q i, p i ) is deterined by an equation siilar to the HJ equation. In this paper, we consider systes with a Hailtonian that can depend on the tie, we show that one can define the action of the one-paraeter roup of canonical transforations enerated by an arbitrary function G(q i, p i, t) on functions defined on the extended confiuration space, in such a way that if G is a constant of otion, then a solution of the correspondin HJ equation is apped into a one-paraeter faily of solutions of this equation. In this anner, each constant of otion adds a continuous paraeter to a iven solution of the HJ equation. We also show that any coplete solution of the HJ equation can be obtained fro a solution without paraeters by eans of the action of the roups of canonical transforations enerated by n constants of otion in involution. Throuhout this paper, several exaples are iven in order to illustrate the definitions and results presented here. 2. The action of a one-paraeter roup of canonical transforations on functions defined on the extended confiuration space We start by reviewin the HJ equation, which will ive us the pattern to follow in the definition of the action of any oneparaeter roup of canonical transforations on functions defined on the extended confiuration space The Hailton Jacobi as an evolution equation Given a Hailtonian H(q i, p i, t) of a syste with n derees of freedo, the correspondin HJ equation is the first-order partial differential equation H(q i, /q i, t) + = 0. (1) Usually one is interested in coplete solutions of the HJ equation (1), that is, solutions S(q i, t, α i ) of Eq. (1), containin n arbitrary paraeters α i, such that ( 2 ) S det 0, (2) q i α j

2 76 G.F. TORRES DEL CASTILLO because they yield the solution of the Hailton equations (see, e.., Refs. 2] and 3]). However, one can consider the HJ equation as an evolution equation, which deterines the function S(q i, t) that reduces to a iven function f(q i ), for t = 0 (or any other initial value, t 0, of t). (This fact does not sound very strane if we consider the relationship between the HJ equation and the Schrödiner equation.) In fact, if we have the solution of the Hailton equations, we can use it to find the solution of the HJ equation that satisfies any initial condition S(q i, t 0 ) = f(q i ). (Siilar ideas are often entioned in the standard textbooks on analytical echanics, without presentin, however, a clear stateent or an explicit procedure.) For exaple, in the case of the Hailtonian H = p p q 2, (3) where and are constants, the solution of the correspondin Hailton equations can be readily obtained and is iven by q 1 = (q 1 ) 0 + (p 1) 0 t, q 2 = (q 2 ) 0 + (p 2) 0 t t2 2, p 1 = (p 1 ) 0, p 2 = (p 2 ) 0 t, (4) where (q 1 ) 0 denotes the value of q 1 at t = 0, and so on. As the initial condition we choose S(q 1, q 2, 0) = α 1 q 1 + α 2 q 2, (5) where α 1, α 2 are arbitrary constants. We want to find the solution of the HJ equation see Eqs. (1) and (3)] ( = 1 ) 2 ( ) ] 2 + q 2, (6) 2 q 1 q 2 that satisfies the initial condition (5). Recallin that /q i = p i, akin use of Eqs. (4) and (5) we have = q 1 q 1 = α 1, = q 2 q 2 t = α 2 t. Substitutin these expressions into the riht-hand side of Eq. (6) we have = 1 2 α1 + (α 2 t) 2] q 2. 2 Cobinin the last three equations one readily finds that, choosin the interation constant so that Eq. (5) is satisfied, S(q 1, q 2, t) = α 1 q 1 + α 2 q 2 tq 2 α 1 2 t 2 + (α 2 t) (7) The expression (7) is a (coplete, R-separable) solution of the HJ equation that reduces to the specified function (5) for t = 0. (A solution is R-separable if it is the su of a function of two or ore variables and functions of one variable.) We close this subsection with a second exaple. The solution of the Hailton equations for the tie-dependent Hailtonian H = p2 ktq, (8) 2 where and k are constants, is iven by q = q 0 + tp 0 + kt3 6, p = p kt2, (9) where q 0 and p 0 denote the values of q and p at t = 0, respectively. Lettin S(q, 0, α) = αq, (10) where α is an arbitrary constant, with the aid of Eqs. (9), we have q = q kt2 = α kt2. (11) Thus, fro Eqs. (1), (8), and (11), we have = 1 2 = 1 2 ( ) 2 + ktq q ( α + 1 ) 2 2 kt2 + ktq. (12) Then, fro Eqs. (10) (12), we readily obtain S(q, t, α) = αq kt2 q 1 (α 2 t αkt3 + k2 t 5 ), 20 which is an R-separable coplete solution of the HJ equation. (We iht start with expressions ore coplicated than (5) and (10), but these siple expressions are enouh to obtain coplete solutions of the HJ equation.) It should be clear that a siilar construction can be devised usin an arbitrary function of (q i, p i, t) instead of a Hailtonian Failies of solutions of the HJ equation and constants of otion Any differentiable function, G(q i, p i, t), defines a (possibly local) one-paraeter roup of canonical transforations, deterined by the (autonoous) syste of first-order ordinary differential equations dq i = G(q j, p j, t), p i dp i = G(q j, p j, t), (13) q i which are of the for of the Hailton equations. For exaple, the function G(q 1, q 2, p 1, p 2, t) = p 1p 2 + q 1, (14)

3 GENERATION OF SOLUTIONS OF THE HAMILTON JACOBI EQUATION 77 where and are constants, leads to the syste of equations dq 1 = p 2, dq 2 = p 1, dp 1 =, dp 2 = 0. (15) Fro the last two equations we find that p 1 = (p 1 ) 0 α, p 2 = (p 2 ) 0, (16) where (p i ) 0 denotes the value of p i for α = 0. Substitutin these expressions into the first pair of equations (15) we et Hence, dq 1 = (p 2) 0, dq 2 = (p 1) 0 α. q 1 = (q 1 ) 0 + (p 2) 0 α, q 2 = (q 2 ) 0 + (p 1) 0 α α2 2. (17) The action of the canonical transforations defined by a function G(q i, p i, t), on functions of (q i, t), will be defined iitatin the HJ equation. Definition. The iae of a iven arbitrary function f(q i, t) under the one-paraeter roup of canonical transforations enerated by G(q i, p i, t) is the function S(q i, t, α) such that G(q i, /q i, t) + = 0, (18) α with the initial condition S(q i, t, 0) = f(q i, t). (Cf. Eq. (1).) The followin exaple shows that this definition produces the expected effect in the case of transforations that only affect the coordinates of the confiuration space. Indeed, the one-paraeter roup of canonical transforations enerated by, e.., G = p 1 is iven by see Eqs. (13)] q 1 = (q 1 ) 0 + α, q i = (q i ) 0, for i 2, p i = (p i ) 0, that is, translations alon the q 1 -axis. Then, Eq. (18) takes the for + q 1 α = 0, whose eneral solution is S(q i, t, α) = F (q 1 α, q 2,..., q n, t), where F is an arbitrary function. By iposin the initial condition S(q i, t, 0) = f(q i, t), we find that S(q i, t, α) = f(q 1 α, q 2,..., q n, t), which is the expected effect of a translation alon the q 1 -axis. As a second exaple, the iaes of the function f(q i, t) = (α 2 t)q 2 + (α 2 t) (19) (which is the function (7) with α 1 = 0) under the transforations enerated by (14) can be obtained proceedin as in the exaples of Sec Makin use of Eqs. (16) and (19) we have with S(q i, t, 0) = f(q i, t)] = q 1 q 1 α = α, α=0 = q 2 q 2 = α 2 t. (20) α=0 Substitutin these expressions into Eq. (18), takin into account Eq. (14), we obtain α = 1 ( α)(α 2 t) q 1. Cobinin the last equations and the initial condition (19) one finds that S(q 1,q 2, t, α 2, α) = (α 2 t)q 2 αq (α 2 t)α 2 + (α 2 t) (21) Note that t is treated here as a constant because the canonical transforations do not affect t. Since the function (19) is a particular case of (7), is a solution of the HJ equation (6), and one can readily verify that (21) is also a solution of Eq. (6), for all values of the paraeter α which is essentially the twoparaeter faily of solutions (7)]. Thus, we have obtained a two-paraeter faily of solutions of the HJ equation (6), startin fro a one-paraeter solution of this equation. As we shall show in the followin Proposition, this is a consequence of the fact that the function (14) is a constant of otion for the Hailtonian (3). Proposition 1. The iae of a solution of the HJ equation, S 0 (q i, t), under the one-paraeter roup of canonical transforations enerated by a constant of otion G(q i, p i, t) is a one-paraeter faily of solutions of the sae HJ equation. Proof. Assuin that S(q i, t, α) is a solution of Eq. (18), akin use of the chain rule repeatedly, we have (here, and in what follows, there is suation over repeated indices) H(q i, /q i, t) + ] = H 2 S + 2 S α p j αq j α = H G(q i, /q i, t) G(q i, /q i, t) p j q j = H p j ( G q j + G p i 2 S q j q i G 2 S G p i q i. (22) Accordin to the hypothesis, G is a constant of otion, that is G + G H G H = 0, (23) q j p j p j q j )

4 78 G.F. TORRES DEL CASTILLO thus, the last line of Eq. (22) aounts to G ( H + H 2 S + 2 S p i q i p j q j q i q i d = G p i q i = dq i ) H(q i, /q i, t) + q i ] H(q i, /q i, t) + see Eqs. (13)]. Hence, we obtain the equation H(q i (α), (q i (α), t, α)/q i, t) + (q ] i(α), t, α) = 0, which involves the iaes of S and the coordinates q i under the transforations enerated by G. Since the expression inside the brackets in the last equation vanishes for α = 0, it is equal to zero for all values of α, thus provin the assertion. A partial converse of this Proposition is true. The steps in the proof show that if the iaes of a solution of the HJ equation under the transforations enerated by G are solutions of the sae HJ equation, and S is a coplete solution of the HJ equation, then G is a constant of otion. The assuption that S is a coplete solution of the HJ equation assures that Eq. (23) holds everywhere. In fact, the followin stroner result holds. Proposition 2. A coplete solution of the HJ equation, S(q i, t, α i ) (not necessarily separable or R-separable), is the iae of S(q i, t, 0) under the transforations enerated by the n constants of otion G j (q i, p i, t) = α j (q i, t, α i (q k, p k, t)) (24) (j = 1, 2,..., n), obtained by eliinatin the α i by eans of p j = q j (q i, α i, t). (25) Furtherore, the functions G i are in involution (i.e., the Poisson bracket of G i and G j is equal to zero). (Note that, accordin to Eq. (2), in principle, one can find the α i fro Eqs. (25).) Proof. We note that, by virtue of Eqs. (25), Eq. (24) is equivalent to Eq. (18). Since, by hypothesis, S(q i, t, α i ) is a coplete solution of the HJ equation, each function G i, defined by (24), is a constant of otion. In order to prove that the functions G i are in involution, akin use of Eq. (24), we calculate the ixed partial derivative 2 S = G j (q i, /q i, t, α k ) α i α j α i = G j p k = G j G i. α i q k p k q k ] Then we see that the coutativity of the partial derivatives of S iplies that the Poisson brackets of the G i are equal to zero. (The assuption that S(q i, t, α i ) is a coplete solution of the HJ equation iplies that the Poisson brackets {G i, G j } vanish everywhere.) Since the Poisson brackets {G i, G j } are equal to zero, the one-paraeter roups of transforations enerated by the functions G i coute. (In fact, the roups of canonical transforations enerated by two functions F and G coute if and only if {F, G} is a trivial constant, not necessarily zero 4].) Let us consider, for exaple, the function S(q i, t, α i ) = α 1 q 1 + α 2 q 2 tq 2 α 1 2 t 2 + (α 2 t) , (26) which is the coplete solution (7) of the HJ equation constructed in Sec One finds that q 1 = α 1, which lead to see Eqs. (25)] On the other hand, α 1 = q 1 α 1t, q 2 = α 2 t, α 1 = p 1, α 2 = p 2 + t. (27) = q 2 + (α 2 t) 2 α 2 2 2, which, takin into account Eqs. (27), are of the for (24) with G 1 = q 1 + p 1t, G 2 = q 2 + p 2t t2. (28) One can readily verify that G 1 and G 2 are constants of otion in involution, and that the function (26) can be obtained by eans of the transforations enerated by G 1 and G 2 fro S(q i, t) = tq 2 2 t 3, (29) 6 which is obtained fro (26) settin α 1 = α 2 = 0. Indeed, the function G α 1 G 1 + α 2 G 2 ( = α 1 q 1 + p ) ( 1t + α 2 q 2 + p 2t + 1 ) 2 t2 2 (30) is a constant of otion for all values of the constants α 1 and α 2. The one-paraeter roup of canonical transforations enerated by G is iven by (denotin by s the correspondin paraeter) q 1 = (q 1 ) 0 + α 1t s, q 2 = (q 2 ) 0 + α 2t s, p 1 = (p 1 ) 0 + α 1 s, p 2 = (p 2 ) 0 + α 2 s,

5 GENERATION OF SOLUTIONS OF THE HAMILTON JACOBI EQUATION 79 hence, akin use of the initial condition (29), q 1 = 0 + α 1 s, q 2 = t + α 2 s, and substitutin these expressions into Eq. (18), with G iven by (30), Thus, s = α 1q 1 α 1t (α 1s) + α 2 q 2 t ( t + α 2s) 1 2 α 2t 2. S(q 1, q 2, t) = α 1 sq 1 + α 2 sq 2 tq 2 (α 1s) 2 t 2 + (α 2s t) 3 (α 2 s) 3 6 2, which reduces to the expression (26) when s = Concludin rearks Accordin to Proposition 2, any coplete solution of the HJ equation is, locally, the iae of a particular solution (without free paraeters) under the action of an Abelian n- diensional roup (that is, a roup locally isoorphic to the additive Lie roup R n ). As pointed out above, these solutions need not be separable or R-separable. It should be noted, however, that soe functions are invariant under the transforations enerated by a function G, even if these transforations are different fro the identity. For instance, a function that does not depend on q 1 is invariant under the translations enerated by p 1. One can readily see that, at least in the case of tieindependent operators, an analo of Proposition 1 holds in the case of the solutions of the Schrödiner equation. In addition to the R-separable coplete solution (7), the HJ equation for the Hailtonian (3) also adits coplete separable solutions in the sae coordinate syste (see also Ref. 5]) 1. G.F. Torres del Castillo, D.A. Rosete Álvarez and I. Fuentecilla Cárcao, Rev. Mex. Fís. 56 (2010) E.T. Whittaker, A Treatise on the Analytical Dynaics of Particles and Riid Bodies, 4th ed. (Cabride University Press, Cabride, 1993). Chap. XII. 3. F. Gantacher, Lectures in Analytical Mechanics (Mir, Moscow, 1975). Chap G.F. Torres del Castillo, Differentiable Manifolds: A Theoretical Physics Approach (Birkhäuser Science, New York, 2012). Sec G.F. Torres del Castillo, Rev. Mex. Fís. 59 (2013) 478.