Mathematics of Classical Mechanics

The mechanics of rigid bodies which I learned in engineering classes was mostly:

• Draw a system.
• Draw a lot of forces and constraints.
• Select a coordinate system. Mostly XYZ.
• Calculate. When something strange happen, check your forces again.

Using this approach for moving parts was pretty annoying.

But maybe there is a more elegant way to do it? I hope to use a more mathematical approach, like Langrangian and Hamiltonian mechanics.