Section 20–2 The rotation equations using cross products

(Angular momentum vector / Angular velocity vector / Velocity vector)

In this section, Feynman discusses an angular momentum vector, angular velocity vector, and velocity vector of a rotating object.

1. Angular momentum vector:

“By the same token, the angular momentum vector, if there is only one particle present, r is the distance from the origin multiplied by the vector momentum… (Feynman et al., 1963, section 20–2 The rotation equations using cross products).”

Feynman expresses an angular momentum vector as L = r × p in which r is a displacement vector (instead of distance) and p is a linear momentum vector. In chapter 52, Feynman provides a better explanation on the angular momentum vector: “when we write a formula which says that the angular momentum is L = r × p, that equation is all right, because if we change to a left-hand coordinate system, we change the sign of L, but p and r do not change; the cross-product sign is changed, since we must change from a right-hand rule to a left-hand rule.” We may clarify that angular quantities are axial vectors, whereas linear quantities are polar vectors in the Gibbs vector system. Alternatively, the angular momentum vector is defined as a bivector in the Clifford vector system.

Feynman states a theorem of angular momentum: if the total external torque is zero, then the total vector angular momentum of the system is a constant. This theorem is also called the law of conservation of angular momentum and he rephrases it as “if there is no torque on a given system, its angular momentum cannot change.” In Chapter 52, Feynman says that “[i]nvariance under rotation through a fixed angle in space corresponds to the conservation of angular momentum.” This is also known as the Noether’s theorem that relates rotational symmetry to the law of conservation of angular momentum. One may add that the angular momentum vector of a rigid body is not definitely in the same direction as its angular velocity vector.

2. Angular velocity vector:

“That is to say, simply, angular velocity is a vector, where we draw the magnitudes of the rotations in the three planes as projections at right angles to those planes (Feynman et al., 1963, section 20–2 The rotation equations using cross products).”

Feynman asks whether angular velocity is a vector and discusses the rotation of an object about two axes simultaneously. The net result of the two rotations is that the object simply rotates about a new axis. However, Feynman did not specify that angular velocity vector is a pseudo vector. One may state a theorem of angular velocity addition: “Given a primed coordinate system rotating with angular velocity  w₁ with respect to an unprimed system, and a starred coordinate system rotating with angular velocity w₂ relative to the primed system, the angular velocity of the starred system relative to the unprimed system is w₁ + w₂ (Symon, 1971, p. 452).” Similarly, a gyroscope rotating about a horizontal axis and vertical axis can be related to the commutative law for addition: w_x + w_z = w_z + w_x.

Feynman says that angular velocity is a vector and we can draw the magnitudes of the rotations in the three planes as projections at right angles to those planes. He adds a footnote that suggests a derivation by compounding the angular displacements of the particles of the body during an infinitesimal time Δt. Importantly, angular displacements can be distinguished as infinitesimal rotations and finite rotations. Physics teachers should illustrate how angular displacements of an object (e.g., a book) about two perpendicular axes in two different orders do not commute. In general, a finite angular displacement is neither a polar vector nor an axial vector because it does not obey the commutative law for addition: θ_xi + θ_yj ¹ θ_yj + θ_xi.

3. Velocity vector:

“We shall leave it as a problem for the student to show that the velocity of a particle in a rigid body is given by v = ω × r… (Feynman et al., 1963, section 20–2 The rotation equations using cross products).”

If a rigid body is rotating about an axis with angular velocity ω, Feynman asks, “What is the velocity of a point at a certain radial position r?” He leaves it as a problem for students to show that the velocity of a particle in a rotating rigid body is given by the equation, v = ω × r, where ω is the angular velocity and r is the displacement vector (instead of position). Although the velocity of an object is usually a real vector, this should not be assumed when it is expressed as a cross product. In a sense, the velocity of the point in the rotating body is an axial vector because it is not moving in a straight line, but rotating about a point. However, Feynman uses the term vector because he classifies a vector as a polar vector and axial vector.

As another example, the Coriolis force can be expressed using a cross product: F_c = 2mv×ω. That is, if a particle is moving with velocity v in a coordinate system that is rotating with angular velocity ω, there is a pseudo force F_c from the perspective of a rotating frame. Physics teachers could elaborate that the cross product of an axial vector v and another axial vector ω is also an axial vector. In other words, the Coriolis force in a rotating coordinate system is not only a pseudo force, but it is also expressed as an axial vector (or pseudo vector). One may conclude that the cross product of two vectors (axial vector × axial vector or axial vector × polar vector) is always an axial vector.

Questions for discussion:

1. How would you define an angular momentum vector?

2. How would you define an angular velocity vector?

3. Is the instantaneous velocity of a particle in a rotating rigid body, v = ω × r, a real vector?

The moral of the lesson: angular momentum vectors, angular velocity vectors and the velocity of a particle in a rotating rigid body can be expressed using the cross product that is an axial vector.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Symon, K. R. (1971). Mechanics (3rd ed.). Addison-Wesley, MA: Reading.