Ignorable generalized coordinate

Since Hamilton s equations imply that pk = 0 if = 0, pk is a constant of motion if qt is such an ignorable coordinate. An ingenious choice of generalized coordinates can produce such constants and simplify the numerical or analytic task of integrating the equations of motion. 

Here s the simplest instance of a Hamiltonian system. Let H p,q) be a smooth, real-valued function of two variables. The variable q is the generalized coordinate and p is the conjugate momentum. (In some physical settings, H could also depend explicitly on time t, but we ll ignore that possibility.) Then a system of the form... 

To understand how electrons with spin interact, it is useful to examine a system consisting of two electrons, such as the helium atom. Let this two-electron system be described by the wave function, in space coordinates, (r), If the electrons are interchanged the wave function will in general be different, ( ), but since the electrons are identical (ignoring spin) the energy of the system will not be affected. The wave functions therefore belong to degenerate levels. 

The antisymmetric tensor is generally not observable in NMR experiments and is therefore ignored. The symmetric tensor is now diagonalized by a suitable coordinate transformation to orient into the principal axis system (PAS). After diagonalization there are still six independent parameters, the three principal components of the tensor and three Euler angles that specify the PAS in the molecular frame. 

If we subtract this zeroth order solution, fourier transform the x coordinates, convert the time coordinate to conformal time, r), defined by dr) = dt/a, and ignore vector and tensor perturbations (discussed in the lectures by J. Bartlett on polarization at this school), the Liouville operator becomes a first-order partial differential operator for /( (k, p, rj), depending also on the general-relativistic potentials, (I> and T. We further define the temperature fluctuation at a point, 0(jfc, p) = f( lj i lodf 0 1 /<9To) 1 where To is the average temperature and )i = cos 6 in the polar coordinates for wavevector k. 

For an isolated system, treatment of the intramolecular Jahn-Teller effect is relatively simple. As the system is isolated, we may ignore molecular rotation and consider a molecule-fixed coordinate system. Within this frame of reference, the electronic and vibrational states can be formulated in terms of the irreducible representations (irreps) of the reference configuration. Overall, the system Hamiltonian is generally written in the form... 

In order to demonstrate how the anomalous relaxation behavior described by the hitherto empirical Eqs. (9)—(11) may be obtained from our fractional generalizations of the Fokker-Planck equation in configuration space (in effect, fractional Smoluchowski equations), Eq. (101), we first consider the fractional rotational motion of a fixed axis rotator [1], which for the normal diffusion is the first Debye model (see Section II.C). The orientation of the dipole is specified by the angular coordinate 4> (the azimuth) constituting a system of one rotational degree of freedom. Electrical interactions between the dipoles are ignored. 

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