Chapter 15

Write down the Lagrangian describing the transverse motion of a uniform stretched membrane, using polar coordinates. Hence find the equations of motion of a circular membrane.

A system is described by the Lagrangian density $$ \mathcal{L}=\frac{1}{2}\left[\psi^{2}-(\nabla \psi)^{2}-\kappa^{2} \psi^{2}\right] $$ Find the field equations. Evaluate the stress-energy tensor, and find the Hamiltonian of the system.