Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions

Abstract

We propose an adaptation of the notion of scaling symmetries for the case of Lie-Hamilton systems, allowing their subsequent reduction to contact Lie systems. As an illustration of the procedure, time-dependent frequency oscillators and time-dependent thermodynamic systems are analyzed from this point of view. The formalism provides a novel method for constructing contact Lie systems on the three-dimensional sphere, derived from recently established Lie-Hamilton systems arising from the fundamental four-dimensional representation of the symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$. It is shown that these systems are a particular case of a larger hierarchy of contact Lie systems on a special class of three-dimensional homogeneous spaces, namely the Cayley-Klein spaces. These include Riemannian spaces (sphere, hyperbolic and Euclidean spaces), pseudo-Riemannian spaces (anti-de Sitter, de Sitter and Minkowski spacetimes), as well as Newtonian or non-relativistic spacetimes. Under certain topological conditions, some of these systems retrieve well-known two-dimensional Lie-Hamilton systems through a curvature-dependent reduction.

49 pages, 2 figures. Minor corrections

References in corpus (13)

• Invariant measures for contact Hamiltonian systems: symplectic sandwiches with contact bread
• Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature
• Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations
• A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory
• Multicontact formulation for non-conservative field theories
• Poisson-Hopf algebra deformations of Lie-Hamilton systems
• Contact Lie systems
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• Poisson-Hopf deformations of Lie-Hamilton systems revisited: deformed superposition rules and applications to the oscillator algebra
• Multidimensional integrable systems from contact geometry
• A representation-theoretical approach to higher-dimensional Lie-Hamilton systems: The symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$
• Lie-Hamilton systems on Riemannian and Lorentzian spaces from conformal transformations and some of their applications
• Lie-Hamilton systems associated with the symplectic Lie algebra $\mathfrak{sp}(6,\mathbb{R})$

Cited by in corpus (1)

• Mixed superposition rules for Lie systems, compatible geometric structures, and applications