Related papers: Symplectic Geometric Methods for Matrix Differenti…
We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and…
This paper investigates the problem of data-driven modeling of port-Hamiltonian systems while preserving their intrinsic Hamiltonian structure and stability properties. We propose a novel neural-network-based port-Hamiltonian modeling…
We consider the geometric numerical integration of Hamiltonian systems subject to both equality and "hard" inequality constraints. As in the standard geometric integration setting, we target long-term structure preservation. We…
Differential equations based on physical principals are used to represent complex dynamic systems in all fields of science and engineering. Through repeated use in both academics and industry, these equations have been shown to represent…
Consider briefly the equations of fluid dynamics-they describe the enormous wealth of detail in all the interacting physical elements of a fluid flow-whereas in applications we want to deal with a description of just that which is…
Our aim is to give a friendly introduction for students to systolic inequalities. We will stress the relationships between the classical formulation for Riemannian metrics and more recent developments related to symplectic measurements and…
In this review paper, we consider three kinds of systems of differential equations, which are relevant in physics, control theory and other applications in engineering and applied mathematics; namely: Hamilton equations, singular…
In this work, a kernel-based surrogate for integrating Hamiltonian dynamics that is symplectic by construction and tailored to large prediction horizons is proposed. The method learns a scalar potential whose gradient enters a…
We consider dynamical systems associated to Lax pairs depending rationnally on a spectral parameter. We show that we can express the symplectic form in terms of algebro--geometric data provided that the symplectic structure on L is of…
The article presents a new perspective on the isomorphism problem for non-ergodic measure-preserving dynamical systems with discrete spectrum which is based on the connection between ergodic theory and topological dynamics constituted by…
Symplectic mappings are discrete-time analogs of Hamiltonian systems. They appear in many areas of physics, including, for example, accelerators, plasma, and fluids. Integrable mappings, a subclass of symplectic mappings, are equivalent to…
In this paper we present a novel approach to the geometric formulation of solid and fluid mechanics within the port-Hamiltonian framework, which extends the standard Hamiltonian formulation to non-conservative and open dynamical systems.…
We generalize symplectic convexity theorems for Hamiltonian actions with proper momentum maps to symplectic actions on orbifolds with mod-$\Gamma$ proper momentum maps.
Despite the advancements in learning governing differential equations from observations of dynamical systems, data-driven methods are often unaware of fundamental physical laws, such as frame invariance. As a result, these algorithms may…
We revisit the geodesic approach to ideal hydrodynamics and present a related geometric framework for Newton's equations on groups of diffeomorphisms and spaces of probability densities. The latter setting is sufficiently general to include…
In this paper we consider the numerical solution of the Hamiltonian wave equation in two spatial dimension. We use the Mimetic Finite Difference (MFD) method to approximate the continuous problem combined with a symplectic integration in…
An outstanding property of any Hamiltonian system is the symplecticity of its flow, namely, the continuous trajectory preserves volume in phase space. Given a symplectic but discrete trajectory generated by a transition matrix applied at a…
We reveal a dynamical SU(2) symmetry in the asymptotic description of supersymmetric matrix models. We also consider a recursive approach for determining the ground state, and point out some additional properties of the model(s).
We derive, and discuss the properties of, a symplectic map for the dynamics of bodies on nearly parabolic orbits. The orbits are perturbed by a planet on a circular, coplanar orbit interior to the pericenter of the parabolic orbit. The map…
The design of navigation observers able to simultaneously estimate the position, linear velocity and orientation of a vehicle in a three-dimensional space is crucial in many robotics and aerospace applications. This problem was mainly dealt…