Velocity dependent force

It must be modified to incorporate velocity-dependent forces or temperature scaling. 

The deterministic force in Eq. (13) is formally assumed to be derived from a potential U(q). However, it will become clear in the following that this assumption is not necessary it would be straightforward to include velocity-dependent forces that arise from, for example, a magnetic field. 

To quantize the dynamics of the particles first requires that we express the velocities of the particles in terms of canonical momenta. In the presence of electromagnetic fields, the canonical momenta are not merely m dx-Jdt). Rather, in order to incorporate Lorentz s velocity-dependent forces into Hamilton s formulation of classical mechanics, the canonical momenta are given by [2]... 

Unless a system contains velocity-dependent forces, its Hamiltonian contains the velocity only through the kinetic energy, and for all configurations the thermodynarruc-equilibriiun density function is an even function of the velocity. Thus zero correlation of position and velocity in the initial distribution is a rather conunon property. 

The assumptions for the Wigner condition do not seem to be valid at the present moment. Jastrow interprets the high energy scattering data in terms of a repulsive core. Feshbach s boundary condition formulation of the two-body data implies velocity dependent forces (in that the boundary conditions are energy independent). Feshbach s solutions are also not in agreement with the Serber mixture ". 

In the discussion above we have neglected the velocity-dependent force in the optical molasses (Sect. 9.1.5). 

A velocity dependent force (magnetic force) is considered in Prob. 2.2. 

There are two key side effects of the velocity dependent force. First, kinematic cooling results in real cooling, not just a rotation of position-momentum phase space, yielding an increased phase space for the cold molecules. Second, since there is dissipation, if the collisions occur in a region containing a trap for the molecules, the trap can be continuously loaded without the worry of how to load pre-cooled molecules into a conservative potential well. 

The dampening coefficient c is the viscous dampening coefficient for a velocity dependent force ... 

This procedure may be used unless the rate-dependence, load history-dependence, or deformation-hardening characteristics of the isolation system necessitate explicit consideration of their nonlinear and/or velocity-dependent force-deflection properties. 

In the absence of velocity-dependent forces this is just the Verlet scheme, but the solid-fluid boundary conditions (200) introduce a hydrodynamic force that depends linearly on the particle velocity [104,134],... 

In the absence of the velocity-dependent forces, i.e., Av = 0, the linearized system equation (3.8) reduces to... 

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