2. Draw a diagram
3. Identify the degrees of freedom
4. Assign a coordinate to each degree of freedom
5. Calculate ${\displaystyle T}$ and ${\displaystyle V}$ in terms of the coordinates
1. Sometimes ${\displaystyle T}$ is difficult; use Cartesian coordinates or whatever system is convenient
2. Convert to the coordinate system in (4)
6. Look for cyclic coordinates
1. Each one gives you a "first integral".
2. A "first integral" is a first-order differential equation. It is usually easier to solve than Lagrange's equations which are second order.
7. If only one coordinate remains, you can either
1. Use Lagrange's equations, or
2. Use the conservation of ${\displaystyle H}$ if ${\displaystyle \partial L/\partial t=0}$
8. If more than one coordinate remains, it is often easier to use Lagrange's equations for each of them, because the Hamiltonian usually couples the degrees of freedom.
9. You should have as many equations as degrees of freedom. Solve them!
10. Apply the boundary conditions.