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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 which 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 first integrals 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 which we call the kinetic metric. This metric via its symmetries (or collineations) brings into the scene the Differential Geometry which provides numerous results and methods concerning the determination of these symmetries. It is apparent that if one manages to provide a systematic way which will relate the determination of the first integrals of a given dynamical system with 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 first integrals 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 the present work is to provide a theorem which realizes this scenario. The method we follow has been considered previously in the literature and consists of the following steps. Consider the generic quadratic first integral of the form I = K ab (t, q c)q aqb + Ka(t, q c)q a + K(t, q c) where K ab (t, q c), Ka(t, q c), K(t, q c) are unknown tensor quantities and require dI/dt = 0. This condition leads to a system of differential equations involving the coefficients K ab (t, q c), Ka(t, q c), K(t, q c) whose solution provides all possible quadratic first integrals 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 which are as important as the autonomous FIs determined by other methods.

A theorem is proved that determines the first integrals of the form I=Kab(t,q)q̇aq̇b+Ka(t,q)q̇a+K(t,q) of autonomous holonomic systems using only the collineations of the kinetic metric that is defined by the kinetic energy or the Lagrangian of the system. It is shown how these first integrals can be associated via the inverse Noether theorem with 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=Kab(t,q)q˙aq˙b+Ka(t,q)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 Kab(t,q), Ka(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.

We determine the conformal algebra of Bianchi III and Bianchi V spacetimes or, equivalently, we determine all Bianchi III and Bianchi V spacetimes which admit a proper conformal Killing vector. The algorithm that we use has been developed in Class. Quantum. Grav. 15, 2909 and concerns the computation of the CKVs of decomposable spacetimes. The main point of this method is that a decomposable space admits a CKV if the reduced space admits a gradient homothetic vector the latter being possible only if the reduced space is flat or a space of constant curvature. We apply this method in a stepwise manner starting from the two dimensional spacetime which admits an infinite number of CKVs and we construct step by step the Bianchi III and V spacetimes by assuming that CKVs survive as we increase the dimension of the space. We find that there is only one Bianchi III and one Bianchi V spacetime which admit at maximum one proper CKV. In each case we determine the conformal Killing vector and the corresponding conformal factor. As a first application in these two spacetimes we study the kinematics of the comoving observers and the dynamics of the corresponding cosmological fluid. As a second application we determine in these spacetimes generators of the Lie symmetries of the wave 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 particular, 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.

We consider the time-dependent dynamical system q¨a=−Γbcaq˙bq˙c−ω(t)Qa(q) where ω(t) is a non-zero arbitrary function and the connection coefficients Γbca 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=Kabq˙aq˙b+Kaq˙a+K where the unknown coefficients Kab,Ka,K are tensors depending on t,qa and impose the condition dIdt=0. This condition leads to a system of partial differential equations (PDEs) involving the quantities Kab,Ka,K,ω(t) and Qa(q). From these PDEs, it follows that Kab is a Killing tensor (KT) of the kinetic metric. We use the KT Kab in two ways: a. We assume a general polynomial form in t both for Kab and Ka; b. We express Kab 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 ω(t) or Qa(q). We consider first that ω(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 Qa(q) to be the generalized time-dependent Kepler potential V=−ω(t)rν and determine the functions ω(t) for which QFIs are admitted. We extend the discussion to the non-linear differential equation x¨=−ω(t)xμ+ϕ(t)x˙(μ≠−1) and compute the relation between the coefficients ω(t),ϕ(t) so that QFIs are admitted. We apply the results to determine the QFIs of the generalized Lane–Emden equation.

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