the Hamilton-Jacobi equation
/ HAM-il-ton yah-KOH-bee /
Suppose you want the shortest travel time from every point to a target, or the cheapest path in a control problem, or the wavefront of light leaving a source. In each case there is a 'value' function whose gradient encodes the optimal direction, and it obeys a single first-order PDE that is nonlinear in the gradient. The Hamilton-Jacobi equation is that master equation, born in classical mechanics and now central to optimal control, geometric optics, and front propagation.
Its form is u_t + H(x, grad u) = 0, where the Hamiltonian H is a given function of position and of the gradient grad u. (A time-independent version, H(x, grad u) = 0, includes the eikonal equation |grad u| = 1/c of geometric optics.) The classical method is characteristics: the PDE for u is equivalent to a system of ODEs — Hamilton's equations dx/dt = H_p, dp/dt = -H_x along bicharacteristic 'rays' — and u is built by integrating along them. But just as for conservation laws, characteristics cross: rays focus, the gradient grad u becomes multivalued, and a classical smooth solution ceases to exist in finite time (think of a wavefront forming a corner, or shortest-path lines colliding behind an obstacle). Past that, one needs a weak notion of solution that is unique and stable. The right one is the viscosity solution of Crandall and Lions, obtained as the vanishing-viscosity limit of u_t + H(x, grad u) = epsilon Laplacian u, which selects the physically meaningful (e.g. first-arrival) solution.
Why it matters: Hamilton-Jacobi is the deep cousin of scalar conservation laws — in one dimension, differentiating u_t + H(u_x) = 0 in x gives a conservation law for v = u_x, and the Hopf-Lax/Lax-Oleinik formulas connect the two. It is the equation behind dynamic programming (the Hamilton-Jacobi-Bellman equation of optimal control), behind level-set methods for moving interfaces, and behind the eikonal equation for travel times. Its honest subtlety is the same as for shocks: smoothness fails, and you must commit to the viscosity-solution framework to get well-posedness.
The eikonal equation |grad u| = 1 is a stationary Hamilton-Jacobi equation; its viscosity solution u(x) is the distance from x to a given target set, and its level sets {u = t} are the wavefronts at time t. Where two parts of the front would collide, the viscosity solution keeps the first-arrival (smaller) value and develops a crease — exactly the analogue of a shock.
Distance/travel-time functions solve Hamilton-Jacobi equations; creases in their level sets are the analogue of shocks.
A persistent misconception is that the classical characteristics always give the solution. They give it only until rays cross; afterward the naive characteristic 'solution' becomes multivalued and wrong, and only the viscosity solution is single-valued and well-posed. Do not confuse this Hamilton-Jacobi PDE for an unknown function u(x,t) with Hamilton's ODEs for the trajectories themselves.