$$\require{cancel}$$

# 9: Hamilton's Action Principle

• 9.1: Introduction to Hamilton's Action Principle
In two papers published in 1834 and 1835, Hamilton announced a dynamical principle upon which it is possible to base all of mechanics, and indeed most of classical physics. Hamilton was seeking a theory of optics when he developed Hamilton’s Principle, plus the field of Hamiltonian mechanics. Hamilton’s Principle can be considered to be the fundamental postulate of classical mechanics. It replaces Newton’s postulated three laws of motion.
• 9.2: Hamilton's Principle of Stationary Action
Hamilton’s crowning achievement was deriving both Lagrangian mechanics and Hamiltonian mechanics, directly in terms of a general form of his principle of least action S.
• 9.3: Lagrangian
Lagrangian mechanics was based on the concepts of kinetic energy and potential energy and was based on d’Alembert’s principle of virtual work. The standard Lagrangian that is the diﬀerence between the kinetic and potential energies. Hamilton extended Lagrangian mechanics by defining Hamilton’s Principle that states that a dynamical system follows a path for which the action functional is stationary, that is, time integral of the Lagrangian.
• 9.4: Application of Hamilton's Action Principle to mechanics
Note that the standard Lagrangian is not unique in that there is a continuous spectrum of equivalent standard Lagrangians that all lead to identical equations of motion. This is because the Lagrangian is a scalar quantity that is invariant with respect to coordinate transformations.
• 9.S: Hamilton's Action Principle (Summary)
This chapter introduced Hamilton’s use of least action to derive Hamilton’s Principle, and its application to Lagrangian and Hamiltonian mechanics. Gauge invariance of the Lagrangian was discussed. The concept of alternative standard, and non-standard, Lagrangians was introduced and their applicability was illustrated.