Two Constants of Motion in the Generalized Damped Oscillator

Transcription

1 Advanced Studies in Theoretical Physics Vol. 10, 2016, no. 2, HIKARI Ltd, Two Constants o Motion in the Generalized Damped Oscillator Eun Ji Jang and Won Sang Chung Department o Physics and Research Institute o Natural Science College o Natural Science, Gyeongsang National University Jinju , Korea Copyright c 2015 Eun Ji Jang and Won Sang Chung. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we consider two types o the generalized damped oscillator. One is governed by a single rictional orce which has a position dependent rictional coeicient and is proportional to the N-th power o the velocity. Another is a quadratically damped harmonic oscillator. For two models, we construct two independent constants o motion. 1 Introduction The one dimentional damped harmonic oscillator called a Caldirola-Kanai Hamiltonian was irst solved by some authors [1-7]. They used the path integral to obtain the exact solution. The Caldirola-Kanai Hamiltonian gives the equation o motion with the linear damping and it depend on the time explicitly. But, i we consider the quadratic damping, the situation becomes more complicated. The Lagrangian or the quadratically damped ree particle was obtained by Denman [8]. He considered the ollowing Lagrangian L = e 2λx 1 2 mẋ2, which gives the equation o motion or the the linearly damped ree particle ẍ = λẋ 2

2 58 Eun Ji Jang and Won Sang Chung In 1993 Lemos[9] used the Nëther s theorem [10] to obtain the conserved charges or simple one-dimensional nonconservative systems such as the linearly damped ree particle, the quadratically damped ree particle and the damped harmonic oscillator problem. Recently, Smith [11] ound two constants o motion or the quadratically damped harmonic oscillator whose equation o motion is given by ẍ = λẋ α 2 x where λ, α are constants. He used the integrating actor to obtain the irst constant o motion depending on the position and velocity but not depending on time explicitly. Using the act that the equation o motion or the damped oscillator is separable in t, he also obtained the second constant o motion depending on the position, velocity and time which is independent o the irst one. He ound that the quadratically damped harmonic oscillator can be solved by eliminating the velocity rom two constant o motions. In this paper we consider two types o the generalized damped oscillator. One is governed by a single rictional orce which has a position dependent rictional coeicient and is proportional to the N-th power o the velocity. Another is a quadratically damped harmonic oscillator. In latter case we adopt the dierent approach rom the re.[11]. For two models, we construct two independent constants o motion. 2 Constant o motion depending on the position and velocity In this section we consider some solvable dissipative systems. We deal with one-dimensional dissipative system which takes the ollowing orm ẍ = (x, ẋ) (1) where γ(x) is a smooth unction in x and we set the mass o a particle to be unity. This equation is related to the damped system because the right hand side involves the velocity. Indeed, or (x, ẋ) = λẋ α 2 x we have a linearly damped oscillator and or (x, ẋ) = λẋ 2 α 2 x we have a quadratically damped oscillator. Recently, Smith [11] ound two constants o motion or the eq.(1) where (x, ẋ) is given by λẋ α 2 x. One is S(x, v) and another T (x, v; t). For S(x, v), we have S(x, v) = S(x 0, v 0 ) (2)

3 Two constants o motion in the generalized damped oscillator 59 and or T (x, v; t), we get T (x, v; t) = T (x 0, v 0 ; t 0 ) (3) where x 0, v 0, t 0 are initial values. The eq.(2) and the eq.(3) constitute two equations in the three independent variables x, v, and t. One equation may be used to eliminate v in the other, yielding the equation o motion x = x(s(x 0, v 0 ), T (x 0, v 0 ; t 0 ); t) (4) In this section we consider more general case than the re.[11]. We ind some solvable choice or (x, ẋ) determining two constants o motion. consider the solvable model The eq.(1) can be written as vdv = (x, v)dx (5) where v = ẋ. Multiplying the integrating actor µ(x) by both side o the eq.(5), we have µvdv µ(x, v)dx = 0 (6) We demand that the eq.(6) implies ds(x, v) = 0, which yields S v = µv, S x The integrability condition is then given by (µv) x = (µ) v = µ (7) In the ollowing subsections we ind the constants o motion depending on the position and velocity. 2.1 (x, v) = γ(x)v N In this case we set Inserting the eq.(9) into the eq.(8), we obtain (8) µ(x, v) = µ 1 (x)µ 2 (v) (9) µ 1 (x) = e α γ(x)dx (10) and µ 2 (v) = v α 2 (11)

4 60 Eun Ji Jang and Won Sang Chung or N = 2 and [ ] µ 2 (v) = v N α exp 2 N v2 N or N 2. Generally, we have (12) Especially, or N = 2, α = 2, we have S = µ 1 (x) v µ2 (v)vdv (13) S = 1 2 e2 γ(x)dx v 2 (14) 2.2 (x, v) = γ(x)v 2 w 2 0x In this case we set µ = µ(x) (15) Inserting the eq.(15) into the eq.(8), we obtain µ(x) = e 2 x γ(x)dx (16) Generally, we have S = 1 2 µ(x)v2 + w 2 0 x dxxµ(x) (17) Especially, or γ(x) = γ 0 = const, we have µ(x) = e 2γ0x, hence we get the ollowing constant o motion S = 1 [ e 2 e2γ 0x v 2 + w0 2 2γ 0 x ] (2γ 0 x 1) + 1 (18) 3 Constant o motion depending on the position, velocity and time In this section we ind the second constant o motion depending on the position, velocity and time. Finding T is, in principle, straightorward or one dimensional systems. The eq.(1) is separable in t with a solution o the orm 4γ 2 0 T (x, v; t) = wt ˆT (x, v), (19) where T t = w, From the eq.(19), (20), we get ˆT x v + ˆT v = w (20) ˆT (x, v) ˆT (x 0, v 0 ) = w(t t 0 ) (21)

5 Two constants o motion in the generalized damped oscillator 61 Without generality we set ˆT (x 0, v 0 ) = 0, hence From the relation ˆT (x, v) = w(t t 0 ) (22) t t 0 = we use the integration by parts to obtain where ˆT (x, v) = w v v 0 dx v, (23) (n + 1)! vn+1 D n D = d dv = v + v x, (24) (25) The solution (24) indeed obeys the second relation o the eq.(20) and the proo is given in the Appendix. 3.1 = γ(x)v N In this case we have D n ( 1 ) = n+1 k=1 k (x)v kn n+2k 2 (26) Using the mathematical induction we have the ollowing recurrence relation: Γ (n+1) k = (kn + n 2k + 2) k Solving the eq.(27), we get where Γ (n+1) 1 = (N + n) 1 (27) 1 γ Γ (n+1) n+2 = 1 γ d k 1 dx d n+1 dx, (k = 2, 3,, n + 1) (28) (29) 1 = ( 1)n+1 (N) n, (30) γ (N) 0 = 1, (N) n = N(N + 1)(N + 2) (N + n 1) (31) Thus, the constant o motion is given by T (x, v; t) = wt w n+1 k (n + 1)! (x)v kn+2k 1 (32) k=1

6 62 Eun Ji Jang and Won Sang Chung The irst ew k s are Γ (1) 1 = N γ, Γ(1) 2 = γ γ 3 Γ (2) N(N + 1) 1 =, Γ (2) 2 = (3N 1) γ γ γ, 3 Γ(2) 3 = 1 γ ( ) γ γ = γ 0 v 2 w 2 0x In this case we have D n ( 1 ) = [ n 2 ] n k k=0 l=0 v n 2k (γ 0 v 2 w 2 0x) 2n+1 2k l (33) where [x] = m or m x < m + 1 ( m is an integer ). For even n and odd n we have m 2m k D 2m = Γ (2m) v 2m 2k (34) k=0 l=0 (γ 0 v 2 w0x) 2 4m+1 2k l m 2m+1 k D 2m+1 = v 2m+1 2k (35) (γ 0 v 2 w0x) 2 4m+3 2k l k=0 l=0 Using the mathematical induction we have the ollowing recurrence relation: 0,0 = (Γ (2m) 0,2m) + w 2 0Γ (2m) 0,0 (36) 0,2m+1 = 2γ 0 (2m + 1)Γ (2m) 0,2m (37) 0,l = 2γ 0 (4m + 2 l)γ (2m) 0,l 1 (Γ(2m) 0,l 1 ) + w 2 0Γ (2m) 0,l, (1 l 2m) (38) k,0 = (2m 2k + 2)Γ (2m) k 1,0 + w2 0Γ (2m) k,0, (1 k m) (39) k,2m+1 k = (2m 2k+2)Γ(2m) k 1,2m+1 k 2γ 0(2m+1 k)γ (2m) k,2m k (Γ(2m) k,2m k ) (1 k m) (40) = (2m 2k + 2)Γ (2m) k 1,l 2γ 0(4m + 2 2k l)γ (2m) 1 (Γ (2m) 1 ) + w 2 0Γ (2m) (1 k m, 1 l 2m) (41) Then, the constant o motion is given by T (x, v; t) = wt w [ n 2 ] n k (n + 1)! k=0 l=0 The irst ew s are Γ (1) 0,0 = w0, 2 Γ (1) 0,1 = 2γ 0 v 2n 2k+1 Γ (2) 0,0 = 3w 4 0, Γ (2) 0,1 = 10γ 0 w 2 0, Γ (2) 0,2 = 8γ 2 0 (γ 0 v 2 w 2 0x) 2n+1 2k l (42) Γ (2) 1,0 = 8γ 2 0, Γ (2) 1,1 = 2γ 0 (43)

7 Two constants o motion in the generalized damped oscillator 63 4 Conclusion In this paper we we considered two types o the generalized damped oscillator. One is governed by a single rictional orce γ(x)v N which has a position dependent rictional coeicient and is proportional to the N-th power o the velocity. Another is a quadratically damped harmonic oscillator governed by the orce γ 0 v 2 w 2 0x. For two types o orce we obtained two constants o motion explicitly; one depends on the position and velocity and another depends on the position, velocity and time, which implies that two models are exactly solvable although the solution is a little ormal. There remains much work in this direction. We doubt whether another solvable damped oscillator exists or not. I hope that this topic and its related problems will be clear in the near uture. Acknowledgements. This work was supported by the National Research Foundation o Korea Grant unded by the Korean Government (NRF- 2015R1D1A1A ). Reerences [1] C. I. Um, K. H. Yeon and W. H. Kahng, The quantum damped driven harmonic oscillator, J. Phys. A: Math. and Gen., 20 (1987), no. 3, [2] C. I. Um, I. H. Kim, K. H. Yeon and T. F. George, Lakshmi N. Pandey, Waveunctions and minimum uncertainty states o the harmonic oscillator with an exponentially decaying mass, J. Phys. A: Math. and Gen., 30 (1997), no. 7, [3] K. H. Yeon, D. H. Kim, C. I. Um and T. F. George, Lakshmi N. Pandey, Relations o canonical and unitary transormations or a general timedependent quadratic Hamiltonian system, Phys. Rev. A, 55 (1997), no. 6, [4] K. H. Yeon, Dan F. Walls, C. I. Um, T. F. George, Lakshmi N. Pandey, Quantum correspondence or linear canonical transormations on general Hamiltonian systems, Phys. Rev. A, 58 (1998), no. 3,

8 64 Eun Ji Jang and Won Sang Chung [5] C. I. Um, S. M. Shin, K. H. Yeon and T. F. George, Reply to Comment on Exact wave unction o a harmonic plus an inverse harmonic potential with time-dependent mass and requency, Phys. Rev. A, 61 (2000), no [6] H. R. Lewis, Classical and Quantum Systems with Time-Dependent Harmonic-Oscillator-Type Hamiltonians, Phys. Rev. Lett., 18 (1967), no. 13, [7] H. R. Lewis, Jr. and W. B. Rieseneld, An Exact Quantum Theory o the Time-Dependent Harmonic Oscillator and o a Charged Particle in a Time-Dependent Electromagnetic Field, J. Math. Phys., 10 (1969), no [8] H. H. Denman, Time-Translation Invariance or Certain Dissipative Classical Systems, Am. J. Phys., 36 (1968), no [9] N. A. Lemos, Symmetries, Noether s theorem and inequivalent Lagrangians applied to nonconservative systems, Revista Mexicana de Fisica, 39 (1993), no. 2, [10] E. Nöther, Invariante Variationsprobleme, Nachr. Akad. Wiss. Gottingen, Math.-Phys. Kl. II, [11] B. R. Smith, The quadratically damped oscillator: A case study o a non-linear equation o motion, Am. J. Phys., 80 (2012), no Appendix In this appendix we show that the solution o (20) indeed obeys the second relation o the eq.(24). +w v ˆT x + ˆT v = w (n + 1)! (n + 1)vn D n (n + 1)! vn+2 x D n + w (n + 1)! vn+1 v D n

9 Two constants o motion in the generalized damped oscillator 65 = w (n + 1)! vn+1 D n+1 + w v n D n n! = w (44) Received: November 17, 2015; Published: February 16, 2016