This section is a review of mechanical systems largely from the point of view of Lagrangian dynamics. In particular, we review the equations of a string as an example of a field theory in one dimension.
We start with the Lagrangian of a discrete system like a single particle.
For a continuous system, like a string, the Lagrangian is an integral of a Lagrangian density function.
The Hamiltonian density can be computed from the Lagrangian density and is a function of the coordinate
In this example of a string, is a simple scalar field. The string has a displacement at each point along it which varies as a function of time.
If we apply the Euler-Lagrange equation, we get a differential equation that the string's displacement will satisfy.
Jim Branson 2013-04-22