Constrained Dynamics in the Hamiltonian formalism

TL;DR

This paper provides pedagogical insights into the Hamiltonian formulation of constrained dynamical systems, focusing on finite-dimensional examples, Dirac brackets, and various types of constraints, serving as an introductory resource for students.

Contribution

It offers a clear, example-driven explanation of Hamiltonian constraints, Dirac brackets, and their applications to finite-dimensional systems, filling a gap in educational resources.

Findings

Explicit construction of Dirac brackets as projections of Poisson brackets

Illustration of constraints in systems like particles in magnetic fields and relativistic particles

Educational framework for understanding constrained Hamiltonian dynamics

Abstract

These are pedagogical notes on the Hamiltonian formulation of constrained dynamical systems. All the examples are finite dimensional, field theories are not covered, and the notes could be used by students for a preliminary study before the infinite dimensional phase space of field theory is tackled. Holonomic constraints in configuration space are considered first and Dirac brackets introduced for such systems. It is shown that Dirac brackets are a projection of Poisson brackets onto the constrained phase space and the projection operator is constructed explicitly. More general constraints on phase space are then considered and exemplified by a particle in a strong magnetic field. First class constraints on phases are introduced using the example of motion on the complex projective space ${\mathbf{C P}}^{n-1}$. Motion of a relativistic particle in Minkowski space with a reparameterisation invariant world-line is also discussed. These notes are based on a short lecture course given at Bhubaneswar Indian Institute of Technology in November 2021.