Hamiltonian principle

Physical Basis for Generalizing Hamiltonian Principles to Non-Hamiltonian Systems... 

This situation is illustrated in Figure 2.3. As in most cases of practical interest the actual path minimizes the action, the Hamiltonian principle defined by Eq. (2.48) is sometimes also denoted the principle of least action. It may be considered as the most fundamental law of Nature, valid in all areas of physics, and all other fundamental laws or equations can be deduced from this principle. 

Varying the generalized variables in exactly the same way as before for discrete systems, the Euler-Lagrange equations of field theory follow from the Hamiltonian principle and read... 

Since the 2/ canonical variables are independent of each other the Hamiltonian principle has to be modified. The variation of the action... 

Note that L2 does not explicitly depend on proper time t, since according to Eq. (3.92) r is uniquely determined by the space-time vector x and the 4-velocity u. For a better comparison between the three-dimensional formulation (Li) and the explicitly covariant formulation (L2) we have employed the velocity v = r (instead of r itself) in Eq. (3.139). Both Lagrangians Li and L2 do not represent physical observables and are therefore not uniquely determined. According to the Hamiltonian principle of least action given by Eq. (2.48), 5S = 0, they only have to yield the same equation of motion. This is in particular guaranteed if even the actions themselves are identical, i.e.. 

The Hartree approximation is usefid as an illustrative tool, but it is not a very accurate approximation. A significant deficiency of the Hartree wavefiinction is that it does not reflect the anti-synnnetric nature of the electrons as required by the Pauli principle [7], Moreover, the Hartree equation is difficult to solve. The Hamiltonian is orbitally dependent because the siumnation in equation Al.3.11 does not include the th orbital. This means that if there are M electrons, then M Hamiltonians must be considered and equation A1.3.11 solved for each orbital. 

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

Linear response theory is an example of a microscopic approach to the foundations of non-equilibrium thennodynamics. It requires knowledge of tire Hamiltonian for the underlying microscopic description. In principle, it produces explicit fomuilae for the relaxation parameters that make up the Onsager coefficients. In reality, these expressions are extremely difficult to evaluate and approximation methods are necessary. Nevertheless, they provide a deeper insight into the physics. 

Although in principle the microscopic Hamiltonian contains the infonnation necessary to describe the phase separation kinetics, in practice the large number of degrees of freedom in the system makes it necessary to construct a reduced description. Generally, a subset of slowly varying macrovariables, such as the hydrodynamic modes, is a usefiil starting point. The equation of motion of the macrovariables can, in principle, be derived from the microscopic... 

Hamiltonian) trajectory in the phase space of the model from which infonnation about the equilibrium dyuamics cau readily be extracted. The application to uou-equilibrium pheuomeua (e.g., the kinetics of phase separation) is, in principle, straightforward. 

A formal derivation of the location of the zeros of Cg t) for a general adiabatic Hamiltonian can be given, following proofs of the adiabatic principle (e.g., [250-252]). The last source, [252] derives an evolution operator U, which is written there, with some slight notational change, in the form... 

If the PES are known, the time-dependent Schrbdinger equation, Eq. (1), can in principle be solved directly using what are termed wavepacket dynamics [15-18]. Here, a time-independent basis set expansion is used to represent the wavepacket and the Hamiltonian. The evolution is then carried by the expansion coefficients. While providing a complete description of the system dynamics, these methods are restricted to the study of typically 3-6 degrees of freedom. Even the highly efficient multiconfiguration time-dependent Hartree (MCTDH) method [19,20], which uses a time-dependent basis set expansion, can handle no more than 30 degrees of freedom. 

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... 

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

Observe that, in principle, it is possible to introduce quaternions in the solution of the free rotational part of a Hamiltonian splitting, although there is no compelling reason to do so, since the rotation matrix is usually a more natural coordinatization in which to describe interbody force laws. 

Variational calculus, Dreyfus (1962), may be employed to obtain a set of differential equations with certain boundary condition properties, known as the Euler-Lagrange equations. The maximum principle of Pontryagin (1962) can also be applied to provide the same boundary conditions by using a Hamiltonian function. 

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