The one-dimensional harmonic oscillator damped with Caldirola-Kanai Hamiltonian

Abstract

In this paper, the solution to the Hamilton-Jacobi equation for the one-dimensional harmonic oscillator damped with the Caldirola-Kanai model is presented. Making use of a canonical transformation, we calculate the Hamilton characteristic function. It was found that the position of the oscillator shows an exponential decay similar to that of the oscillator with damping where the decay is more pronounced when increasing the damping constant γ. It is shown that when γ = 0, the behavior is of an oscillator with simple harmonic motion. However, unlike the damped harmonic oscillator where the linear momentum decays with time, in the case of the oscillator with the Caldirola-KanaiHamiltonian, the momentum increases as time increases due to an exponential growth of the mass m(t) = meγt.

Bateman Oscillators: Caldirola-Kanai and Null Lagrangians and Gauge Functions

“…The results obtained in this paper contradict the previous claims [23,24] that the CK Lagrangian is not valid for the Bateman oscillators but instead it represents a different dynamical system in which mass increases or decreases exponentially in time. As a result of this contradiction, derivations of the Kanai-Caldirola propagator in the de Broglie-Bohm theory [28], which are based on the CK Lagrangian with its mass being time-dependent [29], must be taken with caution.…”

“…The role of the CK Lagrangian in deriving the equation of motion for the Bateman oscillators has been questioned in the literature [23] based on the previous work [24]. The main conclusion of that previous research was that the CK Lagrangian does not describe the Bateman oscillators but instead a different oscillatory system with its mass increasing (b > 0) or decreasing (b < 0) exponentially in time.…”

Comment on ``The one-dimensional harmonic oscillator damped with Caldirola--Kanai Hamiltonian''

“…In Ref. [1] the solution of the equation of motion of a damped one-dimensional harmonic oscillator is obtained by means of the Hamilton-Jacobi equation applied to the so-called Caldirola-Kanai Hamiltonian. As discussed below, throughout that paper there are several conceptual errors related to the Hamilton-Jacobi equation as well as the basic Lagrangian and Hamiltonian formalisms.…”

“…In the third paragraph of the Introduction of Ref. [1] we find the statement that "Dissipative systems are non-Hamiltonian." Unfortunately, a precise definition of a Hamiltonian (or non-Hamiltonian) system is not given there.…”

“…[2,3]); furthermore, in the case of a single secondorder ordinary differential equation (such as the equation of motion considered in Ref. [1]) there exist an infinite number of Lagrangians (a result already known to Darboux in 1894 [4], well before the works of Caldirola and Kanai cited in Ref. [1], see also Ref.…”