More generally, we can work with a set of generalized coordinates and their time derivatives, the generalized velocities: {q_j, q'_j}. r is related to the generalized coordinates by some transformation equation:

The term "generalized coordinates" is really a leftover from the period when Cartesian coordinates were THE coordinate system. Nowaways, we would just call them coordinates.

Consider an arbitrary displacement δr of the particle. The work done by the applied force F is δW = F · δr. Using Newton's second law, we write:

Since work is a physical scalar quantity, we should be able to rewrite this equation in terms of the generalized coordinates and velocities. On the left hand side,

The right hand side is more difficult, but after some shuffling we obtain:

where T = 1/2 m r'² is the kinetic energy of the particle. Our equation for the work done becomes

However, this must be true for any set of generalized displacements δq_i, so we must have

for each generalized coordinate δq_i. We can further simplify this by noting that V is a function solely of r and t, and r is a function of the generalized coordinates and t. Therefore, V is independent of the generalized velocities:

Inserting this into the preceding equation and substituting L = T - V, we obtain Lagrange's equations:

There is one Lagrange equation for each generalized coordinate qi. When q_i = r_i (i.e. the generalized coordinates are simply the Cartesian coordinates), it is straightforward to check that Lagrange's equations reduce to Newton's second law.

The above derivation can be generalized to a system of N particles. There will be 6N generalized coordinates, related to the position coordinates by 3N transformation equations. In each of the 3N Lagrange equations, T is the total kinetic energy of the system, and V the total potential energy.