PHY6200 W07

Rotational Motion of Rigid Bodies


We derived a set of equations for treating the motion of a rigid body in a rotating frame where the principal axes of the body are fixed. These are the Euler equations:

Ldot + ω×L = Γ

or in component form

λ1ω1dot - (λ2 - λ32ω3 = Γ1
λ2ω2dot - (λ3 - λ13ω1 = Γ2
λ3ω3dot - (λ1 - λ21ω2 = Γ3

This is a set of coupled, nonlinear differential equations. We solved these for certain special cases only. Now that we've learned the Lagrangian formulation of mechanics, we can learn how to treat them in the Lagrangian formalism. Since the Lagrangian is simply kinetic minus potential energy in an inertial system, we need a way to express the kinetic energy of a rotating body in an inertial system. This is normally accomplished with by defining 3 angular coordinates that specify the orientation of the body relative to fixed spatial coordinates. The 3 angular coordinates are called Euler angles.

Euler Angles

Since the principal axes represent a rotating coordinate system, they are non-inertial and a Lagrangian cannot be expressed in terms of them. A fixed, inertial coordinate system is needed, but then we need a way to relate the principal axes to it. The Euler angles relate the principal axes of a rotating object to a fixed coordinate system.

The transformation relates (x,y,z) to (e1, e2, e3). First, rotate about the z axis by $phi; until the x axis lies in the plane z-e3 plane. Call this set of axes (x', y', z') and note that z' = z. Second, rotate about the y' axis by $theta; until the z' = z axis is aligned with the e3 axis. Call this set of axes (e1', e2', e3') and note that e3' = e3 and e2' = y'. Finally, rotate about the e3' = e3 axis by ψ until e1' and e2' are aligned with e1 and e2.

Now we can express the angular velocity of the body in terms of the time derivatives of the Euler angles

ω = φdot zhat + θdot e2' + ψdot e3.

For an axially symmetric object, withe the symmetry axis in the 3-direction, e1' and e2' serve as principal axes equally as well as e1 and e2. Therefore, we can express ω in terms of body axes by transforming zhat = e3cosθ - e1'sinθ yielding

ω = -φdot sinθ e1' + θdot e2' +(ψdot + φdot cosθ)e3.
L = -λ1φdot sinθ e1' + λ1θdot e2' + λ3(ψdot + φdot cosθ)e3.
L3 = λ3ω3 = λ3(ψdot + φdot cosθ)
Lz = Lzhat = λ1φdot sin²θ + λ3(ψdot + φdot cosθ) cosθ
Lz = λ1φdot sin²θ + L3 cosθ
T = ½ ωT I ω = ½λ1(φdot² sin²θ + θdot²) + ½λ3(ψdot + φdot cosθ)²

Motion of a Spinning Top

U = MgR cosθ
L = T - U = ½λ1(φdot² sin²θ + θdot²) + ½λ3(ψdot + φdot cosθ)² - MgR cosθ
L/∂θ = (d/dt)(∂L/∂θdot)
λ1φdot² sinθcosθ - λ3(ψdot + φdot cosθ)φdot sinθ + MgR sinθ = (d/dt)(λ1θdot) = λ1θddot
L/∂φ = 0 = (d/dt)(∂L/∂φdot)
λ1φdot sin²θ + λ3(ψdot + φdot cosθ)cosθ = pφ = Lz = const.
L/∂ψ = 0 = (d/dt)(∂L/∂ψdot)
λ3(ψdot + φdot cosθ) = pψ = L3 = const.

Special Case: Steady Precession

Consider the special situation where θ is constant. That means that the angle of the symmetry axis of the top to the vertical is constant. If θ is constant, θdot = θddot = 0. Since both L3 and Lz are constant, the expression for Lz leads to the conclusion that φdot is constant; call this constant φdot = Ω. The expression for L3 leads to the conclusion that ψdot is constant; use the quantity ω3 = ψdot + φdot cosθ. The θ equation becomes, dividing out a common factor sinθ

λ1Ω² cosθ - λ3ω3Ω + MgR = 0

This is a quadratic equation for Ω with roots

Ω = (2λ1cosθ)-13ω3 ± √(λ3²ω3² - 4λ1MgR cosθ)]

Note two things. (i) Since Ω must be real, the factor in the square root must be positive,

λ3²ω3² - 4λ1MgR cosθ ≥ 0


ω3² ≥ (4λ13²)MgR cosθ

Therefore, the top must spin sufficiently fast for steady precession to occur.

(ii) Two values exist for Ω, one large and one small.

A common situation is one where the top spins very rapidly, such that we can take λ3²ω3² >> 4λ1MgR cosθ and expand the square root to obtain approximations for the two values of Ω

Ω ≈ (2λ1cosθ)-13ω3 ± (λ3ω3 - 2λ1MgR cosθ/λ3ω3)].

This yields the approximations

Ω ≈ λ3ω31cosθ - MgR/λ3ω3

for the large solution, and

Ω ≈ MgR/λ3ω3

for the small solution.

General Case: Nutation

In the general case, we must allow θ to vary. Recall that θ is the angle between the axis of symmetry and the z-axis, vertical in this case. As the axis of symmetry precesses in φ a variation of θ produces the nodding motion of the axis. This is called nutation, latin for nodding.

It turns out that for this situation, the energy can be expressed as a function of the angle θ alone. This simplifies the analysis of the problem. Begin with E = T + U, Start with the kinetic energy for a symmetric top as given in Equation~(10.105),

T = ½λ1(φdot² sin²θ + θdot²) + ½λ3(ψdot + φdot cosθ)²,
and potential energy U = MgR cosθ. Use the relations for the constants of the motion

L3 = λ3(ψdot + φdot cosθ)


Lz = &lambda1φdot sin²θ + L3cosθ

to eliminate ψdot and φdot in the expression for the total energy

E = T + U
= ½λ1θdot² + [(Lz - L3cosθ)²]/[2&lambda1sin²θ] + L3²/[2λ3] + U = ½&lambda1θdot² + Ueff(θ)


Ueff = (Lz - L3cosθ)²/[2λ1sin²θ] + L3²/[2λ3] + MgR cosθ.

We are treating an idealized problem of a frictionless top where the energy is constant, or at least working in an approximation where the friction is small and the energy is approximately constant. Therefore, we can write

θdot² = 2[E - Ueff(θ)]/&lambda1

In principal, this can be integrated to find θ as a function of t. It is a little easier if we use the clever substitution u = cosθ which yields udot = -θdot sinθ, or θdot = -udot/sinθ = -udot/√(1 - u²). Upon substituting this into the expression for θdot² we find

udot² = 2(1 - u²)(E - MgRu)/&lambda1 - (Lz - L3u)²/&lambda1²
udot² = f(u).

Turning points of the motion occur for udot=0, or therefore at the roots of the equation f(u) = 0. The equation in udot² can be integrated to yield

t = ∫ du/√(f(u)).

Chapter 11: Coupled Oscillators and Normal Modes

Two Masses and Three Springs

Three Identical Springs and Two Equal Masses

Two Weakly Coupled Oscillators: One Spring Different from the other Two

Lagrangian Approach: The Double Pendulum

The General Case

Three Coupled Pendula

Normal Coordinates*

© 2007 Robert Harr