Question

what is the equivalence of lagrange's and newton's equation of motion

Answer 1

Once you identify a Lagrangian as L(x, v, t) = 1/2 m * v^2 - V(x, t) where V is the potential energy which depends on the position of a particle but not on its velocity (and maybe also on t, as well but that case will only produce an extra term). Then it follows that the stationary action principle gives you m d/dt v = - grad V (where grad stands for the gradient of V, that is, the vector with partial derivatives with respect to the coordinates as components) . It turns out that a conservative force can be expressed as this gradient with negative sign. Also, d/dt v=a, the acceleration vector. Thus, Lagrange's equations of motion are the same as Newton's. Conversely, newton's equations are the same as Lagrange's making the appropriate identifications.

Answer 2

perfect reponse, only L(v, x, t) not L(x, v, t), this can make a difference