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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 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.
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.
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.
With the advent of the Contax by Zeiss-Ikon, the first serious challenge to the Leica in the field of professional 35 mm cameras, both Zeiss-Ikon and Carl Zeiss decided to beat the Leica in every possible way. Bertele's Sonnar series of lenses designed for the Contax were the match in every respect for the Leica for at least two decades. Other ...
The pinhole camera model describes the mathematical relationship between the coordinates of a point in three-dimensional space and its projection onto the image plane of an ideal pinhole camera, where the camera aperture is described as a point and no lenses are used to focus light.
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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.