On a Momentum Based Version of Lagrange’s Equations

Abstract

The present contribution intends to promote an alternative form of Lagrange ’ s Equations, which rests upon the notion of momentum. We first present a short derivation of the proposed momentum based version of Lagrange’s Equations. From this derivation it becomes apparent that the derivatives of the kinetic energy with respect to the generalized coordinates must cancel out in the original kinetic energy based version of Lagrange’s Equations, and thus need not to be computed. The presented momentum based formulation of Lagrange’s Equations is valid for deformable bodies, modeled in the framework of the Ritz approximation technique, where rigid-body degrees-of-freedom may be present. After having stated this momentum based version of Lagrange’s Equations, we restrict to plane motions of rigid bodies, and demonstrate our proposed formulation for the case of a rotational degree of freedom, where we present an additional connection to the notion of momentum of the rigid body, particularly to angular momentum. Finally, we present the exemplary application to systems consisting of two rigid bodies, namely the pendulum with a point mass and movable support, and the Sarazin pendulum consisting of a rigid rotating disc and an attached point mass.

Dedicated to Franz Ziegler, Professor Emeritus of Rational Mechanics, Vienna University of Technology, on the occasion of his 75th birthday.