Hamilton-Jacobi Theory
In this chapter, we present the classical Hamilton-Jacobi theory. This theory has played an enormous role in the development of theoretical and mathematical physics. On the one hand, it builds a bridge between classical mechanics and other branches of phy
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Hamilton-Jacobi Theory
In this chapter, we present the classical Hamilton-Jacobi theory. This theory has played an enormous role in the development of theoretical and mathematical physics. On the one hand, it builds a bridge between classical mechanics and other branches of physics, in particular, optics. On the other hand, it yields a link between classical and quantum theory. In Sect. 12.1 we start with deriving the HamiltonJacobi equation and give a proof of the classical Jacobi Theorem, which yields a powerful tool for solving the dynamical equations of a Hamiltonian system. We interpret the Hamilton-Jacobi equation geometrically as an equation for a Lagrangian submanifold of phase space1 which is contained in the coisotropic submanifold given by a level set of the Hamiltonian. Using this geometric picture, one can extract a general method for solving initial value problems for arbitrary first order partial differential equations of the Hamilton-Jacobi type. This method is based on the fact that solutions are generated by the characteristics of the underlying Hamiltonian system. That is why this procedure is called the method of characteristics. It will be discussed in detail in Sect. 12.2. In Sect. 12.3 we generalize this method to the case of systems of partial differential equations of the Hamilton-Jacobi type. It turns out that one can go beyond the case where a solution is generated by a single function on configuration space. This is interesting both from the mathematical and from the physical point of view. To do so, instead of single generating functions, one must consider families depending on additional parameters. Such families are called Morse families. In Sects. 12.4 and 12.5 we develop the theory of Morse families in a systematic way. In Sect. 12.6 we present the theory of critical points2 of Lagrangian submanifolds in cotangent bundles, including a topological characterization in terms of the Maslov class and a description of the topological data in terms of generating Morse families. In Sects. 12.7 and 12.8 we discuss applications in the spirit of geometric asymptotics. First, we study the short wave asymptotics in lowest order for the Helmholtz 1 In
this chapter, the phase space will always be the cotangent bundle of some configuration space.
2 Points
where the Lagrangian submanifold is not transversal to the fibres.
G. Rudolph, M. Schmidt, Differential Geometry and Mathematical Physics, 641 Theoretical and Mathematical Physics, DOI 10.1007/978-94-007-5345-7_12, © Springer Science+Business Media Dordrecht 2013
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Hamilton-Jacobi Theory
equation. This leads to the eikonal equation of geometric optics. We discuss classes of solutions of this equation including the formation of caustics. In Sect. 12.8, we study the transport equation. We present a detailed study of its geometry and, on this basis, derive first order short wave asymptotic solutions for a class of firstorder partial differential equations. In this analysis a key role is played by a consistency condition of topological
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