The Sussman and Wisdom text discusses some interesting, basic techniques for searching for a path of least action, using a software-experimental approach.

Suppose you have a general path $(t,x(t))$, along with a Lagrangian function, which when integrated over the path, gives the action for that path.

Suppose that $x(t)$ is a point in $n$-dimensional space.

Suppose that we constraint the endpoints of the path to $(t_0,x_0)$ and $(t_1,x_1)$, and we want to search for a path between these points that minimizes the action.

They describe a search approach that is based on interpolation.

Suppose that we have a given interpolation algorithm. (They mention polynomial/spline interpolation.)

For fixed $k$, let $u_1, ..., u_k$ be an increasing sequence of values intermediate between $t_0$ and $t_1$.

Then by choosing values for $f(u_1), ..., f(u_k)$, the interpolation algorithm gives us a path, which we can then feed into the action function, to get a numerical value for the action on that path.

This gives us a $k \cdot n$ dimensional space of paths to search through. We can then apply e.g. a gradient descent search algorithm to find a path of (locally) minimum action.