Section 11–3 Rotations

(Rotation in space / Rotation of axes / Spatial rotational Symmetry)

In this section, Feynman discusses the rotation of an object in space, rotation of axes, and spatial rotation symmetry.

1. Rotation in space:

“In other words, we cannot locate its angular position, but we can tell that it is changing (Feynman et al., 1963, section 11–3 Rotations).”

Feynman explains that a pendulum clock may not work if it is tilted such that the pendulum falls against a side of its case. Thus, physicists can predict that the working of the pendulum clock is dependent on its location and orientation relative to that of the Earth. For example, a pendulum clock in an artificial satellite does not work because there is no sufficient gravitational force. On the other hand, physics teachers may add that a rotation of a wine (cylindrical) bottle about its axis of symmetry (say vertical) does not change its physical appearance. Similarly, if physicists rotate an experimental setup through a fixed angle, the experimental results will remain the same provided the Earth is also rotated.

In a Messenger lecture to the general public, Feynman (1965) says that “[i]f I do some experiments with a piece of equipment built in one place, and then take another one (possibly translated so that it does not get in the way) exactly the same, but turned so that all the axes are in a different direction, it will work the same way. Again we have to turn everything that is relevant. If the thing is a grandfather clock, and you turn it horizontal, then the pendulum will just sit up against the wall of the cabinet and not work. But if you turn the earth too (which is happening all the time) the clock still keeps working (pp. 87-88).” Feynman emphasizes again on the position of the Earth because it affects the magnitude and direction of the gravitational force in an experiment.

2. Rotation of axes:

“This result, which has now been established for both translation and rotation of axes, has certain consequences… (Feynman et al., 1963, section 11–3 Rotations).”

A rotation of axes is a process of replacing a xyz-Cartesian coordinate system to a x'y'z'-Cartesian coordinate system in which the origin is kept fixed and the axes are rotated through an angle. To represent F_x′ in terms of F_x and F_y, we can add their components along the x′-axis, and in a similar manner for F_y′. In summary, we have x′ = x cos θ + y sin θ, y′ = y cos θ – x sin θ, z′ = z and F_x′ = F_x cos θ + F_y sin θ, F_y′ =F_y cos θ − F_x sin θ, F_z′ = F_z. Mathematically, the formulae for x' can be derived by using a basic trigonometric formula, x' = r cos (a - θ) = r cos a cos θ + r sin a sin θ = x cos θ + y sin θ. In addition, we use y' = r sin (a - θ) = r sin a cos θ - r cos a sin θ = y cos θ - x sin θ.

One may prefer Feynman’s (1965) explanations to the general public: “if I locate a point as I have described, by giving its x and y coordinates and someone else, facing a different way, locates the same point in the same way, but calculating the x' and y' in relation to his own position, then you can see that my x coordinate is a mixture of the two co-ordinates calculated by the other man. The connexion of the transformation is that x gets mixed into x' and y' and y into y' and x'. The laws of nature should so be written that if you make such a mixture, and resubstitute in the equations, then the equations will not change their form. That is the way in which the symmetry appears in mathematical form. You write the equations with certain letters, then there is a way of changing the letters from x and y to a different x, x', and a different y, y', which is a formula in terms of the old x and y, and the equations look the same, only they have primes all over them (p. 88).”

3. Spatial rotational symmetry:

“…no one can claim his particular axes are unique, but of course they can be more convenient for certain particular problems (Feynman et al., 1963, section 11–3 Rotations).”

Newton’s laws are applicable to any set of axes and they can be expressed by the same set of equations. An important consequence is that no particular axes are unique, but some axes are more convenient for solving certain particular problems. For example, we can choose an axis that coincides with the direction of gravitational forces. Next, it means that any piece of equipment which is completely self-contained, with all the force-generating equipment completely inside the apparatus, would work in the same way when it is rotated by an angle. In short, physical laws do not have a preferred direction. In a similar sense, a spherical object can be rotated about any axis that passes through its center such that its appearance remains the same.

In a lecture titled Symmetry in Physical Law, Feynman (1965) adds that “[t]he case that rotation in space does not make any difference comes out as the conservation of angular momentum (p. 106).” As a suggestion, physics teachers should mention that symmetry under rotation through a fixed angle corresponds to the law of conservation of angular momentum (based on Noether’s theorem). The condition that physical laws remain invariant with respect to rotation through a fixed angle is also known as the isotropy of space (Landau & Lifshitz, 1976). That is, space has the same mechanical properties in every direction. In other words, physical laws do not distinguish between “up” and “down,” or “forward” and “backward” (Lederman & Hill, 2008).

Questions for discussion:

1. Would experimental results remain the same if an experimental setup is rotated through a fixed angle? Why?

2. What is the meaning of a rotation of axes?

3. What are the implications of spatial rotational symmetry?

The moral of the lesson: an important consequence of rotational symmetry is that no particular axes are unique, but some axes are more convenient for problem-solving.

References:

1. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.

2. 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.

3. Lederman, L. M., & Hill, C. T. (2008). Symmetry and the beautiful universe. New York: Prometheus.

4. Landau, L. D., & Lifshitz, E. M. (1976). Mechanics (3rd ed.). Oxford: Pergamon Press.