Generation of Solutions of the Hamilton–Jacobi Equation

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, αi) of Eq. (1), con- action of the one-parameter group of canonical transforma- taining n arbitrary parameters αi, 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∂αj 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, α) = αq, (10) where m and g are constants, the solution of the correspond- α 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 = α + 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 = − α + kt2 + ktq. (12) 2m 2 S(q1, q2, 0) = α1q1 + α2q2, (5) Then, from Eqs. (10)–(12), we readily obtain where α , α 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, α) = αq + kt2q − α2t + αkt3 + , "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 equation. (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 = ¯ = α1, ∂q1 ∂q1 t=0 Hamiltonian. ¯ ∂S ∂S ¯ = ¯ − mgt = α − 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 = − α 2 + (α − 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, dα ∂pi dα ∂qi which are of the form of the Hamilton equations. S(q1, q2, t) = α1q1 + α2q2 − mgtq2 For example, the function α 2t (α − mgt)3 − α 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 = , = , ¯ dα m dα m ∂S ∂S ¯ = ¯ − mgα = −mgα, dp1 dp2 ¯ = −mg, = 0. (15) ∂q1 ∂q1 α=0 dα dα ¯ ¯ ∂S ∂S ¯ From the last two equations we find that = ¯ = α2 − mgt. (20) ∂q2 ∂q2 α=0 p = (p ) − mgα, p = (p ) , (16) 1 1 0 2 2 0 Substituting these expressions into Eq. (18), taking into account Eq. (14), we obtain where (pi)0 denotes the value of pi for α = 0. Substituting these expressions into the first pair of equations (15) we get ∂S 1 = − (−mgα)(α − mgt) − mgq . dq (p ) dq (p ) − mgα 2 1 1 = 2 0 , 2 = 1 0 . ∂α m dα m dα m Combining the last equations and the initial condition (19) Hence, one finds that 2 (p2)0α (p1)0α gα q1 = (q1)0 + , q2 = (q2)0 + − . (17) m m 2 S(q1,q2, t, α2, α) = (α2 − mgt)q2 − mgαq1 3 3 The action of the canonical transformations defined by a 1 2 (α2 − mgt) − α2 + (α2 − mgt)gα + . (21) function G(qi, pi, t), on functions of (qi, t), will be defined 2 6m2g imitating the HJ equation. Definition. The image of a given arbitrary function f(qi, t) Note that t is treated here as a constant because the canonical under the one-parameter group of canonical transformations transformations do not affect t. Since the function (19) is a generated by G(qi, pi, t) is the function S(qi, t, α) such that 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), ∂S α G(qi, ∂S/∂qi, t) + = 0, (18) for all values of the parameter [which is essentially the two- ∂α parameter family of solutions (7)]. with the initial condition S(qi, t, 0) = f(qi, t).(Cf. Eq. (1).) Thus, we have obtained a two-parameter family of solu- The following example shows that this definition pro- tions of the HJ equation (6), starting from a one-parameter duces the expected effect in the case of transformations that solution of this equation. As we shall show in the following only affect the coordinates of the configuration space. In- Proposition, this is a consequence of the fact that the func- deed, the one-parameter group of canonical transformations tion (14) is a constant of motion for the Hamiltonian (3). generated by, e.g., G = p1 is given by [see Eqs. (13)] Proposition 1. The image of a solution of the HJ equation, S0(qi, t), under the one-parameter group of canonical trans- q = (q ) + α, q = (q ) , for i > 2, p = (p ) , 1 1 0 i i 0 i i 0 formations generated by a constant of motion G(qi, pi, t) is a one-parameter family of solutions of the same HJ equation. that is, “translations” along the q1-axis. Then, Eq. (18) takes the form Proof. Assuming that S(qi, t, α) is a solution of Eq. (18), ∂S ∂S making use of the chain rule repeatedly, we have (here, and + = 0, ∂q1 ∂α in what follows, there is summation over repeated indices) whose general solution is · ¸ ∂ ∂S ∂H ∂2S ∂2S H(qi, ∂S/∂qi, t) + = + S(qi, t, α) = F (q1 − α, q2, . . . , qn, t), ∂α ∂t ∂pj ∂α∂qj ∂α∂t ∂H ∂G(q , ∂S/∂q , t) ∂G(q , ∂S/∂q , t) where F is an arbitrary function. By imposing the initial con- = − i i − i i dition S(qi, t, 0) = f(qi, t), we find that ∂pj ∂qj ∂t µ ¶ ∂H ∂G ∂G ∂2S S(qi, t, α) = f(q1 − α, q2, . . . , qn, t), = − + ∂pj ∂qj ∂pi ∂qj∂qi which is the expected effect of a translation along the q1-axis. ∂G ∂2S ∂G As a second example, the images of the function − − . (22) ∂pi ∂t∂qi ∂t 3 3 (α2 − mgt) − α2 f(qi, t) = (α2 − mgt)q2 + (19) According to the hypothesis, G is a constant of motion, that 6m2g is (which is the function (7) with α1 = 0) under the transfor- ∂G ∂G ∂H ∂G ∂H + − = 0, (23) mations generated by (14) can be obtained proceeding as in ∂t ∂qj ∂pj ∂pj ∂qj

Rev. Mex. Fis. 60 (2014) 75–79 78 G.F. TORRES DEL CASTILLO

thus, the last line of Eq. (22) amounts to Then we see that the commutativity of the partial deriva- µ ¶ tives of S implies that the Poisson brackets of the Gi are ∂G ∂H ∂H ∂2S ∂2S − + + equal to zero. (The assumption that S(qi, t, αi) is a complete ∂pi ∂qi ∂pj ∂qj∂qi ∂t∂qi solution of the HJ equation implies that the Poisson brack- · ¸ ∂G ∂ ∂S ets {Gi,Gj} vanish everywhere.) Since the Poisson brack- = − H(qi, ∂S/∂qi, t) + ets {Gi,Gj} are equal to zero, the one-parameter groups of ∂pi ∂qi ∂t · ¸ transformations generated by the functions Gi commute. (In dqi ∂ ∂S fact, the groups of canonical transformations generated by = − H(qi, ∂S/∂qi, t) + dα ∂qi ∂t two functions F and G commute if and only if {F,G} is a trivial constant, not necessarily zero [4].) [see Eqs. (13)]. Hence, we obtain the equation Let us consider, for example, the function · d H(q (α), ∂S(q (α), t, α)/∂q , t) S(qi, t, αi) = α1q1 + α2q2 − mgtq2 dα i i i ¸ α 2t (α − mgt)3 − α 3 ∂S(q (α), t, α) − 1 + 2 2 , (26) + i = 0, 2m 6m2g ∂t which is the complete solution (7) of the HJ equation con- which involves the images of S and the coordinates qi under structed in Sec. 2.1. One ﬁnds that the transformations generated by G. Since the expression in- side the brackets in the last equation vanishes for α = 0, it is ∂S ∂S = α1, = α2 − mgt, equal to zero for all values of α, thus proving the assertion. ∂q1 ∂q2 A partial converse of this Proposition is true. The steps in which lead to [see Eqs. (25)] the proof show that if the images of a solution of the HJ equation under the transformations generated by G are solutions α1 = p1, α2 = p2 + mgt. (27) of the same HJ equation, and S is a complete solution of the HJ equation, then G is a constant of motion. The assumption On the other hand, that S is a complete solution of the HJ equation assures that 2 2 Eq. (23) holds everywhere. In fact, the following stronger ∂S α1t ∂S (α2 − mgt) − α2 = q1 − , = q2 + 2 , result holds. ∂α1 m ∂α2 2m g Proposition 2. A complete solution of the HJ equation, which, taking into account Eqs. (27), are of the form (24) S(qi, t, αi) (not necessarily separable or R-separable), is the with image of S(qi, t, 0) under the transformations generated by the n constants of motion p1t p2t 1 2 G1 = −q1 + ,G2 = −q2 + + gt . (28) ∂S m m 2 Gj(qi, pi, t) = − (qi, t, αi(qk, pk, t)) (24) ∂αj One can readily verify that G1 and G2 are constants of motion in involution, and that the function (26) can be obtained (j = 1, 2, . . . , n), obtained by eliminating the αi by means by means of the transformations generated by G1 and G2 of from ∂S mg2t3 pj = (qi, αi, t). (25) S(q , t) = −mgtq − , (29) ∂qj i 2 6

Furthermore, the functions Gi are in involution (i.e., the Pois- which is obtained from (26) setting α1 = α2 = 0. son bracket of Gi and Gj is equal to zero). (Note that, ac- Indeed, the function cording to Eq. (2), in principle, one can ﬁnd the αi from Eqs. (25).) G ≡ α1G1 + α2G2 Proof. We note that, by virtue of Eqs. (25), Eq. (24) is equiv- µ ¶ µ ¶ p1t p2t 1 2 alent to Eq. (18). Since, by hypothesis, S(qi, t, αi) is a com- = α −q + + α −q + + gt (30) 1 1 m 2 2 m 2 plete solution of the HJ equation, each function Gi, deﬁned by (24), is a constant of motion. In order to prove that the is a constant of motion for all values of the constants α1 and functions Gi are in involution, making use of Eq. (24), we α2. The one-parameter group of canonical transformations calculate the mixed partial derivative generated by G is given by (denoting by s the corresponding parameter) ∂2S ∂ = − Gj(qi, ∂S/∂qi, t, αk) ∂αi∂αj ∂αi α1t α2t q1 = (q1)0 + s, q2 = (q2)0 + s, ∂G ∂ ∂S ∂G ∂G m m = − j = j i . ∂pk ∂αi ∂qk ∂pk ∂qk p1 = (p1)0 + α1s, p2 = (p2)0 + α2s,

Rev. Mex. Fis. 60 (2014) 75–79 GENERATION OF SOLUTIONS OF THE HAMILTON–JACOBI EQUATION 79 hence, making use of the initial condition (29), 3. Concluding remarks

∂S ∂S According to Proposition 2, any complete solution of the = 0 + α1s, = −mgt + α2s, ∂q1 ∂q2 HJ equation is, locally, the image of a particular solution (without free parameters) under the action of an Abelian n- and substituting these expressions into Eq. (18), with G given dimensional group (that is, a group locally isomorphic to the by (30), additive Lie group Rn). As pointed out above, these solutions need not be separable or R-separable. It should be noted, ∂S α1t = α1q1 − (α1s) + α2q2 however, that some functions are invariant under the transfor- ∂s m mations generated by a function G, even if these transforma- α2t 1 2 tions are different from the identity. For instance, a function − (−mgt + α2s) − α2gt . m 2 that does not depend on q1 is invariant under the translations generated by p . Thus, 1 One can readily see that, at least in the case of time- independent operators, an analog of Proposition 1 holds in S(q , q , t) = α sq + α sq − mgtq 1 2 1 1 2 2 2 the case of the solutions of the Schrodinger¨ equation. (α s)2t (α s − mgt)3 − (α s)3 In addition to the R-separable complete solution (7), the − 1 + 2 2 , 2m 6m2g HJ equation for the Hamiltonian (3) also admits complete separable solutions in the same coordinate system (see also which reduces to the expression (26) when s = 1. Ref. [5])

1. G.F. Torres del Castillo, D.A. Rosete Alvarez´ and I. Fuentecilla 4. G.F. Torres del Castillo, Differentiable Manifolds: A Theoreti- Carcamo,´ Rev. Mex. F´ıs. 56 (2010) 113. cal Physics Approach (Birkhauser¨ Science, New York, 2012). 2. E.T. Whittaker, A Treatise on the Analytical Dynamics of Par- Sec. 8.2. ticles and Rigid Bodies, 4th ed. (Cambridge University Press, 5. G.F. Torres del Castillo, Rev. Mex. F´ıs. 59 (2013) 478. Cambridge, 1993). Chap. XII. 3. F. Gantmacher, Lectures in Analytical Mechanics (Mir, Moscow, 1975). Chap. 4.

Rev. Mex. Fis. 60 (2014) 75–79