A comparative study of the Hannay's angles associated with a damped harmonic oscillator and a generalized harmonic oscillator

Usatenko, O. V.; Provost, Jean‐Pierre; Vallée, G.

doi:10.1088/0305-4470/29/10/035

“…In this adiabatic limit, the invariant I reduces to the adiabatic invariant of the classical DHO [22]. It is quite striking to note that the results obtained herein are similar to the case of the well known and extensively studied [16][17][18][21][22] generalized harmonic oscillator (GHO)…”

Section: Resultssupporting

confidence: 73%

“…We now show that, in the adiabatic limit, the angle (3.5) and the phase (4.8) recover the Hannay angle and the Berry phase [22]. The adiabatic approximation can be obtained by ignoring terms with two or more time derivatives in equation (2.6) and taking for σ (t) the adiabatic solution [15,18,21],…”

Section: Resultsmentioning

confidence: 71%

Igor' Ekhiel'evich Dzyaloshinskii (on his seventieth birthday)

Абрикосов¹,

Andreev²,

Brazovskii³

et al. 2001

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In this paper we use the algebraic and the invariant method to study the time-dependent damped harmonic oscillator from classical and quantum points of view. The solution of the classical equation of motion and the wavefunction solving the time-dependent Schrödinger equation are found explicitly. We show that the original time-dependent quantum-mechanical problem is completely related to the well known time-independent harmonic oscillator. In addition, we elucidate the intimate connection between the damped harmonic oscillator (DHO) and the generalized harmonic oscillator (GHO). More importantly, the evolution of the states of the DHO cannot be cyclic, in contradistinction with the states of the GHO. Explicit expressions for both the dynamical and the geometric angles and phases are deduced in the adiabatic limit. The coherent states describing the invariant-angle variables of the classical DHO are constructed; they allow us to recover the classical evolution and invariant from the quantum evolution.

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“…In this limit the quartic term contributes and one recovers the Hannay's angle of the quartic generalized oscillator determined in the previous section. This later result generalizes to nonlinear systems the result established in [6] for linear oscillators.…”

supporting

confidence: 71%

Comparative study of the adiabatic evolution of a nonlinear damped oscillator and a Hamiltonian generalized nonlinear oscillator

Usatenko

¹

,

Provost

²

,

Vallée

³

et al. 1998

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“…where the first term is the dynamical angle (or phase) and the second term is the geometrical Hannay angle (or Berry phase) [22]. In this adiabatic limit, the invariant I reduces to the adiabatic invariant of the classical DHO [22].…”

Section: Resultsmentioning

confidence: 99%

The Effect of Superthermal Electrons on Mach Probe Diagnostics

Shikama

¹

,

Kado

²

,

Kajita

³

et al. 2004

Jpn. J. Appl. Phys.

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In this paper we use the algebraic and the invariant method to study the time-dependent damped harmonic oscillator from classical and quantum points of view. The solution of the classical equation of motion and the wavefunction solving the time-dependent Schrödinger equation are found explicitly. We show that the original time-dependent quantum-mechanical problem is completely related to the well known time-independent harmonic oscillator. In addition, we elucidate the intimate connection between the damped harmonic oscillator (DHO) and the generalized harmonic oscillator (GHO). More importantly, the evolution of the states of the DHO cannot be cyclic, in contradistinction with the states of the GHO. Explicit expressions for both the dynamical and the geometric angles and phases are deduced in the adiabatic limit. The coherent states describing the invariant-angle variables of the classical DHO are constructed; they allow us to recover the classical evolution and invariant from the quantum evolution.

show abstract

A comparative study of the Hannay's angles associated with a damped harmonic oscillator and a generalized harmonic oscillator

Cited by 14 publications

References 8 publications

Igor' Ekhiel'evich Dzyaloshinskii (on his seventieth birthday)

Igor' Ekhiel'evich Dzyaloshinskii (on his seventieth birthday)

Comparative study of the adiabatic evolution of a nonlinear damped oscillator and a Hamiltonian generalized nonlinear oscillator

The Effect of Superthermal Electrons on Mach Probe Diagnostics

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