Rotation equation of motion

The use of Verlet and Singer s algorithm makes it necessary to use extra care in integrating the equation of motion. Ciccotti et aL have shown how to do it in the case of Verlet s algorithm. As to the rotational equation of motion, we followed a similar procedure using the quantities... 

The equation of motion is derived from the Lagrangian mechanics ( L = T - U ). The rotational motion is derived from the rotational equation of motion for a rigid rod5. 

The rotational equation of motion is given in Eq. (4.11), where 1, represents the moment of inertia, m the rotational speed, and T, the torque ... 

Axes 1,2,3 are principal axes with origin at the center of mass G, and thus Euler s equations of motion for a rigid body can be applied. Under the body coordinate system, the three rotational equations of motion are given by ... 

In the presence of a field H, rotating at the precessional frequency the nuclear system can absorb energy, following which nuclear relaxation occurs. Thus, the equation of motion must include both the precessional and the relaxation contributions ... 

When dealing with the motions of rigid bodies or systems of rigid bodies, it is sometimes quite difficult to directly write out the equations of motion of the point in question as was done in Examples 2-6 and 2-7. It is sometimes more practical to analyze such a problem by relative motion. That is, first find the motion with respect to a nonaccelerating reference frame of some point on the body, typically the center of mass or axis of rotation, and vectorally add to this the motion of the point in question with respect to the reference point. 

However, only the left-hand side of the inequality has a clear, although qualitative, physical meaning. As far as collision time tc is concerned, its evaluation as p/ v) in Eq. (1.58) is rather arbitrary. Alternatively, it may be defined as the correlation time of the collisional processes which modulate the rotation. Using the mechanical equation of motion... 

The change in a molecule s orientation in space as a result of rotation is described by the dynamic equation of motion... 

Consider a rotation of the earth around the z-axis in which every particle, elementary volume, of the earth moves along the horizontal circle with the radius r. Our first goal is to find the distribution of forces inside the earth and with this purpose in mind we will derive an equation of motion for an elementary volume of the fluid. Let us introduce a Cartesian system of coordinates with its origin 0, located on the z-axis of rotation. Since this frame of reference is an inertial one, it does not move with the earth, we can write Newton s second law as... 

Here v, and a, are the velocity and acceleration of the point p in the rotating frame of reference, respectively. Substitution of Equation (2.55) into Newton s second law gives an equation of motion in the non-inertial frame ... 

The set of rotations used in MPC dynamics can be chosen in various ways and the specific choice will determine the values of the transport properties of the system, just as the choice of the intermolecular potential will determine the transport properties in a system evolving by full molecular dynamics through Newton s equations of motion. It is often convenient to use rotations about a randomly chosen direction, n, by an angle a chosen from a set of angles. The... 

Let us consider the rotational dynamics of a two-component neutron star taking into account the pinning and depinning of neutron vortices. Equations of motion of the superfluid and normal components have the following forms [15, 17] ... 

The most expensive parts of a conventional NSE instrument are the main solenoids providing the precession field. A closer look at Bloch s equation of motion for the spins (Eq. 2.11) shows that in a coordinate system that rotates with the precession frequency around Bg the spin is stationary, the coordinate system rotation is equivalent to the addition of - to all magnetic fields. By this means the large precession field inside the main coils may be transformed to zero - zero field spin-echo). The flippers are viewed as elements rotating... 

The first attempt to explain the characteristic properties of molecular spectra in terms of the quantum mechanical equation of motion was undertaken by Born and Oppenheimer. The method presented in their famous paper of 1927 forms the theoretical background of the present analysis. The discussion of vibronic spectra is based on a model that reflects the discovered hierarchy of molecular energy levels. In most cases for molecules, there is a pattern followed in which each electronic state has an infrastructure built of vibrational energy levels, and in turn each vibrational state consists of rotational levels. In accordance with this scheme the total energy, has three distinct components of different orders of magnitude,... 

As a consequence of the transformation, the equation of motion depends on three extra coordinates which describe the orientation in space of the rotating local system. Furthermore, there are additional terms in the Hamiltonian which represent uncoupled momenta of the nuclear and electronic motion and moment of inertia of the molecule. In general, the Hamiltonian has a structure which allows for separation of electronic and vibrational motions. The separation of rotations however is not obvious. Following the standard scheme of the various contributions to the energy, one may assume that the momentum and angular momentum of internal motions vanish. Thus, the Hamiltonian is simplified to the following form. 

Using the equation of motion for a system with differential rotation taking into account the inertial, frictional, torsional and electrostatic forces ... 

Effects such as lift due to particle rotation or fluid velocity gradients can readily be included in Eq. (11-70) if appropriate. The resulting equation of motion is... 

The nature of the dual vector ( ) can be deduced without using any equation of motion, but the dual 4-vector is a fundamental geometric property in the four dimensions of spacetime. The complete description of the electromagnetic field in 0(3) electrodynamics must therefore involve boosts, rotations, and spacetime translations, meaning that is a fundamental geometric property of spacetime. The unit 4-vector i M is orthogonal to the unit 4-vector... 

Many kinds of molecular systems pumped by a strong laser light show chaotic dynamics. Indeed, in a semiclassical model of a multiphoton excitation on molecular vibration, chaos was discovered by Ackerhalt et al. [85] and theoretically and numerically investigated in detail [86,87]. Moreover, the equations of motion that describe a rotating molecule in a laser field can exhibit a chaotic behavior and have been applied in the classical case of a rigid-rotator approximation [87,88]. 

As is the case with all differential equations, the boundary conditions of the problem are an important consideration since they determine the fit of the solution. Many problems are set up to have a high level of symmetry and thereby simplify their boundary descriptions. This is the situation in the viscometers that we discussed above and that could be described by cylindrical symmetry. Note that the cone-and-plate viscometer —in which the angle from the axis of rotation had to be considered —is a case for which we skipped the analysis and went straight for the final result, a complicated result at that. Because it is often solved for problems with symmetrical geometry, the equation of motion is frequently encountered in cylindrical and spherical coordinates, which complicates its appearance but simplifies its solution. We base the following discussion on rectangular coordinates, which may not be particularly convenient for problems of interest but are easily visualized. 

Exercise. The rotation of an ellipsoidal particle suspended in a fluid obeys the macroscopic equation of motion... 

---