Hamilton-Jacobi Approach for Power-Law Potentials

Transcription

1 Brazilian Journal of Physics, vol. 36, no. 4A, Deceber, Hailton-Jacobi Approach for Power-Law Potentials R. C. Santos 1, J. Santos 1, J. A. S. Lia 2 1 Departaento de Física, UFRN, , Natal, RN, Brazil 2 Instituto de Astronoia, Geofísica e Ciências Atosféricas - USP Rua do Matão, Cid. Universitária, 558-9, São Paulo, SP, Brazil Received on 25 August, 26 The classical and relativistic Hailton-Jacobi approach is applied to the one-diensional hoogeneous potential, V (q)=αq n, where α and n are continuously varying paraeters. In the non-relativistic case, the exact analytical solution is deterined in ters of α, n and the total energy E. It is also shown that the non-linear equation of otion can be linearized by constructing a hypergeoetric differential equation for the inverse proble t(q). A variable transforation reducing the general proble to that one of a particle subjected to a linear force is also established. For any value of n, it leads to a siple haronic oscillator if E >, an anti-oscillator if E <, or a free particle if E =. However, such a reduction is not possible in the relativistic case. For a bounded relativistic otion, the first order correction to the period is deterined for any value of n. For n >> 1, it is found that the correction is just twice that one deduced for the siple haronic oscillator (n = 2), and does not depend on the specific value of n. Keywords: Hailton-Jacobi equation; Power-law potentials I. INTRODUCTION The Hailton-Jacobi (HJ) equation is a powerful ethod either to the relativistic and classical fraework, and also plays a proinent role in quantu echanics as a special route for a coprehension of the Schrödinger equation[1, 2]. The associated action variable ethod, which is a slight odification of the HJ approach, is also quite useful for deterining frequencies and energies of periodic systes. Even for the general relativity theory, the iportance of the HJ approach has long been recognized by any authors (see, for instance, [3] and Refs. therein). Indeed, one of the ost elegant ethods for describing geodesics and orbits of test particles in Schwarzchild and Kerr spaceties, as well as for any stationary gravitational configuration, is provided by the relativistic HJ equation[3-5]. In the nonrelativistic doain, the standard applications of the HJ theory are the haronic oscillator, the Kepler proble, and charged particles oving in electro-agnetic fields. Recently, the ethod has also been applied to the rocket proble[6]. Pars[7] studied the otion of a classical particle in a plane under central attraction derived fro the potential, V (r)=αr n, for soe particular values of n. However, to the best of our knowledge, if α and n are continuously varying paraeters the general solution has not been obtained even for the one-diensional case. In this letter we discuss an analytical solution for the power law potential both for the classical and relativistic case. As we shall see, the HJ equation for a particle oving under the action of a one-diensional potential, V (q)=αq n, has a general and unified solution in ters of hypergeoetric functions. The period of the otion as a function of the power index n and of the α paraeter are easily deterined fro the general solution. For copleteness, we also show that such a proble ay also be exactly solved starting fro the nonlinear equation of otion by eploying a slightly odified Euler- Lagrange approach. By changing the coordinate and adopting an auxiliary tie, we show that the otion of the particle in such a potential ay be reduced to the proble of a particle under the action of a linear force: an oscillator if E >, an anti-oscillator if E < or a free-particle if E =, where E is the total energy of the particle. Classically, these results hold regardless of the value of n. Such a reduction cannot be ipleented in the relativistic case. However, the general proble ay be reduced to an integral, which is a natural extension of the earlier Synge s treatent for the relativistic haronic oscillator[8]. II. HAMILTON-JACOBI APPROACH The classical Lagrangian for a particle subject to the onediensional potential, V (q)=αq n, reads L(q, q,t)= 1 2 q2 αq n, (1) where an overdot eans total derivative. The paraeters α, n, vary continuously, and, q are, respectively, the ass of the particle and its generalized coordinate. The Hailtonian is given by (p is the canonical oentu) H(q, p)= p2 + αqn = E, (2) and since it does not depend explicitly on the tie, it represents a conserved quantity which is the energy of the syste. The Hailton-Jacobi equation for the Hailtonian (2) assue the following for [1] 1 ( ) S 2 + αq n + S =, (3) q t where S is the Hailton s principal function. Since the explicit dependence of S on tie is involved only in the last ter,

2 1258 R. C. Santos et al. the variables can be separated. Following standard lines, the solution is assued to have the for S(q,E,t)=W(q,E) Et, (4) where E, the constant of integration, has been identified with the total energy. With this choice, the tie is readily eliinated fro (3), and Hailton s principal function S becoes S = E αq n Et. (5) It should be noticed that if the energy E is positive, the constant α appearing in the potential ay assue either negative or positive values, however, if E, the allowed values of α are necessarily negative. Since H = E, such considerations also follows naturally fro the positivity of the kinetic energy K = 1 2 q2 = E αq n. (6) The second integration constant is a consequence of the Jacobi transforation equation β = S q E = t. (7) 2 E 1 α E qn The root of E arises naturally if one takes for negative values of E, E = E, and copute the partial derivative with respect to E. In order to integrate (7) we ake the change of variable u =(α/e)q n. Thus we have that t + β = 2n 2 E ( ) E 1/n u y 1/n 1 dy. (8) α 1 y It is worth noticing that the integral appearing on the right hand side of the above expression is the incoplete Beta function[9] u B u (a,b)= y a 1 (1 y) b 1 dy, for a = 1/n and b = 1/2. Without loss of generality we ay take q(t = ) =, so that β = in (8). When the otion is finite, the particle oves back and forth (oscillatory otion), and the period T can be easily deterined. We see fro (6) that at the turning points ( q = ) the aplitude of the oscillation is A =(E/α) 1/n, and, therefore, u ax = 1, is the axiu value in the upper liit of the integral (8). The period of the otion is given by T = 4t ax and, taking these considerations into account, we find that equation (8) yields T = 2 αn 2 A1 1/n B(1/n,1/2), (9) where B(1/n,1/2) = πγ ( ) ( 1 n /Γ 1n + 1 2) is the coplete Beta function. The period of this oscillator is clearly dependent on the aplitude A, unless n = 2. As should be expected, for n = 2 equation (9) reduces to T = 2π 2α, which is the well known result for the siple haronic oscillator. It should be recalled that the incoplete Beta function is related to the hypergeoetric Gaussian F(a,b;c;u) by the following identity: B u (a,b)=a 1 u a F(a,1 b;1+ a;u). It thus follows that the result (8) ay be expressed as ( 1 t + β = 2 E qf 2, 1 n ; 1 n + 1; α ) E qn. (1) As a check, we notice that for n = 2 the above expression can be written as ( 1 ω(t + β)=zf 2, 1 2 ; 3 ) 2 ;z2, (11) where we have defined the variable z = α/eq and ω = 2π/T is the frequency of the oscillatory otion. Since F(1/2,1/2;3/2;z 2 )=z 1 arcsinz [9]; by returning to the old variable q, the above equation can be recast as 2E q(t)= sinω(t + β), (12) ω2 which describes the otion of a siple haronic oscillator [1]. III. MODIFIED EULER-LAGRANGE APPROACH We present here another ethod of solution for the proble outlined in the introduction. In principle, by considering that the HJ approach leads to the hypergeoetric function for t(q) as given by (1), it should be possible to obtain the differential hypergeoetric equation starting directly fro the equation of otion. The iportance of this ethod is two-fold: the initial conditions reains arbitrary and the linearization procedure is a guarantee that we get the coplete solution for the nonlinear differential equation governing the otion of the particle. The equation of otion (Euler-Lagrange equation) for the Lagrangian given by (1) is q + nαq n 1 =. (13) Instead to solve the equation of otion above, we will consider an equivalent equation. By ultiplying it by q, and inserting the definition of the total energy of the particle given by (6), we obtain: q q n 2 q2 + ne =. (14) It is worth entioning that, if soe few identifications are ade, (14) is the sae differential equation describing the evolution of the scale factor in the standard Friedan- Robertson-Walker (FRW) cosological odel [1] (in this connection, see also [11]). A first integral of (14) is q 2 = 2E ( 1 α E qn). (15)

3 Brazilian Journal of Physics, vol. 36, no. 4A, Deceber, It should be noted that the signals for the total energy E and for the constant α ust be properly chosen. So, if E > the constant α can have any signal, but if E we ust have a negative α. The integration of (15) is achieved with the introduction of an auxiliary variable defined by u =(α/e)q n. With the aid of this transforation the first integral is written as where du dt = nbu1 1/n (1 u) 1/2, (16) B = 2α ( ) E 1/2 1/n. α At this stage we could integrate (16), as it was done in Sec. (II). Instead, on ay consider the inverse proble, i.e., the solutions for a differential equation expressing the tie as a function of the auxiliary variable u(q). This inversion is readily done through (16), which provides the first derivative of t(u): dt du = 1 nb u1/n 1 (1 u) 1/2, (17) whereas the second derivative ay be put in the for u(1 u) d2 t dt +[c (a + 1)u] =. (18) du2 du where c = 1 1/n and a = 1/2 1/n. This is Gauss s hypergeoetric differential equation. Its general solution consists of a linear cobination of two linearly independent solutions [13]: t(u)=c 1 F(a,;c;u)+C 2 u 1 c F(1 + a c,1 c;2 c;u), (19) where F is the hypergeoetric function, and C 1, C 2 are the arbitrary constants of integration to be deterined by the boundary conditions. It is a property of the hypergeoetric function that if any of the first two paraeters is zero, then the series terinates (i.e., F(a,;c;u)=1). It thus follows that the solution (19) reduces to ( α ) ( 1/n 1 t = C 1 +C 2 qf E 2, 1 n ; 1 n + 1; α ) E qn, (2) where we have substituted the value of the paraeters a and c. The case n = 2gives E q(t)= α sinω(t t ), (21) where the constant ω = 1/C 2 and t = C 1 reain to be deterined. IV. PARAMETRIC SOLUTIONS AND PERIODICITY As we have seen, if E, the general solution t(q) cannot be inverted to obtain explicitly q(t). In such cases, paraetric solutions are usually ore enlightening. Soe siplicity is achieved when we replace the pair (t,q) (τ,q), where the conforal tie τ and the new coordinate Q are defined by and dt = q(τ)dτ, (22) Q = q n 2. (23) Under the above transforations, the expressions for the Lagrangian and energy E, given by (1) and (2), becoe L = n 2 Q 2 Q 2 αq 2, (24) E = n 2 Q 2 Q 2 + αq 2, (25) where a prie denotes derivative with respect to τ. As one ay check, the Euler-Lagrange equations are now given by QQ Q 2 αn2 =, (26) and inserting Q 2 fro the energy conservation law, we obtain Q ( n 2 E ) Q =. (27) This is an interesting result. The above equation describes the otion of a classical particle under the action of a linear force. If E < (E > ) the force is of restoring (repulsive) type, while the otion of a free particle corresponds to E =. The general solution of (27) is Q = Q ε sin ε(ωτ + δ), (28) where ε = 2E/, ω = n/2 is the frequency of the oscillator (anti-oscillator) and Q, δ are integration constants. Since the energy E ay be negative, the phase δ ay assue coplex values. The constant Q can be deterined using the energy equation written in ters of the conforal tie. It turns out that Q = 2α/. Substituting this into (28), and returning to the original coordinate q,wehave q(τ)=( /2α) 1/n ( sin εωτ ε ) 2/n, (29) where we have put δ = in (28). V. RELATIVISTIC HAMILTON-JACOBI APPROACH Let us now consider the relativistic treatent for the hoogeneous potential. For a single-particle the relativistic Lagrangean is L = c 2 1 q2 c 2 αqn. (3)

4 126 R. C. Santos et al. The Hailtonian function for the particle is given by the general forula H = q L q L = p 2 c c 4 + αq n, (31) where p is the canonical oentu. The Hailton-Jacobi equation for S is obtained by replacing in (31) p by S/ q and H by ( S/ t): ( ) S 2 ( ) S 2 c 2 q t + αqn + 2 c 4 =. (32) As before, a solution for (32) can be found separating the variables in the for S(q,E,t)=W(q,E) Et, (33) where the integration constant E is again the total energy. With the above choice one finds fro (32) S = 1 (E αq c n ) 2 2 c 4 Et, (34) which should be copared with its classical version (5). The second integration constant arises out fro the transforation equation (β = S E ) and we have finally t + β = 1 E αq n. (35) c (E αq n ) 2 2 c4 Without loss of generality we take q(t = ) =, so that β = in (35). We next suppose that the potential is a syetric function about the origin, that is, we consider only even values for n. Then the otion will be liited between V ( A) and V (A) where the aplitude is now given by A = [(E c 2 )/α] 1/n, and, fro (35), the period T is deterined by T = 4 c ( ). (36) 1 c 2 2 E αq n Writing the total energy as E = c 2 (1 + ε), and considering that the potential energy is sall copared to the rest ass energy c 2, relativistic corrections to second order in the period T are T = 4 c 2κ(A n q n ) (1 + 3κ 4 (An q n )), (37) where κ = α/c 2. Introducing the variable change y =(q/a) n the above integral can be rewritten as 4 T = ca n 2 1 n 2κ 1 y ( n 1 n ) (1 y) ( 1 2 ) dy n+2 3κA cn ( n 1 y n ) (1 y) 1 2 dy, (38) 2κ whose values are tabulated in ters of the coplete Beta function[9]. Substituting for κ we write the period as [ T = 2 αn 2 A1 n/2 B(1/n,1/2) αa n ( )] 2n 8 c 2. n + 2 (39) Observe that for n = 2 (relativistic haronic oscillator), the above equation reduces to [ T = 2π αa 2 ] 2α 8 c 2, (4) where the ter (3αA 2 )/8c 2 is the first order relativistic correction earlier obtained by Synge[8] following a different approach (see Appendix). The general for (39) is an interesting one. The ter ultiplying the square bracket is just the nonrelativistic period T for the bounded hoogeneous potential as given by (9). The reaining ter represents the relativistic correction T = 3 αa n ( ) 2n T 8 c 2, (41) n + 2 and, although it depends of n, we see that for n 1 it saturates around (3αA n )/4c 2 which is twice the relativistic correction for the haronic oscillator. In conclusion, we have shown that the classical version of hoogeneous potential proble can copletely be solved by the HJ approach. In the relativistic case we not succeed in obtaining an analytic solution, but the earlier Lagrangian Synge s treatent for the haronic oscillator is readily generalized using the HJ transforation. Finally, we stress that although iportant fro their own right, such probles ay also have interest for describing soe excited states appearing in quantu chroodynaics which have been usually described by the relativistic oscillator[14]. The effective hoogeneous potential also appears naturally in the cosological fraework. In this case, the values of the power n is closely related to the nature of the cosic fluid [1, 11]. Its usefulness for providing a possible classification schee for cosological odels has also been recently discussed [15]. Acknowledgent This work was partially supported by CNPq and CAPES (Brazilian Research Agencies). JASL is also grateful to FAPESP No. 4/ VI. APPENDIX: REMARKS ON SYNGE S INTEGRAL The relativistic haronic oscillator (n = 2) was long ago discussed by Synge [8]. In this appendix, we show how the exact Synge s integral for for the period of the otion can be generalized for a power-law index. For an arbitrary n, the period is given by (see (36)) T n = 4 c ( ). (42) 1 c 2 2 E αq n

5 Brazilian Journal of Physics, vol. 36, no. 4A, Deceber, Now, by writing E = c 2 + αa n one finds and, for n = 2wehave T n = 2 αa n [1 + 2χ 2 (1 (q/a) n )] 1 (q/a) n + χ 2 (1 (q/a) n ) 2, (43) where χ 2 = αa n /(c 2 ). Introducing the variable ϕ by q = Asin 2/n ϕ, the above integral becoes T n = 4 αn 2 A1 n/2 π 2 sin 2/n 1 ϕ(1 + 2χ 2 cos 2 ϕ)dϕ, 1 + χ 2 cos 2 ϕ (44) π 2 (1 + 2χ 2 cos 2 ϕ)dϕ T 2 = 4 2α 1 + χ 2 cos 2 ϕ, (45) which is the Synge s integral with a slight different notation (our α = k 2 /2, and A = a in Synge s work[8]). Finally, by expanding (44) in powers of χ, the relativistic correction for the period as given by (39) is recovered. For n=2 it reduces to Synge s expression. [1] H. Goldstein, Classical Mechanics (Addison-Wesley Publ. Co., Reading, Mass. 198). [2] P. R. Holland, The Quantu Theory of Motion (Cabridge UP, Cabridge, 1995). [3] B. Carter, Phys. Lett. A26, 399 (1968); Phys. Rev. 174, 1559 (1968). [4] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergaon Press, New York, 1975). [5] W. H. C. Freire, V. B. Bezerra, and J. A. S. Lia, Gen. Rel. Grav. 33, 147 (21); R. C. Santos, J. A. S. Lia, and V. B. Bezerra, Gen. Rel. Grav. 33, 1969 (22). [6] G. del-valle, I. Capos, and J. L. Jiénez, Eur. J. Phys. 17, 253 (1996); I. Capos, G. del-valle, J. L. Jiénez, Eur. J. Phys. 24, 469 (23). [7] L. A. Pars, A Treatise on Analytical Dynaics (John Wiley & Sons, New York, 1965). [8] J. L. Synge, Classical Dynaics, in Encyclopedia of Physics, Vol. III, (196). [9] M. Abraowitz and I. A. Stegun, Handbook of Matheatical Functions (Dover, New York, 197). [1] J. A. S. de Lia, J. A. M. Moreira, and J. Santos, Gen. Rel. Grav. 3, 425 (1998). [11] M. Szydlowski and W. Czaja, Phys. Rev. D (24). [12] M. J. D. Assad and J. A. S. Lia, Gen. Rel. Grav. 2, 427 (1988). [13] J. B. Seaborn, Hypergeoetric Functions and Their Applications (Springer-Verlag, New York, 1991). [14] C. Dulleond and E. van Beveren, Phys. Rev. D28, 128 (1983). [15] M. Szydlowski and W. Czaja, Phys. Rev. D (24); M. Szydlowski and A. Kurek, gr-qc/6898.