Hamilton-Jacobi theory

Caustics The above formulae can only be valid as long as Eq. (9) describes a unique map in position space. Indeed, the underlying Hamilton-Jacobi theory is only valid for the time interval [0,T] if at all instances t [0, T] the map (QOi4o) —> Q t, qo,qo) is one-to-one, [6, 19, 1], i.e., as long as trajectories with different initial data do not cross each other in position space (cf. Fig. 1). Consequently, the detection of any caustics in a numerical simulation is only possible if we propagate a trajectory bundle with different initial values. Thus, in pure QCMD, Eq. (11), caustics cannot be detected. 

The book contains very little original material, but reviews a fair amount of forgotten results that point to new lines of enquiry. Concepts such as quaternions, Bessel functions, Lie groups, Hamilton-Jacobi theory, solitons, Rydberg atoms, spherical waves and others, not commonly emphasized in chemical discussion, acquire new importance. To prepare the ground, the... 

An obvious approach to the problem was to re-examine the theories of mechanics that failed to predict the observed frequencies and bring them into line with observation. It is therefore not surprising to find that both new theories were modified versions of classical Hamilton-Jacobi theory, adapted so as to incorporate the newly discovered quantum features of atomic spectra. To understand the type of reasoning that produced the quantum theory, it is necessary to make a short excursion into classical HJ theory. Some readers may prefer to skip this section on first reading. For an in-depth discussion more enthusiastic readers are referred to the authoritative text of Goldstein... 

The asymptotic velocity depends explicitly on the shape of the initial conditions, if they do not have compact support. An adaptation of the Hamilton-Jacobi theory from classical mechanics is a usefiil technique to deal with this problem in a very general way, see below. The prototypical example of a concave reaction term is the KPP or logistic term F p) = rp — p). Equation (4.13) implies that v = 2 /rD. Examples for convex reaction functions typically occur in combustion theory, where F p) = — p) is referred to as the Arrhenius reaction term, or F p) =... 

The fact that an infinity of front velocities occurs for pulled fronts gives rise to the problem of velocity selection. In this section we present two methods to tackle this problem. The first method employs the Hamilton-Jacobi theory to analyze the dynamics of the front position. It is equivalent to the marginal stability analysis (MSA) [448] and applies only to pulled fronts propagating into unstable states. However, in contrast to the MSA method, the Hamilton-Jacobi approach can also deal with pulled fronts propagating in heterogeneous media, see Chap. 6. The second method is a variational principle that works both for pulled and pushed fronts propagating into unstable states as well as for those propagating into metastable states. This principle can deal with the problem of velocity selection, if it is possible to find the proper trial function. Otherwise, it provides only lower and upper bounds for the front velocity. 

Crehuet R (2005) The reaction path intrinsic reaction coordinate method and the Hamilton Jacobi theory. J Chem Phys 122 234105... 

Although we shall not deal with Hamilton-Jacobi theory here, the concept of a set of constants of the motion is vital to an understanding of the issue of intramolecular energy flow and statistical vs nonstatistical behavior. In essence, the number of global constants of the motion provides a method for grouping systems into general categories. 

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

David Bohm gave new direction to Madelung s proposal by using the decomposition of the wave equation for a radically new interpretation of quantum theory. He emphasized the similarity between the Madelung and Hamilton-Jacobi equations of motion, the only difference between them being the quantum potential energy term,... 

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. 

After calculating the unperturbed motion of the inner electron, we can find the secular motions of the remaining variables by introducing a new Hamiltonian function, the mean value of Hx taken over the unperturbed motion of the inner electron. The integration of the corresponding Hamilton-Jacobi equation is again performed by the methods of the theory of perturbations. 

The results obtained in the previous subsection of instanton theory can be reproduced in a much easier way with use of the WKB approximation. This is quite helpful to understand the physical meaning of the various quantities and the procedures used. In the same way as we did in the one-dimensional case in Section 2.5.2, we do not need to construct the complex valued Lagrange manifold [7,15,30,37], which constitutes the main obstacle of the conventional WKB theory. Without the energy term, the Hamilton-Jacobi equation can be easily solved and generalization for an... 

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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