Hamiltonian formalism, classical mechanics

In the early 1990s a few classical semimoleculai and molecular models of electron transfer reactions involving bond breaking appeared in the literature. A quantum mechanical treatment of a unified mr el of electrochemical electron and ion transfer reactions involving bond breaking was put forward by Schmickler using Anderson-Newns Hamiltonian formalism (see Section V.2). 

In the following we present the axioms or basic postulates of quantum mechanics and accompany them by their classical counterparts in the Hamiltonian formalism. We begin the presentation with a brief summary of some of the mathematical background essential for the developments in the following. It is by no means a comprehensive presentation, and the reader is supposed to have some basic knowledge about quantum mechanics that may be obtained from any of the many introductory textbooks in quantum mechanics. The focus here is on results of particular relevance to the subjects of this book. We consider, for example, a derivation of a formal expression for the flux density operator in quantum mechanics and its coordinate representation. A systematic way of generating any representation of any combination of operators is set up, and is of immediate usage for the time autocorrelation function of the flux operator used to determine the rate constants of a chemical process. 

In this H is thus now a symbol, not for a quantity but for an operation, to be applied to the wave function cp (see p. 123) H is called the HAMiLTONian operator. This operator H, therefore, has the same form as the function H in classical mechanics when the above-mentioned formal rules are borne in mind. 

Classical Mechanics of Constrained Systems within Lagrangian and Hamiltonian Formalisms... 

It is possible to formulate the classical laws of motion in several ways. Newton s equations are taught in every basic course of classical mechanics. However, especially in the presence of constraint forces, the equations of motion can often be presented in a simpler form by using either Lagrangian or Hamiltonian formalism. In short, in the Newtonian approach, an /V-point particle system is described by specifying the position xa = xa(t) of each particle a as a function of time. The positions are found by solving the equations of motion,... 

Hamiltonian mechanics refers to a mathematical formalism in classical mechanics invented by the Irish mathematician William Rowan Hamilton (1805-1865) during the early 1830 s arising from Lagrangian mechanics which was introduced about 50 years earlier by Joseph-Louis Lagrange (1736-1813). The Hamiltonian equations can however be formulated on the basis of a variational principle without recourse to Lagrangian mechanics [95] [2j. 

In physics treatments, the Liouvillian is often denoted iC. This may seem namral since (a) it allows a formal correspondence with the Schrbdinger equation and between the propagators of quantum mechanics and classical mechanics, and (b) for Hamiltonian dynamics, as we shall see, the Liouvillian is skew-adjoint (in a certain sense skew symmetric) and the inclusion of i explicitly calls attention to this fact. However, we feel it is more natural to omit the i in a treatment that includes study of both stochastic and deterministic models. 

Although it is possible to cast both quantum and classical mechanics in a similar formal language (i.e., distributions in phase space and a Liouville propagator), standard quantum mechanics is based on a mathematical structure that is substantially different from that of classical Hamiltonian mechanics. We first provide a brief qualitative summary of some results of quantum investigations, and then we present details that can be best appreciated by the reader who is well versed in quantum mechanics. 

The third equivalent formulation of classical mechanics to be briefly discussed here is the Hamiltonian formalism. Its main practical importance especially for molecular simulations lies in the solution of practical problems for processes that can be adequately described by classical mechanics despite their intrinsically quantum mechanical character (such as protein folding processes). However, more important for our purposes here is that it can serve as a useful starting point for the transition to quantum theory. The basic idea of the Hamiltonian formalism is to eliminate the / generalized velocities in favor of the canonical momenta defined by Eq. (2.54). This is achieved by a Legendre transformation of the Lagrangian with respect to the velocities. 

This book gives a formal and mathematically challenging presentation of classical mechanics. It may be hard to follow for the beginner, but very enlightening for a second course in mechanics. The symplectic structure of Hamiltonian mechanics is presented in detail and coordinate-free expressions employing differential forms are given. A very detailed appendix of more than 200 pages explains the mathematical foundations. 

The quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

At the very beginning of a study, it is very convenient to perform a purely classical molecular mechanics (MM) energy and geometry minimization procedure MM replaces the Hamiltonians by purely classical potential energies for (i) formal electrostatics for charged atoms in molecules,... 

In the semiclassical model the molecule is treated quantum mechanically whereas the held is represented classically. The held is an externally given function of time F that is not affected by any feedback from the interaction with the molecule. We consider the simplest case of a dipole coupling. The formalism is easily extended to other types of couplings. The time dependence of the periodic Hamiltonian is introduced through the time evolution of the initial phase F = F(0 + oat) = electric held and oa is its frequency. The semiclassical Hamiltonian can be, for example, written as... 

The linear response of a system is determined by the lowest order effect of a perturbation on a dynamical system. Formally, this effect can be computed either classically or quantum mechanically in essentially the same way. The connection is made by converting quantum mechanical commutators into classical Poisson brackets, or vice versa. Suppose that the system is described by Hamiltonian H + where denotes an external perturbation that may depend on time and generally does not commute with H. The density matrix equation for this situation is given by the Bloch equation [32]... 

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