Using older and recent results on the integrability of two-dimensional (2d) dynamical systems, we prove that the results obtained in a recent publication concerning the 2d generalized Ermakov system can be obtained as special cases of a more general
approach. This approach is geometric and can be used to study efficiently similar dynamical systems.
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 metri
c (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 $V=-frac{omega (t)}{r^{ u}}$ and determine the functions $omega(t)$ for which QFIs are admitted. We extend the discussion to the non-linear differential equation $ddot{x}=-omega(t)x^{mu }+phi (t)dot{x}$ $(mu eq -1)$ 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.
The derivation of conservation laws and invariant functions is an essential procedure for the investigation of nonlinear dynamical systems. In this study we consider a two-field cosmological model with scalar fields defined in the Jordan frame. In pa
rticular we consider a Brans-Dicke scalar field theory and for the second scalar field we consider a quintessence scalar field minimally coupled to gravity. For this cosmological model we apply for the first time a new technique for the derivation of conservation laws without the application of variational symmetries. The results are applied for the derivation of new exact solutions. The stability properties of the scaling solutions are investigated and criteria for the nature of the second field according to the stability of these solutions are determined.
A theorem is proved which determines the first integrals of the form $I=K_{ab}(t,q)dot{q}^{a}dot{q}^{b}+K_{a}(t,q)dot{q}^{a}+K(t,q)$ of autonomous holonomic systems using only the collineations of the kinetic metric which is defined by the kinetic en
ergy or the Lagrangian of the system. It is shown how these first integrals can be associated via the inverse Noether theorem to a gauged weak Noether symmetry which admits the given first integral as a Noether integral. It is shown also that the associated Noether symmetry is possible to satisfy the conditions for a Hojman or a form-invariance symmetry therefore the so-called non-Noetherian first integrals are gauged weak Noether integrals. The application of the theorem requires a certain algorithm due to the complexity of the special conditions involved. We demonstrate this algorithm by a number of solved examples. We choose examples from published works in order to show that our approach produces new first integrals not found before with the standard methods.
We consider the generic quadratic first integral (QFI) of the form $I=K_{ab}(t,q)dot{q}^{a}dot{q}^{b}+K_{a}(t,q)dot{q}^{a}+K(t,q)$ and require the condition $dI/dt=0$. The latter results in a system of partial differential equations which involve the
tensors $K_{ab}(t,q)$, $K_{a}(t,q)$, $K(t,q)$ and the dynamical quantities of the dynamical equations. These equations divide in two sets. The first set involves only geometric quantities of the configuration space and the second set contains the interaction of these quantities with the dynamical fields. A theorem is presented which provides a systematic solution of the system of equations in terms of the collineations of the kinetic metric in the configuration space. This solution being geometric and covariant, applies to higher dimensions and curved spaces. The results are applied to the simple but interesting case of two-dimensional (2d) autonomous conservative Newtonian potentials. It is found that there are two classes of 2d integrable potentials and that superintegrable potentials exist in both classes. We recover most main previous results, which have been obtained by various methods, in a single and systematic way.
