Applied Calculus of Variations for Engineers, Third edition, 3rd Edition

10

 

Variational equations of motion

We encountered variational forms of equations of motion in prior chapters, for example, when solving the brachistochrone problem in Section 1.4.2. Several dynamic equations of motion will be derived from variational principles in this chapter using techniques developed by Legendre, Hamilton and Lagrange. Specifically, mechanical systems, electric circuits and orbital motion will be investigated.

 

10.1Legendre’s dual transformation

This transformation invented by Legendre is of fundamental importance. Let us consider the function of n variables

f=f(u1,u2,...,un).

Legendre proposed to introduce a new set of variables by the transformation of

vi=∂f∂ui;i=1,2,...,n.

The Hessian matrix of this transformation ...