Classical mechanics hamiltonian

We next apply these classical relationships to the rigid diatomic molecule. Since the molecule is rotating freely about its center of mass, the potential energy is zero and the classical-mechanical Hamiltonian function H is just the kinetic energy of the two particles,... 

In classical mechanics, Hamiltonian is the sum of kinetic and potential energy... 

With these approximations, the classical-mechanical Hamiltonian function for nuclear motion (omitting translation) is... 

We shall impose the restriction that the potential energy V depends only on the relative coordinates of the particles V= V x,y,z). Substitution of (1.214) and (1.215) into the kinetic-energy expression leads to the following expression for the classical-mechanical Hamiltonian ... 

The first term on the right is the translational kinetic energy of the molecule as a whole this simply adds a constant to the total energy, and we shall omit this term. The second and third terms are ihe rotational and vibrational kinetic energies of the molecule. The final term is the energy of interaction between rotation and vibration. To get the classical-mechanical Hamiltonian function, we add the potential energy V to (5.2), where U is a function of the relative positions of the nuclei. 

This is the classical-mechanical Hamiltonian for the rotation of a rigid body. [Note the similarity of (5.20), (5.22), and (5.23) to the corresponding equations for linear motion we get the equations for rotational motion by replacing the velocity v with the angular velocity to, the linear momentum p with the angular momentum P, and the mass with the principal moments of inertia.)... 

Before considering the quantum-mechanical vibrational wave functions and energies, we must find the classical-mechanical Hamiltonian for vibration. (Quantum mechanics is peculiar in that it depends on classical mechanics to formulate the Hamiltonian, and yet classical mechanics is only a limiting case of the more general theory quantum mechanics.)... 

Computer simulation can employ both quantum and classical mechanical Hamiltonians to study the structure, function, and dynamics at the atomic... 

The classical mechanical problem of two coupled identical harmonic oscillators is described by the classical mechanical Hamiltonian... 

Sir William Rowan Hamilton (1805-1865) devised an alternative form of Newton s equations of motion involving a function H, the Hamiltonian function for the system. For a system where the potential energy is a function of the coordinates only, the total energy remains constant with time that is, E is conserved. We shall restrict ourselves to such conservative systems. For conservative systems, the classical-mechanical Hamiltonian function turns out to be simply the total energy expressed in terms of coordinates and conjugate momenta. For Cartesian coordinates x, y, z, the conjugate momenta are the components of linear momentum in the x, y, and z directions p, Py, and p. ... 

Let us find the classical-mechanical Hamiltonian function for a particle of mass m moving in one dimension and subject to a potential energy V(x). The Hamiltonian function is equal to the energy, which is composed of kinetic and potential energies. The familiar form of the kinetic energy, mvl, will not do, however, since we must express the Hamiltonian as a function of coordinates and momenta, not velocities. Since = Px/m, the form of the kinetic energy we want is p /2m. The Hamiltonian function is... 

Up until now, little has been said about time. In classical mechanics, complete knowledge about the system at any time t suffices to predict with absolute certainty the properties of the system at any other time t. The situation is quite different in quantum mechanics, however, as it is not possible to know everything about the system at any time t. Nevertheless, the temporal behavior of a quantum-mechanical system evolves in a well defined way drat depends on the Hamiltonian operator and the wavefiinction T" according to the last postulate... 

The picture here is of uncoupled Gaussian functions roaming over the PES, driven by classical mechanics. The coefficients then add the quantum mechanics, building up the nuclear wavepacket from the Gaussian basis set. This makes the treatment of non-adiabatic effects simple, as the coefficients are driven by the Hamiltonian matrices, and these elements couple basis functions on different surfaces, allowing hansfer of population between the states. As a variational principle was used to derive these equations, the coefficients describe the time dependence of the wavepacket as accurately as possible using the given... 

A particularly convenient notation for trajectory bundle system can be introduced by using the classical Liouville equation which describes an ensemble of Hamiltonian trajectories by a phase space density / = f q, q, t). In textbooks of classical mechanics, e.g. [12], it is shown that Liouville s equation... 

Rather than solve a Schrodinger equation with the Nuclear Hamiltonian (above), a common approximation is to assume that atoms are heavy enough so that classical mechanics is a good enough approximation. Motion of the particles on the potential surface, according to the laws of classical mechanics, is then the subject of classical trajectory analysis or molecular dynamics. These come about by replacing Equation (7) on page 164 with its classical equivalent ... 

In contrast to dissipative dynamical systems, conservative systems preserve phase-space volumes and hence cannot display any attracting regions in phase space there can be no fixed points, no limit cycles and no strange attractors. There can nonetheless be chaotic motion in the sense that points along particular trajectories may show sensitivity to initial conditions. A familiar example of a conservative system from classical mechanics is that of a Hamiltonian system. 

We recall, from elementary classical mechanics, that symmetry properties of the Lagrangian (or Hamiltonian) generally imply the existence of conserved quantities. If the Lagrangian is invariant under time displacement, for example, then the energy is conserved similarly, translation invariance implies momentum conservation. More generally, Noether s Theorem states that for each continuous N-dimensional group of transformations that commutes with the dynamics, there exist N conserved quantities. 

It is now shown how the abrupt changes in the eigenvalue distribution around the central critical point relate to changes in the classical mechanics, bearing in mind that the analog of quantization in classical mechanics is a transformation of the Hamiltonian from a representation in the variables pR, p, R, 0) to one in angle-action variables (/, /e, Qr, 0) such that the transformed Hamiltonian depends only on the actions 1r, /e) [37]. Hamilton s equations diR/dt = (0///00 j), etc.) then show that the actions are constants of the motion, which are related to the quantum numbers by the Bohr correspondence principle [23]. In the present case,... 

This chapter is organized as follows. In Section II, we briefly summarize the findings of the geometric TST for autonomous Hamiltonian systems to the extent that it is needed for the present discussion. Readers interested in a more detailed exposition are referred to Ref. 35, where the field has recently been reviewed in depth. We restrict our discussion to classical mechanics. Semiclassical extensions of geometric TST have been developed in Refs. 70-75. Section III discusses the notion of the TS trajectory in general and its incarnation in different specific settings. Section IV demonstrates how the TS trajectory allows one to carry over the central concepts of geometric TST into the time-dependent realm. 

A familiar example of Legendre transformation is the relationship that exists between the Lagrangian and Hamiltonian functions of classical mechanics [17]. In thermodynamics the simplest application is to the internal energy function for constant mole number U(S, V), with the differentials... 

The notion of chaos is interwoven with the discussion of time evolution, which we do not pursue in this volume. It is worthwhile, however, to note that it is, by now, well understood that a quantum-mechanical system with a finite Hamiltonian matrix cannot satisfy many of the purely mathematical characterizations of chaos. Equally, however, over long periods of time such systems can manifest many of the qualitative features that one associates with classically chaotic systems. It is not our intention to follow this most interesting theme. Instead we seek a more modest aim, namely, to forge a link between the elementary notions of classical nonlinear dynamics and the algebraic approach. This turns out to be possible using the action-angle variables of classical mechanics. In this section we consider only the nonlinear dynamics aspects. We complete the bridge in Chapter 7. 

In classical mechanics, the Hamiltonian function is the expression of the energy of a molecular system in terms of the momenta of the particles in the system and... 

---