Lie--Hamilton systems: theory and applications

Abstract

This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie--Hamilton systems. We devise methods to study their superposition rules, time independent constants of motion and Lie symmetries, linearisability conditions, etc. Our results are illustrated by examples of physical and mathematical interest.

20 pages

References in corpus (6)

• Superposition rules, Lie theorem and partial differential equations
• (Super)integrability from coalgebra symmetry: formalism and applications
• Superposition rules for higher-order systems and their applications
• Quasi-Lie schemes and Emden--Fowler equations
• Integrability in time-dependent systems with one degree of freedom
• A geometric approach to integrability of Abel differential equations

Cited by in corpus (23)

• Lie-Hamilton systems on the plane: properties, classification and applications
• Dirac--Lie systems and Schwarzian equations
• Lie-Hamilton systems on the plane: Applications and superposition rules
• k-symplectic Lie systems: theory and applications
• Lie-Hamilton systems on curved spaces: A geometrical approach
• Lie symmetries for Lie systems: applications to systems of ODEs and PDEs
• A Lie systems approach to the Riccati hierarchy and partial differential equations
• Mixed superposition rules and the Riccati hierarchy
• Poisson-Hopf algebra deformations of Lie-Hamilton systems
• Contact Lie systems
• Multisymplectic structures and invariant tensors for Lie systems
• Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations
• Poisson-Hopf deformations of Lie-Hamilton systems revisited: deformed superposition rules and applications to the oscillator algebra
• Jacobi-Lie systems: Fundamentals and low-dimensional classification
• Jacobi-Lie Hamiltonian systems on real low-dimensional Jacobi-Lie groups and their Lie symmetries
• Geometry of Riccati equations over normed division algebras
• Jacobi structures on real two- and three-dimensional Lie groups and their Jacobi-Lie systems
• A unified approach to Poisson-Hopf deformations of Lie-Hamilton systems based on sl(2)
• Nonlinear Lie-Hamilton systems: $t$-Dependent curved oscillators and Kepler-Coulomb Hamiltonians
• Geometric models for Lie--Hamilton systems on ${\mathbb{R}}^2$
• Stratified Lie systems: Theory and applications
• Reduction and reconstruction of multisymplectic Lie systems
• Mixed superposition rules for Lie systems, compatible geometric structures, and applications