Hamiltonian systems theory

Nonlinear Hamiltonian system, geometric transition state theory, 200-201 Nonlinear thermodynamics coefficients linear limit, 36 entropy production rate, 39 parity, 28-29... 

It is not possible to discuss highly excited states of molecules without reference to the recent progress in nonlinear dynamics.2 Indeed, the stimulation is mutual. Rovibrational spectra of polyatomic molecules provides both an ideal testing ground for the recent ideas on the manifestation of chaos in Hamiltonian systems and in turn provides many challenges for the theory. 

According to dynamical systems theory, the escape rate is given by the difference (92) between the sum of positive Lyapunov exponents and the Kolmogorov-Sinai entropy. Since the dynamics is Hamiltonian and satisfies Liouville s theorem, the sum of positive Lyapunov exponents is equal to minus the sum of negative ones ... 

A powerful tool for analyzing fluctuations in a nonequilibrium systems is based on the Hamiltonian [57] theory of fluctuations or alternatively on a path-integral approach to the problem [44,58-62]. The analysis requires the solution of two closely interrelated problems. The first is the evaluation of the probability density for a system to occupy a state far from the stable state in the phase space. In the stationary regime, the tails of this probability are determined by the probabilities of large fluctuations. 

The spin-Hamiltonian VB theory is a very simple and easy-to-use semiempi-rical tool that is based on the molecular graph. It is consistent with the VB theory described in Chapter 3, albeit with some simplifying assumptions and a more limited domain of application. Typically, this theory deals with the neutral ground or excited states of conjugated molecules or other homonuclear assemblies with one electron per site. For large systems, it reproduces the results of PPP full Cl, while dealing with a much smaller Hamiltonian matrix. 

The phase space structure of classical molecular dynamics is extensively used in developing classical reaction rate theory. If the quanmm reaction dynamics can also be viewed from a phase-space perspective, then a quantum reaction rate theory can use a significant amount of the classical language and the quantum-classical correspondence in reaction rate theory can be closely examined. This is indeed possible by use of, for example, the Wigner function approach. For simplicity let us consider a Hamiltonian system with only one DOF. Generalization to many-dimensional systems is straightforward. The Wigner function associated with a density operator /)( / is defined by... 

It is a phase space rather than configuration space theory, so it can treat Hamiltonian systems containing unconserved angular momenta like Coriolis interactions which prevent the Hamiltonian from being written as a sum of the kinetic and potential energies [6,18]. The resulting hypersurfaces are dynamical in that they involve momenta as well as coordinates. 

Briefly, the aim of Lie transformations in Hamiltonian theory is to generate a symplectic (that is, canonical) change of variables depending on a small parameter as the general solution of a Hamiltonian system of differential equations. The method was first proposed by Deprit [75] (we follow the presentation in Ref. 76) and can be stated as follows. 

Hamiltonian dynamical system theory is the mathematical framework on which TST rests many textbooks, of various mathematical sophistication, describe this branch of pure/applied mathematics. Some of the various flavors are [20-24]. Very little of this vast information will be needed here, and we shall try to be as self-consistent as possible. 

Extension toward the fully nonlinear case is straightforward for 1-DOF Hamiltonians. The energy conservation relation H p,q) = E allows us to dehne (explicitly or implicitly) p = p q E), thereby reducing the ODE to a simple quadrature. In this procedure there is no problem of principle (unlike the n >2-DOE case). It works in practice also, and it is possible to adapt Eigs. 3-5 to the nonlinear regime. It must be underlined that besides that simple procedure, we present a theorem in dynamical system theory (containing Hamiltonian dynamics as a particular case). This theorem is valid for n DOEs (hence for n = 1) it relates the full dynamics to the linearized dynamics, called tangent dynamics in the mathematical literature. 

Until now, we have discussed NHIMs in general dynamical systems. In this section, we limit our argument to Hamiltonian systems and show how singular perturbation theory works. In particular, we discuss NHIMs in the context of reaction dynamics. First, we explain how NHIMs appear in conventional reaction theory. Then, we will show that Lie permrbation theory applied to the Hamiltonian near a saddle with index 1 acmally transforms the equation of motion near the saddle to the Fenichel normal form. This normal form can be considered as an extension of the Birkhoff normal form from stable fixed points to saddles with index 1 [2]. Finally, we discuss the transformation near saddles with index larger than 1. 

In addition to computer simulations, what drives the research in this direction is elaborated perturbation theories developed almost simultaneously. In particular, the Kolmogorov-Arnold-Moser (KAM) theorem, which has shown the existence of invariant tori under a small perturbation to completely inte-grable systems, and the Nekhoroshev theorem, which has proved exponentially long-time stability of trajectories close to completely integrable ones, are landmarks in this field. Although a lot of works have been done, there still remain unsolved important questions, and the Hamiltonian system is being studied as one of important branches in the theory of dynamical systems [3-5]. 

We thank R. Yasuda and K. Ichiki for their collaborations. One of the author (AS) thanks H. Ito for giving us intense lectures on the perturbation theory of Hamiltonian systems. 

Hamiltonian systems are fundamental to classical mechanics they provide an equivalent but more geometric version of Newton s laws. They are also central to celestial mechanics and plasma physics, where dissipation can sometimes be neglected on the time scales of interest. The theory of Hamiltonian systems is deep and beautiful, but perhaps too specialized and subtle for a first course on nonlinear dynamics. See Arnold (1978), Lichtenberg and Lieberman (1992), Tabor (1989), or Henon (1983) for introductions. 

---