Quadratic first integrals of time-dependent dynamical systems of the form $\ddot{q}^{a}= -\Gamma^{a}_{bc}\dot{q}^{b} \dot{q}^{c} -\omega(t)Q^{a}(q)$

TL;DR

This paper derives quadratic first integrals for time-dependent dynamical systems with variable coefficients, identifying conditions on system functions that admit these integrals, including applications to Kepler and Lane-Emden equations.

Contribution

It introduces a systematic method to find quadratic first integrals for time-dependent systems using Killing tensors and polynomial ansatzes, extending to specific potentials and equations.

Findings

Two independent QFIs for polynomial $\omega(t)$ systems.

Conditions on $\omega(t)$ for Kepler potential QFIs.

Relations between $\omega(t)$ and $\phi(t)$ for nonlinear equations.

Abstract

We consider the time-dependent dynamical system $\ddot{q}^{a}= -\Gamma_{bc}^{a}\dot{q}^{b}\dot{q}^{c}-\omega(t)Q^{a}(q)$ where $\omega(t)$ is a non-zero arbitrary function and the connection coefficients $\Gamma^{a}_{bc}$ are computed from the kinetic metric (kinetic energy) of the system. In order to determine the quadratic first integrals (QFIs) $I$ we assume that $I=K_{ab}\dot{q}^{a} \dot{q}^{b} +K_{a}\dot{q}^{a}+K$ where the unknown coefficients $K_{ab}, K_{a}, K$ are tensors depending on $t, q^{a}$ and impose the condition $\frac{dI}{dt}=0$. This condition leads to a system of partial differential equations (PDEs) involving the quantities $K_{ab}, K_{a}, K,$ $\omega(t)$ and $Q^{a}(q)$. From these PDEs, it follows that $K_{ab}$ is a Killing tensor (KT) of the kinetic metric. We use the KT $K_{ab}$ in two ways: a. We assume a general polynomial form in $t$ both for $K_{ab}$ and $K_{a}$; b. We express $K_{ab}$ in a basis of the KTs of order 2 of the kinetic metric assuming the coefficients to be functions of $t$. In both cases, this leads to a new system of PDEs whose solution requires that we specify either $\omega(t)$ or $Q^{a}(q)$. We consider first that $\omega(t)$ is a general polynomial in $t$ and find that in this case the dynamical system admits two independent QFIs which we collect in a Theorem. Next, we specify the quantities $Q^{a}(q)$ to be the generalized time-dependent Kepler potential $V=-\frac{\omega (t)}{r^{\nu}}$ and determine the functions $\omega(t)$ for which QFIs are admitted. We extend the discussion to the non-linear differential equation $\ddot{x}=-\omega(t)x^{\mu }+\phi (t)\dot{x}$ $(\mu \neq -1)$ and compute the relation between the coefficients $\omega(t), \phi(t)$ so that QFIs are admitted. We apply the results to determine the QFIs of the generalized Lane-Emden equation.