Several Solutions of the Damped Harmonic Oscillator with Time-Dependent Frictional Coefficient and Time-Dependent Frequency

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1 Advanced Studies in Theoretical Physics Vol. 11, 017, no. 6, HIKARI Ltd, Several Solutions of the Damped Harmonic Oscillator with Time-Dependent Frictional Coefficient and Time-Dependent Frequency Eun Ji Jang and Won Sang Chung 1 Department of Physics and Research Institute of Natural Science College of Natural Science, Gyeongsang National University Jinju , Korea Copyright c 017 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 discuss several solutions of the damped harmonic oscillator with time-dependent frictional coefficient and time-dependent frequency. We consider three types of frictional coefficient and frequency. We discuss the matrix formulation for the frictional coefficient γt) = γ 0 1+qt and frequency wt) = w 0 1+qt, q > 0). 1 Introduction The harmonic oscillator is a system playing an important role both in classical and quantum mechanics. It appears in various physical applications running from condensed matter to semiconductors see e.g.[1] for references to such problems). The harmonic oscillator equation with time-dependent parameters [-6] has been discussed for a sudden frequency change using a continuous treatment based on an invariant formalism [7]. This analytic treatment requires that the time-dependent parameter be a monotonic function whose variation is short compared with the typical period of the system. In 1931, Bateman presented [8] the so-called dual or mirror image formalism for damped oscillator. The Lagrangian for the linearly damped free 1 Corresponding author

2 64 Eun Ji Jang and Won Sang Chung particle was first obtained by Caldirola [9] and Kanai [10]. They considered the following Lagrangian L = e λt 1 mẋ, which gives the equation of motion for the the linearly damped free particle ẍ = λẋ In general the damped harmonic oscillator with time-dependent frictional coefficient and time-dependent frequency is governed by the following equation of motion: ẍ + γt)ẋ + w t)x = 0, 1) we set the mass of a particle to be unity. The Lagrangian L = 1 v Sx, t) does not give the eq.1). One can make use of Jacobis Last Multiplier Mt) to derive a suitable Lagrangian giving the eq.1). Consider Lagrangian [ 1 L = Mt) v 1 ] w t)x last multiplier turns out to be the integrating factor of such an equation obeying ) M = L 3) ẋ The canonical momentum p is given by ) p = Mẋ 4) Comparing the equation of motion derived from the eq.) with the eq.1), one get t Mt) = e γs)ds 0 5) By using the standard Legendre transformation it follows that the corresponding Hamiltonian is H = 1 Mt) p + 1 Mt)w t)x 6) In this paper we discuss several solutions of the damped harmonic oscillator with time-dependent frictional coefficient and time-dependent frequency. The paper is organized as follows: In section II we obtain solutions for the damped harmonic oscillator with time-dependent frictional coefficient and time-dependent frequency for γt) = γ 0, wt) = w 0, q > 0) and γt) = γ 1+qt 1+qt 0e ɛt, wt) = w 0 e ɛt, ɛ > 0) and γt) = γ 0 tanh t, wt) = w 0 tanh t. In section III we discuss the matrix formulation for the case of γt) = γ 0, wt) = w 0, q > 0). 1+qt 1+qt

3 Several solutions of the damped harmonic oscillator 65 Solution of the equation of motion In this section we solve the eq.1) for three cases. First we present the general solving technique. If we introduce the function yt) of the form we obtain yt) = e α t 0 γs)ds xt) 7) ÿ + 1 α)γẏ + [αα 1)γ α γ + w ]y = 0 8) Let us change the variable z = wt). If we replace some t-dependent functions with z-dependent ones through the eq.8) reduces to γt) = Λz), ẇ = fz), γ = fλ z) 9) f y + f[f + 1 α)λ]y + [αα 1)Λ αλ f + z ]y = 0 10) In the following subsections we obtain the solutions of the eq.10) for three choices of γt), wt). Here we impose the following initial condition: x0) = A, ẋ0) = 0 11).1 γt) = γ 0 1+qt, wt) = w 0 1+qt, q > 0), α = 1 In this case we have Thus, the eq.10) reduces to The solution to the eq.13) is Λz) = γ 0 w 0 z, fz) = q w 0 z 1) q z y + qγ 0 + q)zy + qγ 0 + w 0)y = 0 13) c 1 z λ + + c z λ γ 0 q > w 0 ) yz) = z λ 0 c 1 + c ln z) γ 0 q = w 0 ), 14) z λ 0 [c 1 cosν ln z) + c sinν ln z)] γ 0 q < w 0 ) λ ± = 1 q [ q + γ 0) ± γ 0 q) w 0 ) ] 15) λ 0 = 1 q q + γ 0) 16)

4 66 Eun Ji Jang and Won Sang Chung ν = 1 q w 0 ) γ 0 q) 17) For γ 0 q > w 0, using the initial condition 11), we have xt) = Aγ 0 + qλ ) qλ λ + ) 1 + qt) λ + γ 0 /q Aγ 0 + qλ + ) qλ λ + ) 1 + qt) λ γ 0 /q, 18) which is a q-deformed over damping case. For γ 0 q = w 0, using the initial condition 11), we have xt) = A1 + qt) λ 0 γ 0 /q 1 + γ 0 + qλ 0 q ln1 + qt) ), 19) which is a q-deformed critical damping case. For γ 0 q < w 0, using the initial condition 11), we have xt) = A1 + qt) λ 0 γ 0 /q [cosν ln1 + qt)) + γ 0 + qλ 0 νq which is a q-deformed under damping case.. γt) = γ 0 e ɛt, wt) = w 0 e ɛt, ɛ > 0) In this case we have Thus, the eq.10) reduces to sinν ln1 + qt))] 0) Λz) = γ 0 w 0 z, fz) = ɛz 1) [ ɛ z y + ɛ z ɛγ ] [ ) 0 γ 1 α)z y + 0 αα 1) + 1 z + αɛγ ] 0 z y = 0 w 0 w0 w 0 ) Now we choose α so that it obeys Then, we get The eq.) can be written as γ0 w0 αα 1) + 1 = 0 3) α = 1 1 ± 1 4 w 0 4) γ0 zy + [1 σz]y + µy = 0, 5) σ = γ 0 w 0 ɛ 1 α), µ = αγ 0 w 0 ɛ 6)

5 Several solutions of the damped harmonic oscillator 67 We set the solution to the eq.5) like yz) = a n z n+λ 7) n=0 Inserting the eq.7) into the eq.5), we have the following characteristic equation for λ: a 0 λ = 0 8) Here we set a 0 = 1 hence λ = 0. The recurrence relation is given by Solving the eq.9) we get a n+1 = σn µ n + 1) a n n 0) 9) a n = σn µ/σ) n n!), 30) a) 0 = 1, a) n = aa + 1)a + ) a + n 1). Thus, we have y 1 = 1 F 1 µ σ ; 1 : σz ) 31) Because the characteristic equation has double root, the second solution takes the following form : y = y 1 ln z + b m z m 3) Inserting the eq.3) into the eq.5) we get y 1 σy 1 + m b m z m 1 + µ σm)b m z m = 0 33) Solving the eq.33), we obtain b 1 = σa 0 a 1 34) and n + 1) b n+1 + µ σn)b n σa n + n + 1)a n+1 = 0, n 0) 35) The first few b n s are b 1 = σa 0 a 1 36) b = 1 4 [σ µ)b 1 + σa 1 4a ] 37)

6 68 Eun Ji Jang and Won Sang Chung Thus, we have b 3 = 1 9 [σ µ)b + σa 6a 3 ] 38) xt) = e αγ 0 ɛw 0 z [c 11 F 1 µ σ ; 1 : σz ) +c ln z 1 F 1 µ σ ; 1 : σz ) +c Because t = 0 gives z = w 0, from x0) = A, we get +c [µ ln w 0 µ 1 F 1 µ σ + 1; : σw 0 we used Because dx dt e αγ 0 ɛ A = c 1 µ 1 F 1 µ ) σ + 1; : σw 0 ) 1 1F 1 µ ) w0 σ ; 1 : σw 0 b m z m ] 39) mb m w m 1 0 ] 40) d dz µ 1F 1 a; 1 : σz) = aσµ 1 F 1 a + 1; : σz) 41) dx = ɛz, from v0) = 0, we obtain dz e αγ 0 ɛ A = c 11 F 1 µ ) σ ; 1 : σw 0 + c [ln w 01 F 1 µ ) σ ; 1 : σw γt) = γ 0 tanh t, wt) = w 0 tanh t In this case we have If we set z = w 0 η, we have b m w0 m ] 4) Λz) = γ 0 w 0 z, fz) = w 0 1 w 0 z 43) 1 η ) y +[1 α)γ 0 ]η1 η )y +[w 0 +αα 1)γ 0 +αγ 0 )η αγ 0 ]y = 0 44) Now we choose α so that it obeys Then, we get w 0 + αα 1)γ 0 = 0 45) α = 1 ) γ 0 ± γ0 4w0, 46) γ 0 α is real when γ0 > 4w0 over-damping case). We set the solution to the eq.44) like yη) = a n η n+λ 47) n=0

7 Several solutions of the damped harmonic oscillator 69 Inserting the eq.47) into the eq.44), we have the following characteristic equation for λ: a 0 λλ 1) = 0, a 1 λλ + 1) = 0 48) Here we set a 0 = 1, a 1 = 0 hence λ = 0, 1. The recurrence relation is given by a n+ = n + λ)n + λ α)γ 0) + αγ 0 a n n 0) 49) n + λ + )n + λ + 1) Solving the eq.49) for λ = 0, we obtain a m = m m)! n +) m n ) m, 50) n ± = 1 4 [1 1 α)γ 0 ± 1 γ 0 ) w 0] 51) Thus, we have the solution for λ = 0 like y 1 = F 1 n+, n ; 1/ : η ) 5) Solving the eq.49) for λ = 1, we obtain a m = Thus, we have the solution for λ = 1 like m m)! n + + 1/) m n + 1/) m 53) y = η F 1 n+ + 1/, n + 1/; 1/ : η ) 54) The general solution of the eq.44) is then given by and y = c 1 y 1 + c y 55) xt) = cosh t) αγ 0 [c 1 F 1 n+, n ; 1/ : η ) +c η F 1 n+ + 1/, n + 1/; 1/ : η ) ] 56) Using the initial condition we get c 1 = A, c = 0, hence xt) = Acosh t) αγ 0 F 1 n+, n ; 1/ : η ) 57)

8 70 Eun Ji Jang and Won Sang Chung 3 Matrix formulation In this section we consider the Matrix formulation for the case of γt) = γ 0, wt) = w 0, q > 0), α = 1. For the eq.1), we set 1+qt 1+qt xt) = e 1 Inserting the eq.58) into the eq.1) we get t 0 γs)ds ψt) 58) ψ + Ω t)ψ = 0, 59) Ω t) = w t) 1 4 γ t) 1 γ 60) In order to convert this second order equation to a first order system we introduce ) ψ1 t) ψt) =, 61) ψ t) ψ 1, ψ are two independent solutions of the eq.1). Now consider the following matrix equation: If we set d dt from the eq.63) we have ) ψ1 t) = ψ t) f1 t) f t) f 3 t) f 4 t) ) ) ψ1 t), 6) ψ t) f 1 f + f + f f 4 = 0 f 1 + f1 + f f 3 = Ω t) f 1 f 3 + f 3 + f 3 f 4 = 0 f 4 + f4 + f f 3 = Ω t) 63) f i t) = α i, i = 1,, 3, 4), 64) 1 + qt α 1 + α 4 = q α 1 qα 1 + α α 3 = w γ 0 1 qγ 0 65) Here we choose α 1 = α 4 = q/. Then, we have and the solution of the eq.6) is ) γ0 q α α 3 = w 0 66) ψt) = e t q) A ψ0) = e ln et q )A ψ0), 67)

9 Several solutions of the damped harmonic oscillator 71 ) α1 α A = α 3 α 4 In the eq.65) we choose α, α 3 as 68) α = α 3 = w = for q-deformed over damping and γ0 q ) w 0 69) ) α = α 3 = w 1 = iw = w0 γ0 q 70) for q-deformed under damping. For the q-deformed over damping case we have Using the relation A n = [n/] j=0 n j ) q ) n j w j I + q ) w A = σ = w [n 1)/] j=0 q ) Inserting the eq.7) into the eq.67), we obtain 71) ) ) n q n 1 j w j+1 σ, 7) j ) ) ψ1 t) = [e t ψ t) q] q/ cosh[w ln e t q] sinh[w ln e t ) ) q] ψ1 0) sinh[w ln e t q] cosh[w ln e t, 74) q] ψ 0) which gives the same result as the eq.18). Similarly, if we replace α = α 3 = w 1 = iw, we can have the same solution for the q-deformed under damping case. Acknowledgements. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government NRF- 015R1D1A1A ). 4 Conclusion In this paper we discussed several solutions of the damped harmonic oscillator with time-dependent frictional coefficient and time-dependent frequency. We considered the damped harmonic oscillator with time-dependent frictional

10 7 Eun Ji Jang and Won Sang Chung coefficient and time-dependent frequency governed by the equation of motion ẍ + γt)ẋ + w t)x = 0. We introduced the function yt) of the form yt) = e α t γs)ds 0 xt) and solved the second order differential equations for three types of time-dependent frictional coefficient and time-dependent frequency. First, we considered the case of γt) = γ 0, wt) = w 0, q > 0), α = 1 1+qt 1+qt for q-deformed over damping case, q-deformed critical damping case and q- deformed under damping case. Second, we considered the case of γt) = γ 0 e ɛt, wt) = w 0 e ɛt, ɛ > 0). In this case we obtained the solutions expressed in terms of the hypergeometric functions. Third, we considered the case of γt) = γ 0 tanh t, wt) = w 0 tanh t. In this case we also obtained the solutions expressed in terms of the hypergeometric functions. Finally, we discussed the Matrix formulation for the case of γt) = γ 0, wt) = w 0, q > 0), α = 1. 1+qt 1+qt We solved the matrix differential equation to obtain the same solution for the q-deformed under damping case. References [1] J. Yu and S. Dong, Exactly solvable potentials for the Schrodinger equation with spatially dependent mass, Phys. Lett. A, ), [] H.R. Lewis, Classical and Quantum Systems with Time-Dependent Harmonic-Oscillator-Type Hamiltonians, Phys. Rev. Lett., ), [3] V.V. Dodonov and V.I. Manko, Coherent states and the resonance of a quantum damped oscillator, Phys. Rev. A, ), [4] J. Janszky and Y.Y. Yushin, Squeezing via frequency jump, Opt. Commun., ), [5] F. Haas and J. Goedert, Dynamical symmetries and the Ermakov invariant, Phys. Lett. A, ), [6] S. Bouquet and H.R. Lewis, A second invariant for one-degree-of-freedom, time-dependent Hamiltonians given a first invariant, J. Math. Phys., ), [7] H. Moya-Cessa and M. Fernandez Guasti, Coherent states for the time

11 Several solutions of the damped harmonic oscillator 73 dependent harmonic oscillator: the step function, Phys. Lett. A, ), [8] H. Bateman, On Dissipative Systems and Related Variational Principles, Phys. Rev., ), [9] P. Caldirola, Forze non conservative nella meccanica quantistica, Il Nuovo Cimento, ), [10] E. Kanai, On the Quantization of the Dissipative Systems, Progress of Theoretical Physics, ), Received: August 1, 016; Published: May 15, 017