Related papers: Variational order for forced Lagrangian systems II…
Spearheaded by the recent efforts to derive stochastic geophysical fluid dynamics models, we present a generic framework for introducing stochasticity into variational principles through the concept of a semi-martingale driven variational…
We describe how geometrical methods can be applied to a system with explicitly time-dependent second-class constraints so as to cast it in Hamiltonian form on its physical phase space. Examples of particular interest are systems which…
In this contribution, we develop a variational integrator for the simulation of (stochastic and multiscale) electric circuits. When considering the dynamics of an electrical circuit, one is faced with three special situations: 1. The system…
We formulate a perturbative approximation to gravitational instability, based on Lagrangian hydrodynamics in Newtonian cosmology. We take account of `pressure' effect of fluid, which is kinematically caused by velocity dispersion, to aim…
We apply methods of the so-called `inverse problem of the calculus of variations' to the stabilization of an equilibrium of a class of two-dimensional controlled mechanical systems. The class is general enough to include, among others, the…
In this paper we study a Hamiltonization procedure for mechanical systems with velocity-depending (nonholonomic) constraints. We first rewrite the nonholonomic equations of motion as Euler-Lagrange equations, with a Lagrangian that follows…
Causal variational principles, which are the analytic core of the physical theory of causal fermion systems, are found to have an underlying Hamiltonian structure, giving a formulation of the dynamics in terms of physical fields in…
This paper presents the Euler-Lagrange equations for fractional variational problems with multiple integrals. The fractional Noether-type theorem for conservative and nonconservative generalized physical systems is proved. Our approach uses…
A method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie-Poisson symmetries is proposed by considering Poisson-Lie groups as deformations of Lie-Poisson (co)algebras. Moreover, the underlying Lie-Poisson…
We study forced oscillations of a rod with a body attached to its free end so that the motion of a system is described by two sets of equations, one of integer and the other of the fractional order. To the constitutive equation we associate…
The Lagrange--Poincar\'e equations for the mechanical system describing the motion of a scalar particle on a Riemannian manifold with a given free and isometric action of a compact Lie group is obtained. In an arising principle fibre…
In the last two decades, significant effort has been put in understanding and designing so-called structure-preserving numerical methods for the simulation of mechanical systems. Geometric integrators attempt to preserve the geometry…
After reviewing the Lagrangian-Hamiltonian unified formalism (i.e, the Skinner-Rusk formalism) for higher-order (non-autonomous) dynamical systems, we state a unified geometrical version of the Variational Principles which allows us to…
The Lagrange, Euler, and Euler-Poincar\'{e} variational principles for the guiding-center Vlasov-Maxwell equations are presented. Each variational principle presents a different approach to deriving guiding-center polarization and…
In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that…
A variational integrator of arbitrarily high-order on the special orthogonal group $SO(n)$ is constructed using the polar decomposition and the constrained Galerkin method. It has the advantage of avoiding the second-order derivative of the…
The $G$-strand equations for a map $\mathbb{R}\times \mathbb{R}$ into a Lie group $G$ are associated to a $G$-invariant Lagrangian. The Lie group manifold is also the configuration space for the Lagrangian. The $G$-strand itself is the map…
Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for…
Forced variational integrators are given by the discretization of the Lagrange-d'Alembert principle for systems subject to external forces, and have proved useful for numerical simulation studies of complex dynamical systems. In this paper…
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional…