Related papers: Integration on $q$-Cosymplectic Manifolds
Since they were introduced in the 1990s, Lie group integrators have become a method of choice in many application areas. These include multibody dynamics, shape analysis, data science, image registration and biophysical simulations. Two…
The aim of this paper is to develop a Hamilton--Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton-Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given…
In this paper we study Hamiltonian systems on contact manifolds, which is an appropriate scenario to discuss dissipative systems. We prove a coisotropic reduction theorem similar to the one in symplectic mechanics.
In this paper, some of formulations of Hamilton-Jacobi equations for Hamiltonian system and regular reduced Hamiltonian systems are given. At first, an important lemma is proved, and it is a modification for the corresponding result of…
Singular theories, characterised by the presence of degeneracies in their Lagrangian or Hamiltonian descriptions, require the systematic implementation of constraints in order to obtain well-defined dynamics. While the symplectic framework…
The Mishchenko-Fomenko theorem on noncommutative integrability of Hamiltonian systems on a symplectic manifold is extended to the case of noncompact invariant submanifolds.
In this paper, we study the extended Hamilton-Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group $G$ on a manifold $M$ and a $G$-invariant vector field $X$ on $M$, we construct complete…
In this paper we study the coisotropic reduction in different types of dynamics according to the geometry of the corresponding phase space. The relevance of the coisotropic reduction is motivated by the fact that these dynamics can always…
This paper explains the recent developments on the symplectic theory of Hamiltonian completely integrable systems on symplectic 4-manifolds, compact or not. One fundamental ingredient of these developments has been the understanding of…
The Hamilton-Jacobi theory is a formulation of Classical Mechanics equivalent to other formulations as Newton's equations, Lagrangian or Hamiltonian Mechanics. It is particulary useful for the identification of conserved quantities of a…
In order to describe the impact of different geometric structures and constraints for the dynamics of a Hamiltonian system, in this paper, for a magnetic Hamiltonian system defined by a magnetic symplectic form, we first drive precisely the…
We generalise the theories of cosymplectic, contact, and cocontact manifolds to the infinite-dimensional setting and calculate model examples of time-dependent and dissipative Hamiltonian systems.
In this paper, we study symplectic integration of canonical Hamiltonian systems with Jacobi polynomials. The relevant theoretical results of continuous-stage Runge-Kutta methods are revisited firstly and then symplectic methods with Jacobi…
Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of…
We introduce the notion of a real form of a Hamiltonian dynamical system in analogy with the notion of real forms for simple Lie algebras. This is done by restricting the complexified initial dynamical system to the fixed point set of a…
The notion of Jacobi-Haantjes manifold, consisting of a Jacobi manifold endowed with an algebra of extended Haantjes operator fields, is proposed as a natural geometric framework which allows us to define the notion of integrability of both…
The aim of this paper is to extend the coisotropic embedding theorem obtained by M. J. Gotay for pre-symplectic manifolds to more general geometric settings: cosymplectic, contact, cocontact, $k$-symplectic, $k$-cosymplectic, $k$-contact,…
The objective of this work is to examine the integrability of Hamiltonian systems in $2D$ spaces with variable curvature of certain types. Based on the differential Galois theory, we announce the necessary conditions of the integrability.…
The notion of integrability is discussed for classical nonautonomous systems with one degree of freedom. The analysis is focused on models which are linearly spanned by finite Lie algebras. By constructing the autonomous extension of the…
While the construction of symplectic integrators for Hamiltonian dynamics is well understood, an analogous general theory for Poisson integrators is still lacking. The main challenge lies in overcoming the singular and non-linear geometric…