How Planetary Motion Equations Are Symmetrical


By Joshua Winn, Princeton University

To understand planetary motion, we can define an effective potential energy, U-effective equal to L-squared over 2mr-squared minus-GMm over r. That way, we can write the energy as E equals 1/2 mv_r-squared plus U-effective. The reason this helps is that even though we live in a three-dimensional world, the motion of the planet is governed by a single-variable equation.

Formulas of classical mechanics such as Newton’s laws
Writing E equals 1/2 mv_r-squared plus U-effective helps us make planetary motion a single-variable equation to be solved. (Image: Lia Koltyrina/Shutterstock)

Difference Between Small-r and Large-r

Let’s plot U-effective as a function of r. For small-r, that one over r-squared is dominant and it’s positive. So, U-effective shoots up to infinity for small-r. For large-r, the one over r is dominant and it’s negative. So, as r grows, the potential dives down to negative values and rises toward zero as r goes to infinity. So, it makes a bowl shape. 

The trajectory of the particle depends on E, how much total energy we give it. First, let’s consider the case in which E is negative, the negative potential energy dominates over the positive kinetic energy. Since the difference between E and U-effective equals 1/2 mv_r-squared, which is always a positive number, the particle’s radius r must be confined to the region where E is bigger than U-effective.

And furthermore, at locations where E minus U-effective is large, that means v_r is large, too, so the particle is moving quickly in the radial direction. Whenever U-effective gets close to E, the particle must be slowing down. When the lines cross, v_r is zero, and r is momentarily standing still.

All this means that the particle’s radial motion can be understood qualitatively by imagining that we drop a marble in a bowl, starting at one of the intersection points. The marble starts at rest, rolls to the bottom and speeds up, rolls up to the same height on the other side, stops briefly, then drops down again, and keeps oscillating. Likewise, the r-value of our particle will grow, then shrink, then grow again, as it’s whirling around. 

How to Make Sense of It?

That makes sense. We already know the particle will follow an ellipse, with a distance to the origin r, that gets bigger and smaller as it goes around. And if we happen to put the particle right at the lowest point in the bowl, it will just stay there. That corresponds to a circular orbit, with an unchanging radius.

We’ve just seen that for a given angular momentum, a circular orbit has the minimum possible energy; it’s the low point in the bowl. Whenever you drain energy out of an orbit, with friction or some other process that leaves angular momentum alone, the orbit will circularize.

It’s impossible for the particle to ever reach r equals zero. That’s because of the first term in the effective potential, L-squared over 2mr-squared, which makes an infinitely high barrier, guarding the origin. The only exception would be if L, the angular momentum, is exactly zero. 

Then there’s no barrier. In plain language, to make a direct hit on the origin, you need to be dropped straight in, with no sideways motion. If you have any angular momentum at all, you’ll orbit the attractor, you won’t hit it.

This article comes directly from content in the video series Introduction to Astrophysics. Watch it now, on Wondrium.

Kepler’s Second Law

Kepler’s second law holds for any central force. The particle approaches the origin, then turns around and flies away, slowing down but never returning. That is an unbounded trajectory. You can define an effective one-dimensional potential for any central force law, whether the force goes like one over r-cubed, or the square root of r, or whatever. 

Illustration of rosette orbits
Almost any power of r leads to infinite rosette loops instead of a fixed shape. (Image: Kes47/Public domain)

In general, what happens is the particle whirls around, going from the minimum to the maximum radius and back again, in accordance with Kepler’s second law. The trajectory makes a beautiful pattern that fills in the space between the minimum and maximum distance. They’re called rosette orbits.

But there’s a remarkable coincidence—the trajectory comes around and repeats, making an ellipse. That is a very special case. Just about any other force law, any other power of r leads to infinitely looping rosettes, not a fixed geometric shape. 

In fact, there’s only one other exception. If the particle is attached to the origin with an ideal spring, with force proportional to r, then its trajectory is also an ellipse, but in that case, the origin is the center of the ellipse instead of the focus.

Noether’s Theorem

 Image of Emmy Noether
Emmy Noether suggested that the universe is mathematically much simpler than it could have been. (Image: Unknown/Public domain)

In advanced classical mechanics, we learn that whenever there is a conserved quantity, like energy, or angular momentum, there’s a corresponding symmetry in nature, a sense in which nature is mathematically simpler than it could have been. This is called Noether’s theorem, after Emmy Noether who published it in 1918.

For example, energy is conserved because the laws of physics don’t change with time: F equals ma forever and always. What we say is that “the equations have time-translational symmetry.” Angular momentum is conserved whenever the situation has rotational symmetry, when things only depend on r, but not theta.

So, what about this third conserved quantity, the eccentricity vector? What’s the corresponding symmetry? It’s very subtle and weird. It turns out the equations governing the motion of a particle under the force of gravity from another particle are mathematically equivalent—through a complicated change of variables—to the equations for a particle moving freely, without any force, on the surface of a 4-dimensional sphere. And it’s the perfect symmetry of that 4-dimensional sphere that leads to the conservation law for the eccentricity vector.

The inverse square law of gravity explains big physics demonstrations in the sky: the motions of the planets. Achieving this understanding was a pivotal development in human history: it was our first real awareness of one of the 4 fundamental forces of nature. It’s also a beautiful connection between mathematics, geometry, and physics.

Common Questions about How Planetary Motion Equations Are Symmetrical

Q: How does U-effective change if we plot it as a function of r?

U-effective for planetary motion shoots up to infinity for small-r. For large-r, as r grows, the potential dives down to negative values and rises toward zero as r goes to infinity. So, it makes a bowl shape.

Q: If a particle is moving towards an attractor what happens if L, the angular momentum, is exactly zero?

If you have any angular momentum at all, you’ll orbit the attractor, you won’t hit it. So in planetary motion to make a direct hit, the particle has to have no sideways motion.

Q: What is Noether’s theorem?

Noether’s theorem tells us that the universe is mathematically simpler than it could have been. This is also true for equations concerning planetary motion.

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