arXiv subjects

Antonios Mitsopoulos

Publications and source records attributed to Antonios Mitsopoulos.

16 recordsLinked to original sources

arXiv2023

Higher order first integrals of autonomous non-Riemannian dynamical systems

We consider autonomous holonomic dynamical systems defined by equations of the form $\ddot{q}^{a}=-Γ_{bc}^{a}(q) \dot{q}^{b}\dot{q}^{c}$ $-Q^{a}(q)$, where $Γ^{a}_{bc}(q)$ are the coefficients of a symmetric (possibly non-metrical) connection and $-Q^{a}(q)$ are the generalized forces. We prove a theorem which for these systems determines autonomous and time-dependent first integrals (FIs) of any order in a systematic way, using the `symmetries' of the geometry defined by the dynamical equations. We demonstrate the application of the theorem to compute linear, quadratic, and cubic FIs of various Riemannian and non-Riemannian dynamical systems.

math-ph↗

arXiv2023

Cubic first integrals of autonomous dynamical systems in $E^2$ by an algorithmic approach

In a recent paper (A. Mitsopoulos and M. Tsamparlis, J. Geom. Phys. 170, 104383, 2021), a general theorem is given which provides an algorithmic method for the computation of first integrals (FIs) of autonomous dynamical systems in terms of the symmetries of the kinetic metric defined by the dynamical equations of the system. In the present work, we apply this theorem to compute the cubic FIs of autonomous conservative Newtonian dynamical systems with two degrees of freedom. We show that the known results on this topic, which have been obtained by means of various different methods, and additional ones derived in this work can be obtained by the single algorithmic method provided by this theorem. The results are collected in four Tables which can be used as an updated reference of this type of integrable and superintegrable potentials. The results we find are for special values of free parameters; therefore, using the methods developed here, other researchers by different suitable choice of the parameters will be able to find new integrable and superintegrable potentials.

math-ph↗

arXiv2023

Integrable and superintegrable 3d Newtonian potentials using quadratic first integrals: A review

The determination of the first integrals (FIs) of a dynamical system and the subsequent assessment of their integrability or superintegrability in a systematic way is still an open subject. One method which has been developed along these lines for second order autonomous dynamical systems is the so-called direct method. According to this method, one assumes a general functional form for the FI I and requires the condition dI/dt=0 along the dynamical equations. This results to a system of partial differential equations (PDEs) to which one adds the necessary integrability conditions of the involved scalar quantities. It is found that the final system of PDEs breaks into two sets: a. One set containing geometric elements only and b. A second set with geometric and dynamical quantities. Then, provided the geometric quantities are known or can be found, one uses the second set to compute the FIs and, accordingly, assess on the integrability of the dynamical system. The solution of the system of PDEs for quadratic FIs (QFIs) has been given in a recent paper J. Math. Phys. 61, 122701 (2020). In the present work, we consider the application of this solution to Newtonian autonomous conservative dynamical systems with three degrees of freedom, and compute integrable and superintegrable potentials whose integrability is determined via autonomous and time-dependent QFIs. The geometric elements of these systems are the ones of the Euclidean space which are known. Setting various values for the parameters determining the geometric elements, we determine in a systematic way all known integrable and superintegrable potentials in E3 together with new ones. For easy reference, the results are collected in tables so that the present work may act as an updated review on the subject of second order integrable/superintegrable potentials in E3.

math-ph↗

arXiv2023

Higher order first integrals of autonomous dynamical systems in terms of geometric symmetries

In general, a system of differential equations is integrable if there exist `sufficiently many' first integrals (FIs) so that its solution can be found by means of quadratures. Therefore, the determination of the FIs is an important issue in order to establish the integrability of a dynamical system. In this work, we consider holonomic autonomous dynamical systems defined by equations $\ddot{q}^{a}= -Γ_{bc}^{a}(q) \dot{q}^{b}\dot{q}^{c} -Q^{a}(q)$ where $Γ^{a}_{bc}(q)$ are the coefficients of a symmetric (possibly non-metrical) connection and $-Q^{a}(q)$ are the generalized forces. We prove a theorem which produces the FIs of any order of such systems in terms of the `symmetries' of the geometry defined by the quantities $Γ_{bc}^{a}(q)$. We apply the theorem to compute quadratic and cubic FIs of various dynamical systems.

math-ph↗

arXiv2022

Quadratic first integrals of constrained autonomous conservative dynamical systems with fixed energy

We consider autonomous conservative dynamical systems which are constrained with the condition that the total energy of the system has a specified value. We prove a theorem which provides the quadratic first integrals (QFIs), time-dependent and autonomous, of these systems in terms of the symmetries (conformal Killing vectors and conformal Killing tensors) of the kinetic metric. It is proved that there are three types of QFIs and for each type we give explicit formulae for their computation. It is also shown that when the autonomous QFIs are considered, then we recover the known results of previous works. For zero potential function, we have the case of constrained geodesics and obtain formulae to compute their QFIs. The theorem is applied in two cases. In the first case, we determine potentials which admit the second of the three types of QFIs. We recover a superintegrable potential of the Ermakov type and a new integrable potential whose trajectories for zero energy and zero QFI are circles. In the second case, we integrate the constrained geodesic equations for a family of two-dimensional conformally flat metrics.

math-ph↗

arXiv2022

Integrability of Dynamical Systems: A Geometrical Viewpoint

The physical phenomena are described by physical quantities related by specific physical laws. In the context of a Physical Theory, the physical quantities and the physical laws are described, respectively, by suitable geometrical objects and relations between these objects. These relations are expressed with systems of (mainly second order) differential equations. The solution of these equations is frequently a formidable task, either because the dynamical equations cannot be integrated by standard methods or because the defined dynamical system is non-integrable. Therefore, it is important that we have a systematic and reliable method to determine their integrability. This has led to the development of several (algebraic or geometric) methods, which determine if a dynamical system is integrable/superintegrable or not. Most of these methods concern the first integrals (FIs), that is, quantities that are constant along the evolution of the system. FIs are important, because they can be used to reduce the order of the system of the dynamical equations and, if there are `enough' of them, even to determine its solution by means of quadratures. In the latter case, the dynamical system is said to be Liouville integrable and it is associated with a canonical Lagrangian, whose kinetic energy defines a metric tensor known as kinetic metric. It is proved that there is a close relation between the geometric symmetries (collineations and Killing tensors) of this metric and the quantities defining the FIs. This correspondence makes it possible to use the results of Differential Geometry in the study of the integrability of dynamical systems. In this thesis, we study this correspondence and geometrize the determination of FIs by developing a new geometric method to compute them.

math-ph↗

arXiv2021

Integrable time-dependent central potentials

The integrable time-dependent central potentials that admit linear and quadratic first integrals other than those constructed from the angular momentum are determined. It is shown explicitly that previous answers to this problem are incomplete. The results are applied in order to find the integrable time-dependent oscillators, the integrable time-dependent generalized Kepler potentials, a class of integrable binary systems with variable mass, and the integrable Yukawa and interatomic potentials with time-dependent parameters. Finally, a new class of integrable potentials is integrated and the corresponding wavefunction is determined.

math-ph↗

arXiv2021

Higher order first integrals of autonomous dynamical systems

A theorem is derived which determines higher order first integrals of autonomous holonomic dynamical systems in a general space, provided the collineations and the Killing tensors -- up to the order of the first integral -- of the kinetic metric, defined by the kinetic energy of the system, can be computed. The theorem is applied in the case of Newtonian autonomous conservative dynamical systems of two degrees of freedom, where known and new integrable and superintegrable potentials that admit cubic first integrals are determined.

math-ph↗

arXiv2021

The generalized Ermakov conservative system: A discussion

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.

math-ph↗

arXiv2021

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

We consider the time-dependent dynamical system $\ddot{q}^{a}= -Γ_{bc}^{a}\dot{q}^{b}\dot{q}^{c}-ω(t)Q^{a}(q)$ where $ω(t)$ is a non-zero arbitrary function and the connection coefficients $Γ^{a}_{bc}$ 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 $K_{ab}, K_{a}, K,$ $ω(t)$ 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 $ω(t)$ or $Q^{a}(q)$. 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 $Q^{a}(q)$ to be the generalized time-dependent Kepler potential $V=-\frac{ω(t)}{r^ν}$ and determine the functions $ω(t)$ for which QFIs are admitted. We extend the discussion to the non-linear differential equation $\ddot{x}=-ω(t)x^{μ}+ϕ(t)\dot{x}$ $(μ\neq -1)$ 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.

math-ph↗

arXiv2021

New conservation laws and exact cosmological solutions in Brans-Dicke cosmology with an extra scalar field

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.

gr-qc↗

arXiv2020

First integrals of holonomic systems without Noether symmetries

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 energy 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.

math-ph↗

arXiv2020

Integrable and Superintegrable Potentials of 2d Autonomous Conservative Dynamical Systems

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.

math-ph↗

arXiv2020

Quadratic first integrals of autonomous conservative dynamical systems

An autonomous 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, it is important that there exists a systematic method to determine the FIs. On the other hand, a system of second order differential equations defines a kinetic energy, which provides a symmetric second order tensor called kinetic metric of the system. This metric via its symmetries brings into the scene the numerous methods of differential geometry and hence it is apparent that one should manage to relate the determination of the FIs to the symmetries of the kinetic metric. The subject of this work is to provide a theorem that realizes this scenario. The method we follow considers the generic quadratic FI of the form $I=K_{ab}(t,q^{c})\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q^{c})\dot{q}^{a} +K(t,q^{c})$ where $K_{ab}(t,q^{c}), K_{a}(t,q^{c}), K(t,q^{c})$ are unknown tensor quantities and requires $dI/dt = 0$. This condition leads to a system of differential equations involving the coefficients of $I$ 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. We also obtain and discuss the time-dependent FIs.

math-ph↗

arXiv2019

Constructing the CKVs of Bianchi III and V spacetimes

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 (1998) 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 an application in the spacetimes we found 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.

gr-qc↗

arXiv2018

Symmetries of spacetimes embedded with an Electromagnetic String Fluid

The electromagnetic string fluid (EMSF) is an anisotropic charged string fluid interacting with a strong magnetic field. In this fluid we consider the double congruence defined by the 4-velocity of the fluid $u^a$ and the unit vector $n^a$ along the magnetic field. Using the standard 1+3 decomposition defined by the vector $u^a$ and the 1+1+2 decomposition defined by the double congruence ${u^a,n^a}$ we determine the kinematic and the dynamic quantities of an EM string fluid in both decompositions. In order to solve the resulting field equations we consider simplifying assumptions in the form of collineations. We decompose the generic quantity $L_{X}g_{ab}$ in a trace $ψ$ and and a traceless part $H_{ab}$. Because all collineations are expressible in terms of the quantity $L_{X}g_{ab}$ it is possible to compute the Lie derivative of all tensors defined by the metric i.e. the Ricci tensor, the Weyl tensor etc. This makes possible the effects of any assumed collineation on the gravitational field equations. This is done as follows. Using relevant identities of Differential Geometry we express the quantity $L_{X}R_{ab}$ where $R_{ab}$ is the Ricci tensor in terms of the two irreducible parts $ψ,H_{ab}$. Subsequently using the gravitational field equations we compute the same quantity $L_{X}R_{ab}$ in terms of the Lie derivative of the dynamic variables. We equate the two results and find the field equations in the form $L_{X}{\text{Dynamic variable}}=F(ψ,H_{ab},{\text{ dynamic variables}})$. This result is general and holds for all gravitational systems and in particular for the EMSF. Subsequently we specialize our study at two levels. We consider the case of a Conformal Killing Vector (CKV) parallel to $u^a$ and a CKV parallel to $n^a$.

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