Related papers: Dependent coordinates in the Lagrange-Poincar\'e e…
We study the structure and dynamics of the infinite sequence of extensions of the Poincar{\'e} algebra whose method of construction was described in a previous paper [1]. We give explicitly the Maurer-Cartan (MC) 1-forms of the extended Lie…
A geometric version of the Poincar\'e Lemma is established for the topological vector space of differential chains. In particular, every differential k-cycle with compact support in a contractible open subset U of a smooth n-manifold M is…
Stable estimation of rigid body pose and velocities from noisy measurements, without any knowledge of the dynamics model, is treated using the Lagrange-d'Alembert principle from variational mechanics. With body-fixed optical and inertial…
We construct an operational formulation of classical mechanics without presupposing previous results from analytical mechanics. In doing so, several concepts from analytical mechanics will be rediscovered from an entirely new perspective.…
In this paper, we propose the concept of $(\pm)$-discrete Dirac structures over a manifold, where we define $(\pm)$-discrete two-forms on the manifold and incorporate discrete constraints using $(\pm)$-finite difference maps. Specifically,…
The equations for the critical points of the action functional defined by a Lagrangian depending on higher-order derivatives of admissible curves on a Lie algebroid are found. The relation with Euler-Poincar\'e and Lagrange Poincar\'e type…
We consider free and proper cotangent-lifted symmetries of Hamiltonian systems. For the special case of G = SO(3), we construct symplectic slice coordinates around an arbitrary point. We thus obtain a parametrisation of the phase space…
For the Jacobian resulting from the previously considered problem of the path integral reduction in Wiener path integrals for a mechanical system with symmetry describing the motion of two interacting scalar particles on a manifold that is…
The classical dynamics of a particle that is driven by a rapidly oscillating potential (with frequency $\omega$) is studied. The motion is separated into a slow part and a fast part that oscillates around the slow part. The motion of the…
I introduce an extended configuration space for classical mechanical systems, called pair-space, which is spanned by the relative positions of all the pairs of bodies. To overcome the non-independence of this basis, one adds to the…
We single out a class of Lagrangians on a group manifold, for which one can introduce non-canonical coordinates in the phase space, which simplify the construction of the Poisson structure without explicitly calculating the Dirac bracket.…
An explicit Lagrangian description is given for the Heisenberg equation on the algebra of operators of a quantum system, and for the Landau-von Neumann equation on the manifold of quantum states which are isospectral with respect to a fixed…
The Dirac equation with chiral symmetry is derived using the irreducible representations of the Poincar\'{e} group, the Lagrangian formalism, and a novel method of projection operators that takes as its starting point the minimal assumption…
With the help of Weil polynomials on the Lie algebra of the Poincar\'e group in D dimensions, we construct a Lagrangian form for higher dimensional gravity on a principal fibre bundle whose base space is an even-dimensional Riemann-Cartan…
Any given system of ordinary differential equations in $n$-dimensional configuration space can be obtained from a peculiar variational problem with one local symmetry. The obtained action functional leads to the Hamiltonian formulation in…
The inverse problem of the calculus of variations consists in determining if the solutions of a given system of second order differential equations correspond with the solutions of the Euler-Lagrange equations for some regular Lagrangian.…
A methodology on making the variational principle well-posed in degenerate systems is constructed. In the systems including higher-order time derivative terms being compatible with Newtonian dynamics, we show that a set of position…
The properties of the equation of Dirac type in three-dimensional and five-dimensional Minkowski space-time with respect to time reflection (in sense of Pauli and Wigner) as well as to the operation of charge conjugation are investigated.…
This note presents an attempt to provide a conceptual framework for variational formulations of classical physics. Variational principles of physics have all a common source in the {\it principle of virtual work} well known in statics of…
The spin degrees of freedom for the relativistic particle are described by either Poincar\'{e} group variables (classically) or Grassmann variables (pseudo-classically). The relationship between those two descriptions are given. In doing…