Simple Mechanical Systems and Fields

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.

$$\bgroup\color{black} L(q,\dot{q})=T-V \egroup$$

Lagrange's equations are

$$\bgroup\color{black}{d \over dt}\left({\partial L\over\partial\dot{q}_i}\right)-{\partial L\over\partial q_i}=0 \egroup$$

where the $q_i$ are the coordinates of the particle. This equation is derivable from the principle of least action.

$$\bgroup\color{black} \delta \int\limits_{t_1}^{t_2}L(q_i,\dot{q}_i)dt=0 \egroup$$

Similarly, we can define the Hamiltonian

$$\bgroup\color{black} H(q_i,p_i)=\sum\limits_ip_i\dot{q}_i-L \egroup$$

where $p_i$ are the momenta conjugate to the coordinates $q_i$.

$$\bgroup\color{black} p_i={\partial L\over\partial\dot{q}_i} \egroup$$

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

$$\bgroup\color{black} L=\int {\cal L} dx \egroup$$

For example, for a string,

$$\bgroup\color{black} {\cal L}={1\over 2}\left[\mu\dot{\eta}^2-Y\left({\partial\eta\over\partial x}\right)^2\right] \egroup$$

where $Y$ is Young's modulus for the material of the string and $\mu$ is the mass density. The Euler-Lagrange Equation for a continuous system is also derivable from the principle of least action states above. For the string, this would be.

$$\bgroup\color{black} {\partial\over\partial x}\left({\partial... ...rtial t)}\right) -{\partial{\cal L}\over\partial\eta}=0 \egroup$$

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

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

$$\bgroup\color{black} {\cal H}=\dot{\eta}{\partial{\cal L}\over\partial\dot{\eta}}-{\cal L} \egroup$$

In this example of a string, $\eta(x,t)$ 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.

$$\begin{eqnarray*} {\cal L}&=&{1\over 2}\left[\mu\dot{\eta}^2-Y\left({\partial\et... ...0 \\ \ddot{\eta}&=&{Y\over\mu}{\partial^2\eta\over\partial x^2} \end{eqnarray*}$$

This is the wave equation for the string. There are easier ways to get to this wave equation, but, as we move away from simple mechanical systems, a formal way of proceeding will be very helpful.