Quadratic first integrals of autonomous conservative dynamical systems

Abstract

An autonomous holonomic dynamical system is described by a system of second order differential equations whose solution gives the trajectories of the system. The solution is facilitated by the use of first integrals (FIs) that are used to reduce the order of the system of differential equations and, if there are enough of them, to determine the solution. Therefore, in the study of dynamical systems, it is important that there exists a systematic method to determine the FIs of second order differential equations. On the other hand, a system of second order differential equations defines (as a rule) a kinetic energy (or Lagrangian), which provides a symmetric second order tensor that we call the kinetic metric. This metric via its symmetries (or collineations) brings into the scene the differential geometry that provides numerous results and methods concerning the determination of these symmetries. It is apparent that if one manages to provide a systematic way that will relate the determination of the FIs of a given dynamical system to the symmetries of the kinetic metric defined by this very system, then one will have at his/her disposal the powerful methods of differential geometry in the determination of the FIs and, consequently, the solution of the dynamical equations. This was also a partial aspect of Lie’s work on the symmetries of differential equations. The subject of this work is to provide a theorem that realizes this scenario. The method we follow has been considered previously in the literature and consists of the following steps: Consider the generic quadratic FI of the form I=Kab(t,qc)q̇aq̇b+Ka(t,qc)q̇a+K(t,qc), where Kab(t, qc), Ka(t, qc), and K(t, qc) are unknown tensor quantities and require dI/dt = 0. This condition leads to a system of differential equations involving the coefficients Kab(t, qc), Ka(t, qc), and K(t, qc) whose solution provides all possible quadratic FIs of this form. We demonstrate the application of the theorem in the classical cases of the geodesic equations and the generalized Kepler potential in which we obtain all the known results in a systematic way. We also obtain and discuss the time-dependent FIs that are as important as the autonomous FIs determined by other methods.