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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.
Haselgrove developed her equations at Cambridge University in the 1950s, as a student under Kenneth Budden, by re-applying the earlier work of William Rowan Hamilton and Hamilton's principle in geometrical optics [4] to radio propagation in a plasma. [5] Indeed, the application of Haselgrove's equations is often termed Hamiltonian ray tracing.
Hamilton's principle (or the action principle) states that the motion of a conservative holonomic (integrable constraints) mechanical system is such that the action integral = (, ˙,) is stationary with respect to variations in the path ().
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Numerous other concepts and objects in mechanics, such as Hamilton's principle, Hamilton's principal function, the Hamilton–Jacobi equation, Cayley-Hamilton theorem are named after Hamilton. The Hamiltonian is the name of both a function (classical) and an operator (quantum) in physics, and, in a different sense, a term from graph theory .