Syllabus

• Lagrangian formulation of acoustic field in gases
• Hamiltonian formulation for continuous systems
• Canonical equations from a variational principle
• Poisson’s brackets and canonical field variables

Basic Introduction

In the first part of the course on classical mechanics, you have mainly studied problems where the focus is on a single particle (or a pair of particles). Many problems in physics cannot be described in terms of a finite collection of particles. Such problems include the study of elastic solids, propagation of sound in gases, the study of electromagnetic radiation and many more.

When we have one degree of freedom (e.g., a particle moving on a line), the system is described by a Lagrangian, $L(q,\dot{q},t),$ which is a function of a generalized coordinate (such as the position of the particle), a generalized velocity (such as the velocity of the particle), and time. For two degrees of freedom, $L(q_1,q_2,\dot{q}_1,\dot{q}_2,t).$ In general, we may write