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Concept
Hamiltonian optics
Summary
Hamiltonian optics and Lagrangian optics are two formulations of geometrical optics which share much of the mathematical formalism with Hamiltonian mechanics and Lagrangian mechanics.
Hamilton's principle
In physics, Hamilton's principle states that the evolution of a system described by 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 and is the Lagrangian. Condition is valid if and only if the Euler-Lagrange equations are satisfied, i.e.,
with .
The momentum is defined as
and the Euler–Lagrange equations can then be rewritten as
where .
A different approach to solving this problem consists in defining a Hamiltonian (taking a Legendre transform of the Lagrangian) as
for which a new set of differential equations can be derived by looking at how the total differential of the Lagrangian depends on parameter σ, positions and their derivatives relative to σ. This derivation is the same as in Hamiltonian mechanics, only with time t now replaced by a general parameter σ. Those differential equations are the Hamilton's equations
with . Hamilton's equations are first-order differential equations, while Euler-Lagrange's equations are second-order.
The general results presented above for Hamilton's principle can be applied to optics. In 3D euclidean space the generalized coordinates are now the coordinates of euclidean space.
Fermat's principle
Fermat's principle states that the optical length of the path followed by light between two fixed points, A and B, is a stationary point. It may be a maximum, a minimum, constant or an inflection point. In general, as light travels, it moves in a medium of variable refractive index which is a scalar field of position in space, that is, in 3D euclidean space. Assuming now that light travels along the x3 axis, the path of a light ray may be parametrized as starting at a point and ending at a point .
Official source
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