User:Johnjbarton/sandbox/least action

Notes, for Draft only below here

Ref:

when the equations of motion are derivable from a variation principle (Hamilton's principle), a general and systematic procedure for the establishment of the conservation theorems can be developed from a direct study of the variational integral. Since the general equations of mechanics, electromagnetic theory, etc. in use at the present time are derivable from such variational principles, this procedure furnishes the most suitable basis for the systematic study of the conservation theorems.

Noether from https://www.eftaylor.com/pub/BibliogLeastAction12.pdf

Yourgrau:

• QM with action: one postulate rather than two; p127.
• Intrinsically covariant based on Lagrangian
• Feynman like Huygen's but in configuration space. p136.
• Feynman not a variational principle but reduces to least action (Hamiltons) for h to zero. p137.
• Feynman integral QM vs Wave/Matrix differential. p137.
• Appl to hydrodynamics, incl superfluids. p147

Key resource: Edwin Taylor's site on least action

• As an suitable topic for undergraduates.

Gray's article and references therein:

• "For conservative (time-invariant) systems the Hamilton and Maupertuis principles are related by a Legendre transformation (Gray et al. 1996a, 2004)."
• "Note that E is fixed but T is not in Maupertuis' principle (4), the reverse of the conditions in Hamilton's principle (2)."

An attempt to sort out terminology

• Principle of least action
• Stationary-action principle
• Hamilton's principle.
• Hamilton's principle function.

Hamilton's optico-mechanical analogy should also be discussed.

Hamilton's optio-mechanical analogy

Modern formulation based on Fermat varational principle.

Goldstein

Hamilton's principle.

$\delta \int _{t{1}}^{t{2}}L(q_{1},...q_{n},{\dot {q}}_{1},...{\dot {q}}_{n},t)dt=0$

The $\delta$ variation corresponds to virtual displacements with time fixed and coordinates varied consistent with constraints.

Goldstein pg 356 "Another variational principle associated with the Hamiltonian formulation is known as the principle of Least Action". It involves a new type of variation.

Goldstein, bottom of page 359: modern custom: the integral in Hamilton's principle is the action, where as the one in least action is the abbreviated action.

Goldstein calls the indefinite integral arising from Hamilton-Jacoci differential equations "Hamiltons principal function", not as Feynman the "action".

Goldstein discusses the optico-mechanical analogy in section 9-8 of the first ed. and Fermat as a special case of least action in 7-5