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 ( q 1 ( σ ) , … , q N ( σ ) ) {\displaystyle \left(q_{1}{\left(\sigma \right)},\dots ,q_{N}{\left(\sigma \right)}\right)} 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
δ S = δ ∫ σ A σ B L ( q 1 , ⋯ , q N , q ˙ 1 , ⋯ , q ˙ N , σ ) d σ = 0 {\displaystyle \delta S=\delta \int _{\sigma _{A}}^{\sigma _{B}}L\left(q_{1},\cdots ,q_{N},{\dot {q}}_{1},\cdots ,{\dot {q}}_{N},\sigma \right)\,d\sigma =0}
where q ˙ k = d q k / d σ {\displaystyle {\dot {q}}_{k}=dq_{k}/d\sigma } and L {\displaystyle L} is the Lagrangian. Condition δ S = 0 {\displaystyle \delta S=0} is valid if and only if the Euler-Lagrange equations are satisfied, i.e.,
∂ L ∂ q k − d d σ ∂ L ∂ q ˙ k = 0 {\displaystyle {\frac {\partial L}{\partial q_{k}}}-{\frac {d}{d\sigma }}{\frac {\partial L}{\partial {\dot {q}}_{k}}}=0}
with k = 1 , … , N {\displaystyle k=1,\dots ,N} . The momentum is defined as
p k = ∂ L ∂ q ˙ k {\displaystyle p_{k}={\frac {\partial L}{\partial {\dot {q}}_{k}}}}
and the Euler–Lagrange equations can then be rewritten as
p ˙ k = ∂ L ∂ q k {\displaystyle {\dot {p}}_{k}={\frac {\partial L}{\partial q_{k}}}}
where p ˙ k = d p k / d σ {\displaystyle {\dot {p}}_{k}=dp_{k}/d\sigma } . A different approach to solving this problem consists in defining a Hamiltonian (taking a Legendre transform of the Lagrangian) as
H = ∑ k q ˙ k p k − L {\displaystyle H=\sum _{k}{{\dot {q}}_{k}}p_{k}-L}
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