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A characterization of isochronous centres in terms of symmetries.

Emilio Freire, Gasull, Armengol, Guillamon, Antoni 2 (2004)

Revista Matemática Iberoamericana

We present a description of isochronous centres of planar vector fields X by means of their groups of symmetries. More precisely, given a normalizer U of X (i.e., [X,U]= µ X, where µ is a scalar function), we provide a necessary and sufficient isochronicity condition based on µ. This criterion extends the result of Sabatini and Villarini that establishes the equivalence between isochronicity and the existence of commutators ([X,U]= 0). We put also special emphasis on the mechanical aspects of isochronicity;...

A simple method for constructing non-liouvillian first integrals of autonomous planar systems

Axel Schulze-Halberg (2006)

Czechoslovak Mathematical Journal

We show that a transformation method relating planar first-order differential systems to second order equations is an effective tool for finding non-liouvillian first integrals. We obtain explicit first integrals for a subclass of Kukles systems, including fourth and fifth order systems, and for generalized Liénard-type systems.

A version of non-Hamiltonian Liouville equation

Celina Rom (2014)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we give a version of the theorem on local integral invariants of systems of ordinary differential equations. We give, as an immediate conclusion of this theorem, a condition which guarantees existence of an invariant measure of local dynamical systems. Results of this type lead to the Liouville equation and have been frequently proved under various assumptions. Our method of the proof is simpler and more direct.

Algebraic integrability for minimum energy curves

Ivan Yudin, Fátima Silva Leite (2015)

Kybernetika

This paper deals with integrability issues of the Euler-Lagrange equations associated to a variational problem, where the energy function depends on acceleration and drag. Although the motivation came from applications to path planning of underwater robot manipulators, the approach is rather theoretical and the main difficulties result from the fact that the power needed to push an object through a fluid increases as the cube of its speed.

Equivalence and symmetries of first order differential equations

V. Tryhuk (2008)

Czechoslovak Mathematical Journal

In this article, the equivalence and symmetries of underdetermined differential equations and differential equations with deviations of the first order are considered with respect to the pseudogroup of transformations x ¯ = ϕ ( x ) , y ¯ ...

Exact solutions of generalized Lane-Emden equations of the second kind

Kısmet Kasapoğlu (2024)

Applications of Mathematics

Contact and Lie point symmetries of a certain class of second order differential equations using the Lie symmetry theory are obtained. Generators of these symmetries are used to obtain first integrals and exact solutions of the equations. This class of equations is transformed into the so-called generalized Lane-Emden equations of the second kind y ' ' ( x ) + k x y ' ( x ) + g ( x ) e n y = 0 . Then we consider two types of functions g ( x ) and present first integrals and exact solutions of the Lane-Emden equation for them. One of the considered...

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