Introduction to Classical Mechanics · Problem 6.34

The Lagrangian method — Morin

Physics — Mechanics · principle-of-stationary-action

Problem

\ddot{x}

dependence * Assume that there is

\ddot{x}

dependence (in addition to

x

\dot{x}

t

dependence) in the Lagrangian in Theorem 6.1. There will then be the additional term

(\partial L/\partial \ddot{x}_a)\ddot{\beta}

in Eq. (6.19). It is tempting to integrate this term by parts twice, and then arrive at a modified form of Eq. (6.22):

\frac{\partial L}{\partial x_0} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}_0}\right) + \frac{d^2}{dt^2}\left(\frac{\partial L}{\partial \ddot{x}_0}\right) = 0.

Is this a valid result? If not, where is the error in the reasoning?