Inequivalent Lagrangians for the damped harmonic oscillator

2004 ◽  
Vol 82 (7) ◽  
pp. 561-567 ◽  
Author(s):  
S Ghosh ◽  
J Shamanna ◽  
B Talukdar
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Higher Order ◽  
Space Structure ◽  
Phase Space Structure ◽  
Damped Harmonic Oscillator ◽  
Constants Of Motion ◽  
Order Degenerate ◽  
Degenerate Lagrangian ◽  
Dynamical Origin

A constant of the motion, in addition to what exists in the literature, is presented for the damped harmonic oscillator and its dynamical origin is investigated. These two constants of motion are used to construct expressions for a hierarchy of inequivalent Lagrangians. It is shown that each inequivalent Lagrangian may be related to a higher order degenerate Lagrangian. The hierarchical Lagrangians tend to pose some characteristic problems for discussing the corresponding phase-space structure. PACS Nos.: 47.20.Ky, 42.81.Dp

Explicit symplectic-like integration with corrected map for inseparable Hamiltonian

2020 ◽  
Vol 501 (1) ◽  
pp. 1511-1519
Author(s):  
Junjie Luo ◽  
Weipeng Lin ◽  
Lili Yang
Keyword(s):  
Phase Space ◽  
Space Structure ◽  
Eighth Order ◽  
Time Step ◽  
Extended Phase ◽  
Energy Error ◽  
Phase Space Structure ◽  
Compact Binary ◽  
Three Body

ABSTRACT Symplectic algorithms are widely used for long-term integration of astrophysical problems. However, this technique can only be easily constructed for separable Hamiltonian, as preserving the phase-space structure. Recently, for inseparable Hamiltonian, the fourth-order extended phase-space explicit symplectic-like methods have been developed by using the Yoshida’s triple product with a mid-point map, where the algorithm is more effective, stable and also more accurate, compared with the sequent permutations of momenta and position coordinates, especially for some chaotic case. However, it has been found that, for the cases such as with chaotic orbits of spinning compact binary or circular restricted three-body system, it may cause secular drift in energy error and even more the computation break down. To solve this problem, we have made further improvement on the mid-point map with a momentum-scaling correction, which turns out to behave more stably in long-term evolution and have smaller energy error than previous methods. In particular, it could obtain a comparable phase-space distance as computing from the eighth-order Runge–Kutta method with the same time-step.

scholarly journals Dirac’s Method for the Two-Dimensional Damped Harmonic Oscillator in the Extended Phase Space

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 180 ◽  
Author(s):  
Laure Gouba
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Canonical Quantization ◽  
Extended Phase Space ◽  
Two Dimensional ◽  
Class Constraint ◽  
Extended Phase ◽  
Damped Harmonic Oscillator ◽  
Points Of View ◽  
Space Description

The system of a two-dimensional damped harmonic oscillator is revisited in the extended phase space. It is an old problem that has already been addressed by many authors that we present here with some fresh points of view and carry on a whole discussion. We show that the system is singular. The classical Hamiltonian is proportional to the first-class constraint. We pursue with the Dirac’s canonical quantization procedure by fixing the gauge and provide a reduced phase space description of the system. As a result, the quantum system is simply modeled by the original quantum Hamiltonian.

scholarly journals Discrete phase-space structure of n-qubit mutually unbiased bases

2009 ◽  
Vol 324 (1) ◽  
pp. 53-72 ◽  
Author(s):  
A.B. Klimov ◽  
J.L. Romero ◽  
G. Björk ◽  
L.L. Sánchez-Soto
Keyword(s):  
Phase Space ◽  
Space Structure ◽  
Mutually Unbiased Bases ◽  
Phase Space Structure ◽  
Discrete Phase

scholarly journals Phase space structure of generalized Gaussian cat states

2010 ◽  
Vol 374 (43) ◽  
pp. 4385-4392 ◽  
Author(s):  
Fernando Nicacio ◽  
Raphael N.P. Maia ◽  
Fabricio Toscano ◽  
Raúl O. Vallejos
Keyword(s):  
Phase Space ◽  
Space Structure ◽  
Phase Space Structure ◽  
Cat States
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