Hamilton - Jacobi differential

This equation is known as the Hamilton-Jacobi differential equation. The problem is now to find a complete solution, i.e. a solution which involves at and/—1 other constants of integration a2, as. . . at, apart from the purely additive constant in S. This function S provides a transformation (1) of the kind desired at the same time... 

A general rule for the rigorous solution of the Hamilton-Jacobi differential equation (5) cannot be given. In many cases a solution is obtained on the supposition that S can be represented as the sum of/functions, each of which depends on only one of the co-ordinates q (and, of course, on the integration constants ax. . . af) ... 

It is easy to see that the Hamilton-Jacobi differential equation is separable neither in rectangular nor in polar co-ordinates. It may, however, be made separable by introducing parabolic co-ordinates. We put... 

It is the Hamiltonian function of a system in which all co-ordinates but one are cyclic. The motion may be found in the usual way by solving a Hamilton-Jacobi differential equation for one degree of freedom. Since and ij (like ° and rj°) must vanish with A, we need only consider small motions, that is, those belonging to a system whose Hamiltonian function is... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

In the first chapter no attempt will be made to give any parts of classical dynamics but those which are useful in the treatment of atomic and molecular problems. With this restriction, we have felt justified in omitting discussion of the dynamics of rigid bodies, non-conservative systems, non-holonomic systems, systems involving impact, etc. Moreover, no use is made of Hamilton s principle or of the Hamilton-Jacobi partial differential equation. By thus limiting the subjects to be discussed, it is possible to give in a short chapter a thorough treatment of Newtonian systems of point particles. 

Suffixes p and a both refer to accidentally degenerate variables.) This is a partial differential equation of the Hamilton-Jacobi type. It does not admit of integration in all cases, and the method fails, therefore, for the determination of the motion for arbitrary values of the Jfc s. We can show, however, as in the example of 44, that the motions for which the wp° s are constant to zero approximation, and remain constant also to a first approximation, are stationary motions in the sense of quantum theory. 

This differential equation of the Hamilton-Jacobi type for one degree of freedom can always be solved by the method of quadratures and we find... 

The set of partial differential equations (8.37), dp- /dt = F3, and dpi/dt = Fj form a closed system. We calculate the velocity of a propagating front, connecting the state (0, pg, 0) to (py, 0,0) from initial conditions with compact support, from the Hamilton-Jacobi equation in dimensionless units ... 

Eventually the partial differential equation, of tjrpe Hamilton-Jacobi [20], for updating the level set function is produced. 

The determination of the good actions describing vibration-rotation motion requires the solution of the molecular Hamilton-Jacobi equation, which is a nonlinear partial differential equation in 3Na"5 variables (including rotation), where is the number of atoms. Even for = 3 (a triatomic molecule) an exact solution to this equation is extremely complex computationally, and it is not practical for collisional applications. Several approximations can be used to simplify this treatment, however, including (i) the separation of vibration from rotation (valid in the limit of an adequate vibration-rotation time scale separation), and (ii) the use of classical perturbation theory (in 2nd and 3rd order) to solve the three-dimensional vibrational Hamilton-Jacobi equation which remains after the separation of rotation. Details of both the separation procedures and the perturbation-theory solution are discussed elsewhere. For the present application, the validity of the first... 

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