Hamiltonian optics

Hamiltonian optics and Lagrangian optics are two formulations of geometrical optics which share much of the mathematical formalism with Hamiltonian mechanics and Lagrangian mechanics.However, the meaning of Liouville’s theorem in mechanics is rather different from the theorem of conservation of étendue. Liouville’s theorem is essentially statistical in nature, and it refers to the evolution in time of an ensemble of mechanical systems of identical properties but with different initial conditions. Each system is represented by a single point in phase space, and the theorem states that the average density of points in phase space is constant in time. An example would be the molecules of a perfect classical gas in equilibrium in a container. Each point in phase space, which in this example has 2N dimensions, where N is the number of molecules, represents one of an ensemble of identical containers, an ensemble large enough to permit taking a statistical average of the density of representative points. Liouville’s theorem states that if all the containers remain in equilibrium, the average density of points remains constant. Hamiltonian optics and Lagrangian optics are two formulations of geometrical optics which share much of the mathematical formalism with Hamiltonian mechanics and Lagrangian mechanics. In physics, Hamilton's principle states that the evolution of a system ( q 1 ( σ ) , ⋯ , q N ( σ ) ) {displaystyle left(q_{1}left(sigma ight),cdots ,q_{N}left(sigma ight) ight)} described by N {displaystyle N} generalized coordinates between two specified states at two specified parameters σA and σB is a stationary point (a point where the variation is zero), of the action functional, or where q ˙ k = d q k / d σ {displaystyle {dot {q}}_{k}=dq_{k}/dsigma } . Condition δ S = 0 {displaystyle delta S=0} is valid if and only if the Euler-Lagrange equations are satisfied with k = 1 , ⋯ , N {displaystyle k=1,cdots ,N} .