Comments on the Hamiltonian formulation for linear and non-linear oscillators including dissipation

✍ Scribed by M.A.F. Sanjuán

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 223 KB
• Volume: 185
• Category: Article
• ISSN: 0022-460X
• DOI: 10.1006/jsvi.1995.0413

✦ Synopsis

The Hamiltonian formalism for the damped harmonic oscillator has recently received some attention [1][2][3][4], in which a canonical transformation has been used in order to remove the damping term in the original term. The original idea goes back, as far as the author knows, to the article of Bateman [5], where he introduces the transformation in the Lagrangian formulation for the linear damped harmonic oscillator, while trying to prove that a linear dissipative system can be derived from a variational principle. Later, Havas [6] studied the range of application of the Lagrangian and Hamiltonian formalism. This was also used by Denman and Buch [7] in order to study the Hamilton-Jacobi equation and to analyze dissipative systems for its possible treatment in quantum mechanics.

In reference [4], Nagem and Sandri found, in the case in which the natural frequency is zero, that the energy E T is a constant of motion and that the two quantities K 1 and K 2 , K 1 =x˙+gx=e -gt p x +gx/2, K 2 =x˙e gt =p x -(g/2)x e gt , (1, 2)