Dirac--Lie systems and Schwarzian equations

Abstract

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.

41 pages

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Cited by in corpus (19)

• Lie-Hamilton systems on the plane: Applications and superposition rules
• k-symplectic Lie systems: theory and applications
• Lie symmetries for Lie systems: applications to systems of ODEs and PDEs
• Schwarzian derivative treatment of the quantum second-order supersymmetry anomaly, and coupling-constant metamorphosis
• Poisson-Hopf algebra deformations of Lie-Hamilton systems
• Contact Lie systems
• A geometric Hamilton--Jacobi theory for a Nambu--Poisson structure
• Multisymplectic structures and invariant tensors for Lie systems
• A Hamilton-Jacobi theory for implicit differential systems
• Jacobi structures on real two- and three-dimensional Lie groups and their Jacobi-Lie systems
• Jacobi-Lie systems: Fundamentals and low-dimensional classification
• Quasi-Lie schemes for PDEs
• A new application of k-symplectic Lie systems
• Lie systems and Schrödinger equations
• Nonlinear Lie-Hamilton systems: $t$-Dependent curved oscillators and Kepler-Coulomb Hamiltonians
• Homotopy Poisson algebras, Maurer-Cartan elements and Dirac structures of CLWX 2-algebroids
• Invariance of second order ordinary differential equations under two-dimensional affine subalgebras of EP Lie algebra
• Reduction and reconstruction of multisymplectic Lie systems
• Mixed superposition rules for Lie systems, compatible geometric structures, and applications