Hamilton'S Principle and Noether's Theorem

Abstract This chapter presents extended forms of Hamilton's Principle and the phase space Hamilton's principle based on the extended Lagrangian and Hamiltonian methods. Hamilton's Principle has already been treated in the context of traditional Lagrangian and Hamiltonian mechanics. Noether's theorem, a method for using symmetries of the extended Lagrangian to identify quantities that are conserved during the motion of the system, is also presented here. Noether's theorem is a powerful technique for discovering conserved quantities in complex Lagrangian systems. The chapter presents the basics of the method in the simple context of Lagrangian systems with a finite number of degrees of freedom. Both the traditional and the extended Hamilton's Principles are an application of the calculus of variations to mechanics. The traditional Hamilton's Principle uses the coordinate parametric method, while the extended Hamilton's Principle uses the general parametric method.