Wlodzimierz M. Tulczyjew

Time reflection and the dynamics of particles and antiparticles

In this paper we continue our analysis of a formulation of electrodynamics fully covariant under the full Poincare group. Transformations under the four different components of the group force on us the introduction of antiparticles, either in the identification by Feynman or in the identification of Dirac.

A Note on Holonomic Constraints

This note is a preliminary account of research undertaken jointly with G. Marmo of Napoli and P. Urbanski of Warsaw.

A Variational Formulation of Analytical Mechanics in an Affine Space

Abstract Variational formulations of statics and dynamics of mechanical systems controlled by external forces are presented as examples of variational principles.

Differential forms on vector bundles

Abstract A bigraded algebra of polynomial forms on a vector bundle is introduced and a bidegree to each multivector-valued form is assigned. Numerous examples of polynomial forms appearing in various constructions in differential geometry are given. The Poincare lemma for polynomial forms is proved.

A variational formulation of electrodynamics with external sources

We present a variational formulation of electrodynamics using de Rham even and odd differential forms. Our formulation relies on a variational principle more complete than the Hamilton principle and thus leads to field equations with external sources and permits the derivation of the constitutive relations. We interpret a domain in space-time as an odd de Rham 4-current. This permits a treatment of different types of boundary problems in an unified way. In particular we obtain a smooth transition to the infinitesimal version by using a current with a one point support.

DYNAMICS WITH EXTERNAL FORCES REVISITED

The study of isolated system is of little interest to physicists since it is not applicable to practical situations. Analytical mechanics should be brought closer to control theory. Behavior of a system coupled to external controlling devices should be examined. There is a class of controlling devices representable by external forces. We proposed a framework for describing the dynamics of mechanical devices subject to control by external forces. The original version of the theory was presented in G. Marmo, W. Tulczyjew and P. Urbanski, Dynamics of autonomous systems with external forces, Acta Phys. Polon. B33 (2002) 1181–1240. The geometric constructions used in original paper are clarified. The Legendre–Fenchel transformation is described and the concept of external force is introduced. Liouville structures are defined.