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.

Lagrange's equations are

where the are the coordinates of the particle. This equation is derivable from the

Similarly, we can define the

where are the momenta conjugate to the coordinates .

For a continuous system, like a **string**, the Lagrangian is an integral of a Lagrangian density function.

For example, for a string,

where is Young's modulus for the material of the string and is the mass density. The

Recall that the Lagrangian is a function of and its space and time derivatives.

The **Hamiltonian density** can be computed from the Lagrangian density and is a function of the coordinate
and its
conjugate momentum.

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.

This is the

Jim Branson 2013-04-22