Dynamics on Leibniz manifolds

Abstract

This paper shows that various relevant dynamical systems can be described as vector fields associated to smooth functions via a bracket that defines what we call a Leibniz structure. We show that gradient flows, some dissipative systems, and nonholonomically constrained simple mechanical systems, among other dynamical behaviors, can be described using this mathematical construction that generalizes the standard Poisson bracket currently used in Hamiltonian mechanics. The symmetries of these systems and the associated reduction procedures are described in detail. A number of examples illustrate the theoretical developments in the paper.

19 pages

References in corpus (2)

• Algebroids - general differential calculi on vector bundles
• The Volterra lattice as a gradient flow

Cited by in corpus (23)

• Singular reduction of implicit Hamiltonian systems
• Brackets
• AV-differential geometry: Euler-Lagrange equations
• Boundary controlled irreversible port-Hamiltonian systems
• Reviewing the Geometric Hamilton-Jacobi Theory concerning Jacobi and Leibniz identities
• Geometrical dissipation for dynamical systems
• A Hamilton-Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds
• Formation and Evolution of Singularities in Anisotropic Geometric Continua
• Geometric Hamilton-Jacobi theory for systems with external forces
• Dynamical Systems on Leibniz Algebroids
• Leibniz bialgebras, classical Yang-Baxter equations and dynamical systems
• Geometric dynamics of Vlasov kinetic theory and its moments
• On Locally Conformally Cosymplectic Hamiltonian Dynamics and Hamilton-Jacobi Theory
• Fractional Dynamical Systems on Fractional Leibniz Algebroids
• Jacobi Structures in ${\mathbb{R}}^3$
• Invariant critical sets of conserved quantities
• On Extensions, Lie-Poisson Systems, and Dissipations
• Geometrical Mechanics on algebroids
• Leibniz Dynamics with Time Delay
• On Geometrical Couplings of Dissipation and Curl Forces
• On differential operators generated by geometric structures
• Asymptotic and Lyapunov stability of Poisson equilibria
• Leibniz-Dirac structures and nonconservative systems with constraints