ABSTRACT
For a chemist, the typical first introduction to the Hamiltonian function takes place in an undergraduate class, perhaps physical chemistry, in the context of quantum mechanics. Historically, however, the concepts are introduced into classical mechanics first, and are linked to two principles more profound than Newton’s three laws. The two principles are the conservation of energy principle for autonomous mechanical systems and the least action principle. The latter is expressed typically using a Legendre transform of the Hamiltonian function, called the Lagrangian. In formulating a quantum theory, it is necessary to embed Heisenberg’s uncertainty principle into the equations of motion. Feynman derives the path integral formalism by departing from the least action principle. Learning about the path integral as the element of the time evolution Lie group is easier if the reader is familiar with the principle of least action and the Lagrangian concept. With the exception of some special applications in quantum field theory, path integrals can be formulated using configuration space and the Lagrangian function. The conservation of energy principle has survived the physics revolution of the last century, and generalizes to other situations, such as the laws of thermodynamics.
