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


  • Francis Segovia-Chaves Facultad de Ciencias Exactas y Naturales Universidad Surcolombiana, Huila-Colombia



Hamilton-Jacobi equations, Caldirola-Kanai Hamiltonian, damped harmonic oscillator


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-Kanai
Hamiltonian, the momentum increases as time increases due to an exponential growth of the mass m(t) = meγt.