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Brownian MotionFluctuations, Dynamics, and Applications$

Robert M. Mazo

Print publication date: 2008

Print ISBN-13: 9780199556441

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780199556441.001.0001

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(p.264) Appendix D Euler Angles

(p.264) Appendix D Euler Angles

Source:
Brownian Motion
Publisher:
Oxford University Press

In this appendix we discuss the Euler angles specifying the orientation of a rigid body in space. Erect an orthogonal coordinate system (x′′′, y′′′, z′′′) fixed in the body, and a coordinate system (x, y, z) fixed in space. The orientation of the body is specified by the angles relating these two sets of coordinates. A particularly useful set of three independent angles has been found by Euler. They are defined as follows.

Start with the body fixed axes (triple primed) coinciding with the space fixed axes (unprimed). We move the body to its final orientation in three steps. First, rotate the body, and consequently the body fixed axes, by an angle α about the z axis. Call the new position of the body fixed axes (x′, y′, z′). The second step is to rotate the body about the y′ axis by an angle β. The new position of the body fixed axes are called x′′, y′′, z′′). Note that the y′ and y′′ axes are the same. Finally, rotate the body about the z′′ axis by an angle γ. The new position, (x′′′, y′′′, z′′′) is the final position of the body fixed axes. Thus the overall effect is to express the total rotation in terms of the product (sequential rotations) of rotations by the three angles (α, β, γ). Figure D.1 shows the relative positions of the initial, intermediate, and final coordinate systems.

The rotations α, β, and γ are carried out about body fixed axes, although the body rotates and the body fixed axes change their positions with respect to the space fixed axes during the course of the overall rotation.

We now want to outline the transition from eqn (15.3.1) to eqn (15.3.2). The operators δ/δξi are the generators of infinitesimal rotations about the body fixed (triple primed) axes. They transform like vectors. δ/δα, δ/δβ, and δ/δγ generate infinitesimal rotations about the z, y′ and z′′′ axes respectively. They also transform like vectors. From Fig. D.1 it can be seen that

(D.1) Appendix D Euler Angles
(D.2) Appendix D Euler Angles

These equations can be solved for the δ/δξi, yielding

(D.3) Appendix D Euler Angles

(p.265)

Appendix D Euler Angles

Fig. D.1. The Euler angles α, β, and γ specifying the position of the body-fixed axes (x′′′, y′′′, z′′′) with respect to the space-fixed axes (x, y, z).

It is now a routine though tedious matter to compute the right-hand side of eqn (15.3.1) in terms of the variables α, β and γ. The result is eqn (15.3.2) of the main text when D1 = D2D3. The general case, when all of the Ds are unequal, can be handled by the same method.