Hamilton–Jacobi equation

In mathematics, the Hamilton–Jacobi equation (HJE) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi. − ∂ S ∂ t = H ( q , ∂ S ∂ q , t ) . {displaystyle -{frac {partial S}{partial t}}=Hleft(q,{frac {partial S}{partial q}},t ight);.} S ( q , t ) ≐ ∫ ( q , t ) L d t , {displaystyle S(q,t)doteq int ^{(q,t)}{mathcal {L}},mathrm {d} t;,} ∂ S ∂ q = p . {displaystyle {frac {partial S}{partial q}}=p;.} ∂ S ∂ t = − H , {displaystyle {frac {partial S}{partial t}}=-H;,} In mathematics, the Hamilton–Jacobi equation (HJE) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi. In physics, the Hamilton–Jacobi equation is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely. The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 18th 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, Schrödinger's equation, as described below; for this reason, the HJE is considered the 'closest approach' of classical mechanics to quantum mechanics. Boldface variables such as q {displaystyle mathbf {q} } represent a list of N {displaystyle N} generalized coordinates, A dot over a variable or list signifies the time derivative (see Newton's notation), e.g., The dot product notation between two lists of the same number of coordinates is a shorthand for the sum of the products of corresponding components, e.g., Given the Hamiltonian H ( q , p , t ) {displaystyle H(q,p,t)} of a mechanical system (where q {displaystyle q} , p {displaystyle p} are coordinates and momenta of the system and t {displaystyle t} is time) the Hamilton–Jacobi equation is written as a first-order, non-linear partial differential equation for the Hamilton's principal function S ( q , t ) {displaystyle S(q,t)} , The Hamilton's principal function is defined as the function of the upper limit of the action integral taken along the minimal action trajectory of the system, where L {displaystyle {mathcal {L}}} is the Lagrangian of the system and where the trajectory satisfies the Euler–Lagrange equation of the system,