as the surface that the light emitted at time has reached at time . Light rays and wave fronts are dual: if one is known, the other can be deduced. More precisely, geometrical optics is a variational problem where the “action” is the travel time along a path, where is the medium's index of refraction and is an infinitesimal arc length. From the above formulation, one can compute the ray paths using the Euler–Lagrange formulation; alternatively, one can compute the wave fronts by solving the Hamilton–Jacobi equation. Knowing one leads to knowing the other. The above duality is very general and applies to all systems that derive from a variational principle: either compute the trajectories using Euler–Lagrange equations or the wave fronts by using Hamilton–Jacobi equation. The wave front at time , for a system initially at at time , is defined as the collection of points such that . If is known, the momentum is immediately deduced. Once is known, tangents to the trajectories are computed by solving the equationfor , where is the Lagrangian. The trajectories are then recovered from the knowledge of . Relationship to the Schrödinger equation The isosurfaces of the function can be determined at any time t. The motion of an -isosurface as a function of time is defined by the motions of the particles beginning at the points on the isosurface. The motion of such an isosurface can be thought of as a wave moving through -space, although it does not obey the wave equation exactly. To show this, let S represent the phase of a wave where is a constant (Planck's constant) introduced to make the exponential argument dimensionless; changes in the amplitude of the wave can be represented by having be a complex number. The Hamilton–Jacobi equation is then rewritten as which is the Schrödinger equation. Conversely, starting with the Schrödinger equation and our ansatz for , it can be deduced that The classical limit () of the Schrödinger equation above becomes identical to the following variant of the Hamilton–Jacobi equation, Applications HJE in a gravitational field Using the energy–momentum relation in the form for a particle of rest mass travelling in curved space, where are the contravariant coordinates of the metric tensor (i.e., the inverse metric) solved from the Einstein field equations, and is the speed of light. Setting the four-momentum equal to the four-gradient of the action , gives