The equations of motion in Lagrangian mechanics are Lagrange's equations, also known as Euler-Lagrange equations. Below, we sketch out the derivation of Lagrange's equation from Newton's laws of motion. See the references for more detailed and more general derivations.

Consider a single particle with mass m and position vector r. The applied force, F, can be expressed as the gradient of a scalar potential energy function V(r, t):

Such a force is independent of third- or higher-order derivatives of r, so Newton's second law forms a set of 3 second-order ordinary differential equations. Therefore, the motion of the particle can be completely described by 6 independent variables, or degrees of freedom. An obvious set of variables is { rj, r?j | j = 1, 2, 3}, the Cartesian components of r and their time derivatives, at a given instant of time (ie. position (x,y,z) and velocity (Vx,Vy,Vz) ).