One-Dimensional Autonomous Systems and Dissipative Systems

Using the modified Prelle-Singer approach, we point out that explicit time independent first integrals can be identified for the damped linear harmonic oscillator in different parameter regimes. Using these constants of motion, an appropriate Lagrangian and Hamiltonian formalism is developed and the resultant canonical equations are shown to lead to the standard dynamical description. Suitable canonical transformations to standard Hamiltonian forms are also obtained. It is also shown that a possible quantum mechanical description can be developed either in the coordinate or momentum representations using the Hamiltonian forms.

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“…The reason for not getting the Lagrangian and generalized linear momentum from the first integral (A.1) reported in Refs. [23,24] is as follows:…”

Section: Discussionmentioning

confidence: 99%

On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator

Chandrasekar

Senthilvelan

Lakshmanan

2007

Journal of Mathematical Physics

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“…The equation(23) (for any b(t), c(t)) admits a Lagrangian description with the reciprocal Lagrangian of the form L = (ẋf (t) + xg(t)) −1 , where f is given by(26) and g = 2f b −ḟ .…”

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confidence: 99%

A direct approach to the construction of standard and non-standard Lagrangians for dissipative-like dynamical systems with variable coefficients

Cieśliński¹,

Nikiciuk²

2010

J. Phys. A: Math. Theor.

150

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We present a direct approach to the construction of Lagrangians for a large class of one-dimensional dynamical systems with a simple dependence (monomial or polynomial) on the velocity. We rederive and generalize some recent results and find Lagrangian formulations which seem to be new. Some of the considered systems (e.g., motions with the friction proportional to the velocity and to the square of the velocity) admit infinite families of different Lagrangian formulations.

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“…It is very well known that for non dissipative systems the Lagrangian and Hamiltonian can be easily obtained by subtracting or adding respectively the kinetic and potential energy of the system (Goldstein, 1980), but when we have dissipation in our dynamical system this construction is not useful and the corresponding Lagrangian and Hamiltonian is certainly not trivial to obtain (González, 2004;López, 1996). The reason is that there is not yet a consistent Lagrangian and Hamiltonian formulation for dissipative systems.…”

Section: Introductionmentioning

confidence: 96%