Solvable Structures Associated to the Nonsolvable Symmetry Algebra sl(2,R)\mathfrak{sl}(2,\mathbb{R})

Third-order ordinary differential equations with Lie symmetry algebras isomorphic to the nonsolvable algebra $\mathfrak{sl}(2,\mathbb{R})$ admit solvable structures. These solvable structures can be constructed by using the basis elements of these algebras. Once the solvable structures are known, the given equation can be integrated by quadratures as in the case of solvable symmetry algebras

Dirac--Lie systems and Schwarzian equations

A Lie system is a system of differential equations admitting a superposition rule, i.e., a function describing its general solution in terms of any generic set of particular solutions and some constants. Following ideas going back to the Dirac's description of constrained systems, we introduce and analyse a particular class of Lie systems on Dirac manifolds, called Dirac--Lie systems, which are associated with `Dirac--Lie Hamiltonians'. Our results enable us to investigate constants of the motion, superposition rules, and other general properties of such systems in a more effective way. Several concepts of the theory of Lie systems are adapted to this `Dirac setting' and new applications of Dirac geometry in differential equations are presented. As an application, we analyze traveling wave solutions of Schwarzian equations, but our methods can be applied also to other classes of differential equations important for Physics.Comment: 41 page

Generating Finite Dimensional Integrable Nonlinear Dynamical Systems

In this article, we present a brief overview of some of the recent progress made in identifying and generating finite dimensional integrable nonlinear dynamical systems, exhibiting interesting oscillatory and other solution properties, including quantum aspects. Particularly we concentrate on Lienard type nonlinear oscillators and their generalizations and coupled versions. Specific systems include Mathews-Lakshmanan oscillators, modified Emden equations, isochronous oscillators and generalizations. Nonstandard Lagrangian and Hamiltonian formulations of some of these systems are also briefly touched upon. Nonlocal transformations and linearization aspects are also discussed.Comment: To appear in Eur. Phys. J - ST 222, 665 (2013

Canonical Forms and Their Integrability for Systems of Three 2nd-Order ODEs

Differential invariants and their corresponding canonical forms for systems of three 2nd-order ODEs possessing three-dimensional Lie algebras are constructed. Their extension up to kth-order system of three 2nd-order ODEs is presented. Furthermore singularity in invariant structure for the canonical forms is investigated. In addition integrability of these canonical forms is discussed. Illustrative physical examples from mechanics of system of particles are provided