the angular momentum is

™

L = T. (15.42)

In general, T depends on the choice of the origin of the coordinate system,

except when the total external force is zero.

If there is no external torque, the angular momentum is a constant of the

motion.

15.7 Rigid body motion

We now consider the special case of a system of N particles in which the

mutual interactions keeping the particles together con¬ne the relative posi-

tions of the particles so strongly that the whole system may be considered as

one rigid body. We now transform r to generalized coordinates that consist

of three c.o.m. coordinates rcm , three variables that de¬ne a real orthogonal

3 — 3 matrix R with determinant +1 (a proper rotation matrix) and 3N ’ 6

internal coordinates. The latter are all constrained to a constant value and

therefore do not ¬gure in the Lagrangian.

Use the vector multiplication rule a · (b — c) = (a — b) · c.

7

406 Lagrangian and Hamiltonian mechanics

Z

Z´

X´ k

c

a

Y

j

i

b

X

Y´

Figure 15.1 Body-¬xed coordinate system X Y Z with base vectors a, b, c, rotated

with respect to a space-¬xed coordinate system XY Z with base vectors i, j, k.

Using a body-¬xed coordinate system X , Y , Z (see Fig. 15.1, the posi-

tions of the particles are speci¬ed by 3N coordinates ri , which are all con-

stants. There are, of course, only 3N ’ 6 independent coordinates because

six functions of the coordinates will determine the position and orientation

of the body-¬xed coordinate system. The rotation matrix R transforms

the coordinates of the i-th particle, relative to the center of mass, in the

space-¬xed system to those in the body-¬xed system:8

ri = R(ri ’ rcm ). (15.43)

The positions in the space-¬xed coordinate system X, Y, Z are given by

ri = rcm + RT ri . (15.44)

8 When consulting other literature, the reader should be aware of variations in notation that are

commonly used. In particular the transformation matrix is often de¬ned as the transpose of

our R, i.e., it transforms from the body-¬xed to space-¬xed coordinates, or the transformation

is not de¬ned as a rotation of the system of coordinate axes but rather as the rotation of a

vector itself.

15.7 Rigid body motion 407

In Section 15.6 we have seen that the c.o.m. (de¬ned in (15.34)) behaves

independent of all other motions according to (15.35). We disregard the

c.o.m. motion from now on, i.e., we choose the c.o.m. as the origin of both

coordinate systems, and all vectors are relative to rcm .

The rotation matrix transforms the components v in the space-¬xed sys-

tem of any vector v to components v in the body-¬xed system as

v = RT v .

v = Rv, (15.45)

The columns of RT , or the rows of R, are the components of the three body-

¬xed unit vectors (1,0,0), (0,1,0) and (0,0,1) in the space-¬xed system. They

are the direction cosines between the axes of the two systems. Denoting the

orthogonal unit base vectors of the space-¬xed coordinate system by i, j, k

and those of the body-¬xed system by a, b, c, the rotation matrix is given

by

⎛ ⎞⎛ ⎞

a·i a·j a·k

ax ay az

R = ⎝ bx by bz ⎠ = ⎝ b · i b · j b · k ⎠ . (15.46)

c·i c·j c·k

cx cy cz

The rotational motion is described by the time-dependent behavior of the

rotation matrix:

ri (t) = R(t)ri . (15.47)

Therefore we need di¬erential equations for the matrix elements of R(t)

or for any set of variables that determine R. There are at least three of

those (such as the Euler angles that describe the orientation of the body-

¬xed coordinate axes in the space-¬xed system), but one may also choose a

redundant set of four to nine variables, which are then subject to internal

constraint relations. If one would use the nine components of R itself, the

orthogonality condition would impose six constraints. Expression in terms

of Euler angles lead to the awkward Euler equations, which involve, for

example, division by the sine of an angle leading to numerical problems for

small angles in simulations. Another possibility is to use the homomorphism

between 3D real orthogonal matrices and 2D complex unitary matrices,9

leading to the Caley“Klein parameters10 or to Wigner rotation matrices

that are often used to express the e¬ect of rotations on spherical harmonic

functions. But all these are too complex for practical simulations. There

are two recommended techniques, with the ¬rst being the most suitable one

for the majority of cases:

9 See, e.g., Jones (1990).

10 See Goldstein (1980).

408 Lagrangian and Hamiltonian mechanics

• The use of cartesian coordinates of (at least) two (linear system), three

(planar system) or four (general 3D system) “particles” in combination

with length constraints between them (see Section 15.8). The “particles”

are dummy particles that have mass and fully represent the motion, while

the points that are used to derive the forces (which are likely to be at the

position of real atoms) now become virtual particles.

• The integration of angular velocity in a principal axes system, preferably

combined with the use of quaternions to characterize the rotation. This

method of solution is described below.

15.7.1 Description in terms of angular velocities

We know that the time derivative of the angular momentum (not the angular

velocity!) is equal to the torque exerted on the body (see (15.42)), but that

relation does not give us a

straightforward equation for the rate of change of the rotation matrix,

which is related to the angular velocity.

First consider an in¬nitesimal rotation dφ = ω dt of the body around an

axis in the direction of the vector ω. For any point in the body dr = ω —r dt

and hence v = ω — r. Inserting this into the expression for the angular

momentum L and using the vector relation

a — (b — c) = (a · c)b ’ (a · b)c, (15.48)

we obtain

mi r i — v i

L=

i

mi r i — (ω — r)

=

i