From constants of motion to superposition rules for Lie-Hamilton systems

Abstract

A Lie system is a nonautonomous system of first-order differential equations possessing a superposition rule, i.e. a map expressing its general solution in terms of a generic finite family of particular solutions and some constants. Lie-Hamilton systems form a subclass of Lie systems whose dynamics is governed by a curve in a finite-dimensional real Lie algebra of functions on a Poisson manifold. It is shown that Lie-Hamilton systems are naturally endowed with a Poisson coalgebra structure. This allows us to devise methods to derive in an algebraic way their constants of motion and superposition rules. We illustrate our methods by studying Kummer-Schwarz equations, Riccati equations, Ermakov systems and Smorodinsky-Winternitz systems with time-dependent frequency.

30 pages

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Cited by in corpus (23)

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• Symmetry of stochastic non-variational differential equations
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• 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
• Integrable deformations of Rössler and Lorenz systems from Poisson-Lie groups
• Contact 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 structures on real two- and three-dimensional Lie groups and their Jacobi-Lie systems
• Jacobi-Lie Hamiltonian systems on real low-dimensional Jacobi-Lie groups and their Lie symmetries
• Geometry of Riccati equations over normed division algebras
• Jacobi-Lie systems: Fundamentals and low-dimensional classification
• Quasi-Lie schemes for PDEs
• A representation-theoretical approach to higher-dimensional Lie-Hamilton systems: The symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$
• A unified approach to Poisson-Hopf deformations of Lie-Hamilton systems based on sl(2)
• Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions
• Lie-Hamilton systems on Riemannian and Lorentzian spaces from conformal transformations and some of their applications
• Geometric models for Lie--Hamilton systems on ${\mathbb{R}}^2$
• Lie systems and Schrödinger equations
• Nonlinear Lie-Hamilton systems: $t$-Dependent curved oscillators and Kepler-Coulomb Hamiltonians
• Reduction and reconstruction of multisymplectic Lie systems