Constrined hamiltonian dynamics geometric structure and application to relativistic particle machanics

Abstract

The study of constrained dynamical systems has gained importance in recent years, following the realisation that they are often found in nature. The subject developed out of efforts to recast Einstein's theory of gravitation in Hamiltonian form, as a prelude to quantisation. It has since turned out that constraints are fairly common in physically interesting field theories and also appear in finite?dimensional systems. This thesis deals with the classical dynamics of constrained systems and treats two topics in this area. In Part I we discuss the geometric structure of these systems and in Part II, we use constraints to describe interacting relativistic particles. Part I: The aim of this part is to develop a detailed geometrical picture of the dynamics and continuous symmetries of singular Lagrangian systems. We keep closely in touch with physical ideas and motivation, keeping the geometric picture and the physical one side by side. Practical applications of constraint theory usually use local coordinates. We try to get the most out of such a local description before recasting the formalism in intrinsic terms. Chapter II treats the dynamics of a singular Lagrangian system and Chapter III their symmetries. The geometrical analysis of the dynamics begins on the tangent bundle over the configuration manifold with the Lagrangian function given on TQ. The Lagrangian then determines the Legendre map from TQ to the cotangent bundle. This map corresponds physically to the definition of the canonical momenta. Working on TQ we express the Euler–Lagrange equations intrinsically, as an equation for the dynamical vector field ?. This is regarded as an equation to be solved for ?. In the singular case the solution emerges only after several stages of analysis and exists only on a part of TQ. We discuss the steps involved in this analysis, keeping a close parallel with Dirac’s theory of constrained systems. Secondary constraints emerge geometrically as a necessary condition for a solution to exist. In the course of the analysis we demand that ? be second order and tangent to its domain of definition. At the end of the analysis on TQ we have an acceptable dynamical vector field defined on some sub?region. We then use the Legendre map to transfer the discussion to the cotangent bundle. In the singular case the image of TQ is a submanifold ?? and the image of ? is a submanifold of TQ. The dynamical vector field found on TQ is then transferred to TQ and expressed with the help of the embedding of ?? in the symplectic manifold TQ. The final form of the dynamics on TQ involves a certain amount of arbitrariness. We then discuss the physical interpretation of the formalism, pointing out different ways of identifying the physical states with points of ??, their relative merits and drawbacks. Of particular interest is the elucidation in intrinsic terms of Dirac’s suggestion that one add to the Hamiltonian all the secondary first?class constraints, multiplied by arbitrary functions. Then we discuss some special cases to illustrate the general discussion.