In classical mechanics
the path chosen by a system is the path that minimises the action, which is the time integral
of the lagrangian, L(q, qdot, t) = T - V. T is the kinetic energy
and V the potential energy
, q is some set of coordinates
and qdot indicates their time derivatives
. The q may be any set of coordinates, euclidian
or anything else that seems convenient.
Note that there might be time dependence in T and V.
Minimizing the action leads to a set of differential equations of motion for q, the Euler-Lagrange equations.
Solving them is usually much simpler than writing down all the forces in the system, since it is possible in the Lagrangian to choose any convenient set of coordinates and to include time dependence. Also it is possible to include constraints in the equations, to keep a cart on a rollercoaster track, for instance.