arXiv subjects

Tamás Milkovszki

Publications and source records attributed to Tamás Milkovszki.

4 recordsLinked to original sources

arXiv2019

The Lie symmetry group of the general Lienard-type equation

We consider the general Lienard-type equation $\ddot{u} = \sum_{k=0}^n f_k \dot{u}^k$ for $n\geq 4$. This equation naturally admits the Lie symmetry $\frac{\partial}{\partial t}$. We completely characterize when this equation admits another Lie symmetry, and give an easily verifiable condition for this on the functions $f_0, \dots , f_n$. Moreover, we give an equivalent characterization of this condition. Similar results have already been obtained previously in the cases $n=1$ or $n=2$. That is, this paper handles all remaining cases except for $n=3$.

math.DS↗

arXiv2017

About the projective Finsler metrizability: First steps in the non-isotropic case

We consider the projective Finsler metrizability problem: under what conditions the solutions of a given system of second-order ordinary differential equations (SODE) coincide with the geodesics of a Finsler metric, as oriented curves. SODEs with isotropic curvature have already been thoroughly studied in the literature and have proved to be projective Finsler metrizable. In this paper, we investigate the non-isotropic case and obtain new results by examining the integrability of the Rapcsák system extended with curvature conditions. We consider the $n$-dimensional generic case, where the eigenvalues of the Jacobi tensor are pairwise different and compute the first and the higher order compatibility conditions of the system. We also consider the three-dimensional case, where we find a class of non-isotropic sprays for which the PDE system is integrable and, consequently, the corresponding SODEs are projective metrizable.

math.DG↗

arXiv2016

Invariant metrizability and projective metrizability on Lie groups and homogeneous spaces

In this paper we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left-invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of these results for geodesic orbit spaces are given.

math.DG↗

arXiv2015

About the integrability of the Rapcsák equation

In [17] A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential equations. In this paper we investigate the integrability of the Rapcsák system, consisting of the Rapcsák equations and the homogeneity condition, by using the Spencer version of the Cartan-Kähler theorem. We also consider the extended Rapcsák system, where the first integrability conditions are included.

math.DG↗