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The general results presented above for Hamilton's principle can be applied to optics using the Lagrangian defined in Fermat's principle.The Euler-Lagrange equations with parameter σ =x 3 and N=2 applied to Fermat's principle result in ˙ = with k = 1, 2 and where L is the optical Lagrangian and ˙ = /.
Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q 1, q 2, ..., q N) between two specified states q 1 = q(t 1) and q 2 = q(t 2) at two specified times t 1 and t 2 is a stationary point (a point where the variation is zero) of the action functional [] = ((), ˙ (),) where (, ˙,) is the Lagrangian function for the system.
Hamilton's optico-mechanical analogy is a conceptual parallel between trajectories in classical mechanics and wavefronts in optics, introduced by William Rowan Hamilton around 1831. [1] It may be viewed as linking Huygens' principle of optics with Maupertuis' principle of mechanics.
In this formulation, particles travel every possible path between the initial and final states; the probability of a specific final state is obtained by summing over all possible trajectories leading to it. In the classical regime, the path integral formulation cleanly reproduces Hamilton's principle, and Fermat's principle in optics.
The relativistic Lagrangian can be derived in relativistic mechanics to be of the form: = (˙) (, ˙,). Although, unlike non-relativistic mechanics, the relativistic Lagrangian is not expressed as difference of kinetic energy with potential energy, the relativistic Hamiltonian corresponds to total energy in a similar manner but without including rest energy.
Hamilton's equations above work well for classical mechanics, but not for quantum mechanics, since the differential equations discussed assume that one can specify the exact position and momentum of the particle simultaneously at any point in time.
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Principles of Optics, colloquially known as Born and Wolf, is an optics textbook written by Max Born and Emil Wolf that was initially published in 1959 by Pergamon Press. [1] After going through six editions with Pergamon Press, the book was transferred to Cambridge University Press who issued an expanded seventh edition in 1999. [ 2 ]