Analytical Dynamics Development of the Canonical Equations

TL;DR

This paper introduces a novel approach to deriving Hamiltonian functions and canonical equations directly from d'Alembert's equation, providing a more fundamental connection to classical analytical dynamics than traditional methods.

Contribution

It presents a new method for developing Hamiltonian functions by investigating d'Alembert's equation, offering a more direct link to classical dynamics.

Findings

Provides a direct derivation of Hamiltonian from d'Alembert's equation

Establishes a clearer connection between Hamiltonian mechanics and classical dynamics

Offers an alternative to Legendre transformation approaches

Abstract

It is most common to construct the Hamiltonian function and Hamilton's canonical equations through a Legendre transformation of the Lagrangean function or through the central equation. These common perspectives, however, seem abstract and detached from classical analytical dynamics. A new and different approach is presented in which the Hamiltonian function is created as one investigates d'Alembert's equation of motion. This formulation directly ties the Hamiltonian function and Hamilton's canonical equations to the root of classical analytical dynamics more than any other approach.