# 13: Hamilton’s Principle of Least Action

- 13.1: Introduction to Hamilton’s Principle of Least Action
- 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.

- 13.2: Principle of Least 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.

- 13.3: Standard 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.

- 13.4: Gauge Invariance of the Lagrangian
- 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.

- 13.5: Non-Standard Lagrangians
- The flexibility and power of Lagrangian mechanics can be extended to a broader range of dynamical systems by employing an extended definition of the Lagrangian that is based on Hamilton’s Principle, which provides a general definition of the Lagrangian that applies to standard Lagrangians, which are expressed as the diﬀerence between the kinetic and potential energies, as well as to non-standard Lagrangians where there may be no clear separation into kinetic and potential energy terms.

- 13.6: Inverse Variational Calculus
- Non-standard Lagrangians and Hamiltonians are not based on the concept of kinetic and potential energies. Therefore, development of non-standard Lagrangians and Hamiltonians require an alternative approach that ensures that they satisfy Hamilton’s Principle, which underlies the Lagrangian and Hamiltonian formulations. One useful alternative approach is to derive the Lagrangian or Hamiltonian via an inverse variational process based on the assumption that the equations of motion are known.

- 13.7: Dissipative Lagrangians
- Energy dissipation is an irreversible process that plays an important role for most physical systems encountered in nature. This irreversibility contrasts with the reversible nature of the basic models employed to describe conservative systems. Dissipation for an observed system usually arises from interactions between the observed system and a bath of unobserved systems that absorb the energy. Usually the detailed structure of the absorb systems is irrelevant for the calculations.

- 13.8: Linear Velocity-Dependent Dissipation
- The special case of linear velocity-dependent dissipation is used below to illustrate the potential capabilities of standard and non-standard Lagrangians to derive the equations of motion for dissipative dynamical systems.

- 13.S: Hamilton’s Principle of Least Action (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.