Hamilton–Jacobi equation

Comparison with other formulations of mechanics[edit]

The Hamilton–Jacobi equation is a single, first-order partial differential equation for the function of the ${\displaystyle N}$ generalized coordinates ${\displaystyle q_{1},\,q_{2},\dots ,q_{N}}$ and the time ${\displaystyle t}$. The generalized momenta do not appear, except as derivatives of ${\displaystyle S}$, the classical action.


For comparison, in the equivalent Euler–Lagrange equations of motion of Lagrangian mechanics, the conjugate momenta also do not appear; however, those equations are a system of ${\displaystyle N}$, generally second-order equations for the time evolution of the generalized coordinates. Similarly, Hamilton's equations of motion are another system of 2N first-order equations for the time evolution of the generalized coordinates and their conjugate momenta ${\displaystyle p_{1},\,p_{2},\dots ,p_{N}}$.


Since the HJE is an equivalent expression of an integral minimization problem such as Hamilton's principle, the HJE can be useful in other problems of the calculus of variations and, more generally, in other branches of mathematics and physics, such as dynamical systems, symplectic geometry and quantum chaos. For example, the Hamilton–Jacobi equations can be used to determine the geodesics on a Riemannian manifold, an important variational problem in Riemannian geometry. However as a computational tool, the partial differential equations are notoriously complicated to solve except when is it possible to separate the independent variables; in this case the HJE become computationally useful.^[5]: 444