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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 ...
The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers. [4] [5] [6] The connection to the Hamilton–Jacobi equation from classical physics was first drawn by Rudolf Kálmán. [7] In discrete-time problems, the analogous difference equation is usually referred to as the ...
It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in dynamic programming (the Hamilton–Jacobi–Bellman equation), differential games (the Hamilton–Jacobi–Isaacs equation) or front evolution problems, [1] [2] as well as ...
This is a partial differential equation, in particular a Hamilton–Jacobi equation, and can be solved numerically, for example, by using finite differences on a Cartesian grid. [2] [3] However, the numerical solution of the level set equation may require advanced techniques. Simple finite difference methods fail quickly.
Using the energy given above as the action, one may choose to solve either the Euler–Lagrange equations or the Hamilton–Jacobi equations. Both methods give the geodesic equation as the solution; however, the Hamilton–Jacobi equations provide greater insight into the structure of the manifold, as shown below.
The Hamilton's principal function satisfies the Hamilton–Jacobi equation, a formulation of classical mechanics. Due to a similarity with the Schrödinger equation, the Hamilton–Jacobi equation provides, arguably, the most direct link with quantum mechanics.
Adopting the radial distance r and the azimuthal angle φ as the coordinates, the Hamilton-Jacobi equation for a central-force problem can be written + + = where S = S φ (φ) + S r (r) − E tot t is Hamilton's principal function, and E tot and t represent the total energy and time, respectively.
The Hamilton–Jacobi equations provide a commonly used and straightforward method for identifying constants of motion, particularly when the Hamiltonian adopts recognizable functional forms in orthogonal coordinates. Another approach is to recognize that a conserved quantity corresponds to a symmetry of the Lagrangian.