Rotations of a rigid body

It is easy to see that the orientation of a vector in space can be described by the two spherical polar angles, 0 and 4 . From a reference position parallel to the Z axis, the required orientation is obtained as follows  

For a general body, however, a third angle is needed. The three angles are known 

The symbol co is commonly used as a short-hand for the orientation ( / , 6, x). The volume element for integration is 

It can be shown that a sequence of rotations about axes fixed in space is equivalent to the same sequence of rotations but performed in the reverse order about axes which rotate with the body (provided that the body-fixed and space-fixed axes coincide initially). 

The seeds of this idea can be seen in equation (5.10). For our general rotation, this means 

For a general body, however, a third angle is needed. The three angles are known as Euler angles and are usually labelled 4 , 9 and x and 9 are used to define the 

We note that the three Euler angles have the following ranges  

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This result shows that the funetions that deseribe the rotation of a rigid body through angles Q, < >, X naust be a eombination of rotation matriees (aetually D l,m, m(0, X) funetions). 

Vector spaces which occur in physical applications are often direct products of smaller vector spaces that correspond to different degrees of freedom of the physical system (e.g. translations and rotations of a rigid body, or orbital and spin motion of a particle such as an electron). The characterization of such a situation depends on the relationship between the representations of a symmetry group realized on the product space and those defined on the component spaces. 

This result shows that the functions that describe the rotation of a rigid body through angles 0, < ),% must be a combination of rotation matrices (actually D l,M, m(0, ( ),%) functions). Because of the normalization of the Dl,m,M (0, < ), %) functions ... 

To consider the quantum mechanics of rotation of a polyatomic molecule, we first need the classical-mechanical expression for the rotational energy. We are considering the molecule to be a rigid rotor, with dimensions obtained by averaging over the vibrational motions. The classical mechanics of rotation of a rigid body in three dimensions is involved, and we shall simply summarize the results.2... 

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.)... 

The energy levels from the quantum mechanical solution of the rotation of a rigid body have the characteristic feature of increasing separation with angular momentum. The energy levels are given by the expression ... 

The determination of accurate molecular structure from molecular rotational resonance (MRR) spectra has always been a great challenge to this branch of spectroscopy [/]. There are three basic facts which make this task feasible (1) the free rotation of a rigid body is described in classical as well as in quantum mechanics by only three parameters, the principal inertial moments of the body, Ig, g = x, v, z ... 

The Hamilton-Jacobi phase is thereby identified as the angle of rotation of a rigid body about an axis in the direction of the spin. 

These angles are simply related to the Euler angles describing the rotation of a rigid body (Brink... 

H. B. G. Casimir, The Rotation of a Rigid Body in Quantum Mechanics, Leyden thesis, J. B. Wolters, the Hague, Netherlands (1931). 

The general case of the rotator free to move in three dimensions is more complicated, but is treated according to similar principles. The Schrodinger equation is first expressed in spherical polar coordinates, r, 6, and . For the rotation of a rigid body about its centre of gravity, r is constant and is included in a term representing the moment of inertia, /. The conditions for physically admissible solutions lead to the result... 

Moment of inertia n. A measure of the effectiveness of mass in rotation. In the rotation of a rigid body not only the body s mass, but also the distribution of the mass about the axis of rotation determines the change in the angular velocity resulting from the action of a given torque for a given time. Moment of inertia in rotation is analogous to mass (inertia) in simple translation. The cgs unit is gcm. Dimensions - [ML ]. If mi, m2, m3, etc., represent the masses of infinitely small particles of a body ri, r2, r, etc., their respective distances from an axis of rotation, the moment of inertia about this axis will be... 

In a number of astronomical problems, it is necessary to analyze rotation of a rigid body T around the centre of mass in a gravitational field of other bodies. Particularly important is the case where the distance to an attracting object is much... 

Theorem 4.3.3 (Bogoyavlensky [309]). The translational-rotary motion of an arbitrary rigid body T in a Newtonian held with an arbitrary (inhomogeneous) quadratic potential is determined by a Liouville-integrable Hamiltonian system. The dynamics of the centre of mass O is integrated in elementary functions rotation of a rigid body around the centre of mass is integrated in Riemannian theta-functions. 

An arbitrary three-dimensional rotation of a rigid body may be described [1] using the Euler angles , 0, x (Fig- 5.4). The body-fixed a, f>, and c axes are initially aligned with the space-fixed x, y, and z axes, respectively. The body is first rotated counterclockwise by the angle about the z axis this rotation does not affect the orientation of the c axis, but rotates the a and b axes in the xy... 

Moment of Inertia n A measure of the effectiveness of mass in rotation. In the rotation of a rigid body not only the body s mass, but the distribution of the mass about the axis of rotation determines the change in the angular velocity resulting from the action of a given torque for a given time. Moment of inertia in rotation... 

We now assume that all bond lengths and bond angles of a polyatomic molecule are locked at their equilibrium values, so that the molecule cannot vibrate and rotates as a rigid body. The classical rotation of a rigid body is described in terms of moments of inertia taken relative to three mutually perpendicular axes that pass through the center of mass of the object. For an object consisting of n mass points, the moment of inertia about an axis is defined to be... 

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