Variational principles Jacobi

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. 

This method has been used successfully for the treatment of quantum solids. We point out, however, that an alternative approach, providing an explicit quantum-mechanical treatment of excited states, is possible. In this approach we write exciton-like vibrationally excited wavefunctions, having the translational symmetry of the crystal. Using these wavefunctions as a basis, a secular equation for the excited states is obtained by applying the variational principle to the excited states. Work along these lines is now in progress and will be reported separately (Jacobi and Schnepp, 1971). 

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