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Hamilton–Jacobi equation
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<p>In physics, the Hamilton–Jacobi equation, named after <a href="/facts/William_Rowan_Hamilton/U86CqUlt">William Rowan Hamilton</a> and <a href="/facts/Carl_Gustav_Jacob_Jacobi/4Syeauo5">Carl Gustav Jacob Jacobi</a>, is an alternative formulation of <a href="/facts/Classical_mechanics/ykG8upsN">classical mechanics</a>, equivalent to other formulations such as <a href="/facts/Newton%2527s_laws_of_motion/XKzGTXK1">Newton's laws of motion</a>, <a href="/facts/Lagrangian_mechanics/sTxyqWz8">Lagrangian mechanics</a> and <a href="/facts/Hamiltonian_mechanics/2ReerlNA">Hamiltonian mechanics</a>. </p><p>The Hamilton–Jacobi equation is a formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, it fulfilled a long-held goal of theoretical physics (dating at least to <a href="/facts/Johann_Bernoulli/U8kRr2Bn">Johann Bernoulli</a> in the eighteenth century) of finding an analogy between the propagation of light and the motion of a particle. The wave equation followed by mechanical systems is similar to, but not identical with, <a href="/facts/Schr%25C3%25B6dinger%2527s_equation/3csgSIWF">Schrödinger's equation</a>, as described below; for this reason, the Hamilton–Jacobi equation is considered the "closest approach" of <a href="/facts/Classical_mechanics/ykG8upsN">classical mechanics</a> to <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>. The qualitative form of this connection is called <a href="/facts/Hamilton%2527s_optico-mechanical_analogy/6TohRf3s">Hamilton's optico-mechanical analogy</a>. </p><p>In mathematics, the Hamilton–Jacobi equation is a <a href="/facts/Necessity_and_sufficiency/G45aDCY9">necessary condition</a> describing extremal <a href="/facts/Geometry/wTQf5ffQ">geometry</a> in generalizations of problems from the <a href="/facts/Calculus_of_variations/Ya5Lsma3">calculus of variations</a>. It can be understood as a special case of the <a href="/facts/Hamilton%25E2%2580%2593Jacobi%25E2%2580%2593Bellman_equation/6EoqXTgo">Hamilton–Jacobi–Bellman equation</a> from <a href="/facts/Dynamic_programming/CLcabtmk">dynamic programming</a>. </p>