arXiv subjects
Publications and source records attributed to Wlodzimierz M. Tulczyjew.
A conceptual framework for variational formulations of physical theories is proposed. Such a framework is displayed here just for statics, but it is designed to be subsequently adapted to variational formulations of static field theories and dynamics.
We describe the geometric structures involved in the variational formulation of physical theories. In presence of these structures, the constitutive set of a physical system can be generated by a family of functions. We discuss conditions, under which a family of functions generates an immersed Lagrangian submanifold. These conditions are given in terms of the Hessian of the family.
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.
It is widely believed that statistical interpretation of quantum mechanics requires that density operators representing quantum states be normalized. We present a description of selective measurements in terms of density operators. The description is inspired by Schwinger's Algebra of Microscopic Measurements. Density operators used are not normalized. We do not know applications of density operators requiring normalization.
An analysis of the concept of orientation used in electrodynamics is presented. At least two different versions are encountered in the literature. Both are clearly identified and comparisons are made.
In this paper we continue our analysis of a formulation of electrodynamics fully covariant under the full Poincaré group. Transformations under the four different components of the group force on us the introduction of particles, either in the identification by Feynman or in the identification of Dirac.
Variational formulations of statics and dynamics of mechanical systems controlled by external forces are presented as examples of variational principles.
We propose a new description of dynamics of autonomous mechanical systems which includes the momentum-velocity relation. This description is formulated as a variational principle of virtual action more complete than the Hamilton Principle. The inclusion of constraints in this description is the main topic of the present note. We give examples and models of constraints in variational formulations of statics and dynamics of autonomous systems. Holonomic constraints are usually interpreted as restrictions on configurations of a mechanical system. Nonholonomic constraints are additional restrictions imposed on velocities. This traditional terminology is not adapted to our concept of constraints as imposed on virtual displacements and only indirectly affecting configurations and velocities. Our concept of nonholonomic constraints makes perfect sense for static systems even if velocities do not appear in the description of such systems.
Standandard Hamiltonian mechanics in its homogeneous formulation is applied to the study of discontinuities representing rapid changes of Hamiltonians. Different formulations of Hamiltonian mechanics are reviewed. An original representation of a positive homogeneous Hamiltonian system by an outer oriented coisotropic submanifold of the phase space is proposed.
Categorial methods for generating new local algebras from old ones are presented. A direct proof of the differential structure of the prolongations of a manifold is proposed.
Within the framework of mappings between affine spaces, the notion of $n$-th polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as linearity, iterability, Leibniz and chain rules, are shared -- at the finite level -- by the polarization operators. We give these results by means of explicit general formulae, which are valid at any order $n$, and are based on combinatorial identities. The infinitesimal limits of the $n$-th polarizations of a function will yield its $n$-th derivatives (without resorting to the usual recursive definition), and the above mentioned properties will be recovered directly in the limit. Polynomial functions will allow us to produce a coordinate free version of Taylor's formula.
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 mechanical systems. This principle is presented here as the first step in characterizing local stable equilibria of static systems. An extended analysis of local equilibria is given for systems with configuration manifolds of finite dimensions. Numerous examples of the principle of virtual work and the Legendre transformation applied to static mechanical systems are provided. Configuration spaces for the dynamics of autonomous mechanical systems and for statics of continua are constructed in the final sections. These configuration spaces are not differential manifolds.
The application of the Legendre transformation to a hyperregular Lagrangian system results in a Hamiltonian vector field generated by a Hamiltonian defined on the phase space of the mechanical system. The Legendre transformation in its usual interpretation can not be applied to homogeneous Lagrangians found in relativistic mechanics. The dynamics of relativistic systems must be formulated in terms of implicit differential equations in the phase space and not in terms of Hamiltonian vector fields. The constrained Hamiltonian systems introduced by Dirac [1] are not general enough to cover some important cases. We formulate a geometric framework which permits Lagrangian and Hamiltonian descriptions of the dynamics of a wide class of mechanical systems. Lagrangians and Hamiltonians are presented as families of functions. The Legendre transformation and the inverse Legendre transformation are described as transitions between these families. Two examples, the dynamics of a relativistic particle and a space-time formulation of geometric optics (relativistic massless particle), are given.
A comparative analysis of two different versions of the Legendre transformation is presented. We provide an almost complete although somewhat superficial review of the geometric background for analytical mechanics. Complete coordinate characterizations of all structures are provided. Intrinsic constructions of most of the objects are given. Examples of applications to a number of physical systems is given