Noether Symmetries of Time Dependent Damped Dynamical Systems: A Geometric Approach

Two damped dynamical systems have been studied for a long time in the literature: The damped Kepler system and the damped Harmonic oscillator. The damped factor is usually in the first order in the velocities and it is either time dependent or coordinate dependent whereas the potential is either autonomous or time and space dependent. One example is the Bateman-Caldirola-Kanai(BCK) model [1] which uses a time dependent Lagrangian/Hamiltonian for studying different aspects of dissipative systems at classical and quantum level. In all damped dynamical systems the role of first integrals is crucial because it simplifies the dynamics and allows for the integration or in general the study of the dynamical equations. To determine the first integrals of a dynamical system there are various methods. To mention a few, the Noether theorem, the specification of the functional form of the first integral I and the subsequent requirement dI/dt = 0 which leads to a system of differential equations whose solution provides the first integral, the Prelle-Singer method, and the δ−formalism developed by Katzin and Levin. For most of the damped systems, at least for the cases of the Kepler and the Harmonic Oscillator, there is a Lagrangian which describes the dynamical equations. This means that for these systems one may use the Noether theorem to compute the first integrals. This is the approach of the present work. We consider the time dependent damped dynamical systems described by a Lagrangian L = A(t)L0 where L0 is an autonomous Lagrangian - usually associated with the undamped system- and prove a Proposition which allows the computation of the Noether integrals in a systematic way using the geometric properties of a Riemannian space. The method can be applied to curved spaces where not much has been done. It is important to note that the metric of this Riemannian space is defined by the kinetic energy of the dynamical system and it is not the metric of the space where the motion of the system occurs. In this way the dynamics of the system ”locks” with its own geometry. The purpose of the Proposition is to present an algorithm which applies in definite steps and determines the Noether integrals using standard calculations. The application of the algorithm is demonstrated in the case of Kepler motion. We consider the simple case of Kepler motion with constant damping and a complex one with a time dependent damping of the form A(t) = γ t . For the time dependent case we find new results. For example the homothetic vector produces Noether integrals only for the specific values of γ = −1,−1 3 .