RESEARCH Revista Mexicana de F´ısica 60 (2014) 75–79 JANUARY-FEBRUARY 2014 Generation of solutions of the Hamilton–Jacobi equation G.F. Torres del Castillo Departamento de F´ısica Matematica,´ Instituto de Ciencias Universidad Autonoma´ de Puebla, 72570 Puebla, Pue., Mexico.´ Received 17 September 2013; accepted 7 October 2013 It is shown that any function G(qi; pi; t), defined on the extended phase space, defines a one-parameter group of canonical transformations which act on any function f(qi; t), in such a way that if G is a constant of motion then from a solution of the Hamilton–Jacobi (HJ) equation one obtains a one-parameter family of solutions of the same HJ equation. It is also shown that any complete solution of the HJ equation can be obtained in this manner by means of the transformations generated by n constants of motion in involution. Keywords: Hamilton–Jacobi equation; canonical transformations; constants of motion. Se muestra que cualquier funcion´ G(qi; pi; t), definida en el espacio fase extendido, define un grupo uniparametrico´ de transformaciones canonicas´ las cuales actuan´ sobre cualquier funcion´ f(qi; t), de tal manera que si G es una constante de movimiento entonces a partir de cualquier solucion´ de la ecuacion´ de Hamilton–Jacobi (HJ) uno obtiene una familia uniparametrica´ de soluciones de la misma ecuacion´ de HJ. Se muestra tambien´ que cualquier solucion´ completa de la ecuacion´ de HJ puede obtenerse de esta manera por medio de las transformaciones generadas por n constantes de movimiento en involucion.´ Descriptores: Ecuacion´ de Hamilton–Jacobi; transformaciones canonicas;´ constantes de movimiento. PACS: 45.20.Jj; 02.30.Jr; 02.20.Qs 1. Introduction way that if G is a constant of motion, then a solution of the corresponding HJ equation is mapped into a one-parameter In the Hamiltonian formulation of classical mechanics, the family of solutions of this equation. In this manner, each canonical transformations, given locally by expressions of constant of motion adds a continuous parameter to a given the form Qi = Qi(qj; pj; t), Pi = Pi(qj; pj; t), act on the ex- solution of the HJ equation. We also show that any complete tended phase space (the Cartesian product of the phase space solution of the HJ equation can be obtained from a solution and the time axis) and, therefore, there is a naturally defined without parameters by means of the action of the groups of action of a canonical transformation on any function defined canonical transformations generated by n constants of mo- on the extended phase space (that is, on any function of qi, tion in involution. pi, and t). On the other hand, Hamilton’s principal function, Throughout this paper, several examples are given in or- usually denoted by an S, is a function defined on the extended der to illustrate the definitions and results presented here. configuration space (that is, a function of qi and t), just like the wave function in the elementary Schrodinger¨ equation, 2. The action of a one-parameter group of but there is no obvious way to define the action of a canoni- canonical transformations on functions de- cal transformation on a function of qi and t. fined on the extended configuration space However, in Ref. [1] it was shown that, under certain conditions, one can define an action of a time-independent We start by reviewing the HJ equation, which will give us the canonical transformation [that is, a canonical transforma- pattern to follow in the definition of the action of any one- tion that does not involve the time, Qi = Qi(qj; pj), parameter group of canonical transformations on functions Pi = Pi(qj; pj)] on functions defined on the configuration defined on the extended configuration space. space, in such a way that if a (time-independent) Hamilto- nian is invariant under the canonical transformation, a solu- 2.1. The Hamilton–Jacobi as an evolution equation tion of the corresponding time-independent Hamilton–Jacobi (HJ) equation is mapped into another solution. Furthermore, Given a Hamiltonian H(qi; pi; t) of a system with n degrees it was shown that the action of the one-parameter group of of freedom, the corresponding HJ equation is the first-order canonical transformations generated by an arbitrary function partial differential equation G(q ; p ) @S i i is determined by an equation similar to the HJ equa- H(q ; @S=@q ; t) + = 0: (1) tion. i i @t In this paper, we consider systems with a Hamiltonian Usually one is interested in complete solutions of the HJ that can depend on the time, we show that one can define the equation (1), that is, solutions S(qi; t; ®i) of Eq. (1), con- action of the one-parameter group of canonical transforma- taining n arbitrary parameters ®i, such that µ 2 ¶ tions generated by an arbitrary function G(qi; pi; t) on func- @ S det 6= 0; (2) tions defined on the extended configuration space, in such a @qi@®j 76 G.F. TORRES DEL CASTILLO because they yield the solution of the Hamilton equations The expression (7) is a (complete, R-separable) solution of (see, e.g., Refs. [2] and [3]). However, one can consider the the HJ equation that reduces to the specified function (5) for HJ equation as an evolution equation, which determines the t = 0. (A solution is R-separable if it is the sum of a function function S(qi; t) that reduces to a given function f(qi), for of two or more variables and functions of one variable.) t = 0 (or any other initial value, t0, of t). (This fact does not We close this subsection with a second example. The sound very strange if we consider the relationship between solution of the Hamilton equations for the time-dependent the HJ equation and the Schrodinger¨ equation.) In fact, if Hamiltonian p2 we have the solution of the Hamilton equations, we can use H = ¡ ktq; (8) it to find the solution of the HJ equation that satisfies any 2m initial condition S(qi; t0) = f(qi). (Similar ideas are often where m and k are constants, is given by mentioned in the standard textbooks on analytical mechanics, 3 tp0 kt 1 2 without presenting, however, a clear statement or an explicit q = q0 + + ; p = p0 + kt ; (9) procedure.) m 6m 2 For example, in the case of the Hamiltonian where q0 and p0 denote the values of q and p at t = 0, re- 2 2 spectively. Letting p1 + p2 H = + mgq2; (3) 2m S(q; 0; ®) = ®q; (10) where m and g are constants, the solution of the correspond- ® ing Hamilton equations can be readily obtained and is given where is an arbitrary constant, with the aid of Eqs. (9), we have ¯ by ¯ @S @S ¯ 1 2 1 2 = ¯ + kt = ® + kt : (11) (p ) t (p ) t gt2 @q @q t=0 2 2 q = (q ) + 1 0 ; q = (q ) + 2 0 ¡ ; 1 1 0 m 2 2 0 m 2 Thus, from Eqs. (1), (8), and (11), we have p = (p ) ; p = (p ) ¡ mgt; (4) µ ¶2 1 1 0 2 2 0 @S 1 @S = ¡ + ktq @t 2m @q where (q1)0 denotes the value of q1 at t = 0, and so on. As µ ¶ the initial condition we choose 1 1 2 = ¡ ® + kt2 + ktq: (12) 2m 2 S(q1; q2; 0) = ®1q1 + ®2q2; (5) Then, from Eqs. (10)–(12), we readily obtain where ® , ® are arbitrary constants. We want to find the 1 2 µ ¶ solution of the HJ equation [see Eqs. (1) and (3)] 1 1 1 k2t5 S(q; t; ®) = ®q + kt2q ¡ ®2t + ®kt3 + ; "µ ¶ µ ¶ # 2 2m 3 20 @S 1 @S 2 @S 2 = ¡ + ¡ mgq2; (6) R @t 2m @q1 @q2 which is an -separable complete solution of the HJ equa- tion. (We might start with expressions more complicated than that satisfies the initial condition (5). Recalling that (5) and (10), but these simple expressions are enough to ob- @S=@qi = pi, making use of Eqs. (4) and (5) we have tain complete solutions of the HJ equation.) ¯ It should be clear that a similar construction can be de- @S @S ¯ ¯ vised using an arbitrary function of (qi; pi; t) instead of a = ¯ = ®1; @q1 @q1 t=0 Hamiltonian. ¯ @S @S ¯ = ¯ ¡ mgt = ® ¡ mgt: 2.2. Families of solutions of the HJ equation and con- @q @q ¯ 2 2 2 t=0 stants of motion Substituting these expressions into the right-hand side of Any differentiable function, G(q ; p ; t), defines a (possibly Eq. (6) we have i i local) one-parameter group of canonical transformations, de- @S 1 £ ¤ termined by the (autonomous) system of first-order ordinary = ¡ ® 2 + (® ¡ mgt)2 ¡ mgq : @t 2m 1 2 2 differential equations Combining the last three equations one readily finds that, dqi @G(qj; pj; t) dpi @G(qj; pj; t) = ; = ¡ ; (13) choosing the integration constant so that Eq. (5) is satisfied, d® @pi d® @qi which are of the form of the Hamilton equations. S(q1; q2; t) = ®1q1 + ®2q2 ¡ mgtq2 For example, the function ® 2t (® ¡ mgt)3 ¡ ® 3 ¡ 1 + 2 2 : (7) p p 2m 6m2g G(q ; q ; p ; p ; t) = 1 2 + mgq ; (14) 1 2 1 2 m 1 Rev. Mex. Fis. 60 (2014) 75–79 GENERATION OF SOLUTIONS OF THE HAMILTON–JACOBI EQUATION 77 where m and g are constants, leads to the system of equations the examples of Sec. 2.1. Making use of Eqs. (16) and (19) we have [with S(qi; t; 0) = f(qi; t)] dq1 p2 dq2 p1 = ; = ; ¯ d® m d® m @S @S ¯ = ¯ ¡ mg® = ¡mg®; dp1 dp2 ¯ = ¡mg; = 0: (15) @q1 @q1 ®=0 d® d® ¯ ¯ @S @S ¯ From the last two equations we find that = ¯ = ®2 ¡ mgt: (20) @q2 @q2 ®=0 p = (p ) ¡ mg®; p = (p ) ; (16) 1 1 0 2 2 0 Substituting these expressions into Eq.