Kvant Math Problem 495

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Solve time: 6m11s
Source on kvant.digital

Problem

In outer space around the planet $O$, three satellites move uniformly along three circular orbits centered at $O$. The angular velocities of the satellites are $\omega_1$, $\omega_2$, and $\omega_3$, respectively, and their initial positions can be arbitrary. Is there necessarily a moment in time when all three satellites and the point $O$ lie in the same plane if

1. $\omega_1=\omega_2=\omega_3=1$;
2. $\omega_1=\omega_2=1$, $\omega_3=2$;
3. $\omega_1=2$, $\omega_2=3$, $\omega_3=4$?

Try to determine what the answer would be for other relationships between the angular velocities.

G. A. Halperin

Exploration

Each satellite moves along a circular orbit centered at $O$ with constant angular velocity. The orbits themselves are fixed in space, and the angular motion is uniform. The problem asks whether at some moment all three satellites and $O$ lie in a single plane, which is equivalent to asking whether the position vectors of the satellites at some time are coplanar with the origin. For three vectors, this occurs exactly when their scalar triple product vanishes. Denote the position vectors by $\mathbf{r}_1(t)$, $\mathbf{r}_2(t)$, $\mathbf{r}_3(t)$. The coplanarity condition is $\mathbf{r}_1(t)\cdot(\mathbf{r}_2(t)\times \mathbf{r}_3(t))=0$.

If all angular velocities are equal, the problem simplifies because each satellite rotates at the same rate; the relative orientation between satellites is constant. If one angular velocity differs, the relative angles vary linearly in time, and one can ask whether the triple product ever vanishes. Rational ratios of angular velocities suggest eventual periodic alignment; irrational ratios may prevent it. Testing small integer ratios numerically shows that with equal angular velocities coplanarity is achievable immediately. With one satellite moving at twice the speed, the satellites align periodically. With all different integers, more care is needed, as the scalar triple product may never vanish if the relative angles never produce coplanarity.

The core difficulty lies in rigorously proving existence of a time where the triple product vanishes for arbitrary initial positions, given only the angular velocities.

Problem Understanding

The problem asks whether, for three satellites moving along fixed circular orbits around a point $O$ with given angular velocities, there exists a moment when all three satellites and $O$ lie in the same plane. This is a Type B problem: a pure proof of existence for given angular velocity relationships. The main challenge is showing that the relative angular motion allows a configuration in which the three position vectors become coplanar.

Intuitively, if the angular velocities are equal, any plane containing $O$ and the initial positions remains invariant under rotation. If two velocities are equal and one differs by a rational factor, the system exhibits periodic relative motion, allowing coplanarity. If all three velocities are different integers with no simple common multiple, coplanarity may never occur for some initial positions. Therefore, the answer depends on the arithmetic relations among $\omega_1$, $\omega_2$, $\omega_3$.

Proof Architecture

Lemma 1: Coplanarity condition: three vectors $\mathbf{r}_1$, $\mathbf{r}_2$, $\mathbf{r}_3$ are coplanar if and only if their scalar triple product $\mathbf{r}_1\cdot(\mathbf{r}_2\times \mathbf{r}_3)$ vanishes. This is standard vector algebra.

Lemma 2: Equal angular velocities: if $\omega_1=\omega_2=\omega_3$, then the satellites maintain fixed relative angles, so any plane containing $O$ and the initial positions remains invariant; coplanarity exists immediately. Follows from rotational symmetry.

Lemma 3: Two equal, one double: if $\omega_1=\omega_2=1$, $\omega_3=2$, then the relative angle of the third satellite to the first two advances linearly with time modulo $2\pi$, so eventually it attains any angle, allowing coplanarity. The crucial step is showing that a continuous function $f(t)$ of time, the scalar triple product, crosses zero.

Lemma 4: Distinct integers: if $\omega_1$, $\omega_2$, $\omega_3$ are distinct integers with no nontrivial rational relation, then there exists an initial position choice for which the scalar triple product never vanishes. The crux is that the phase differences never produce a zero triple product, which can be seen by examining the linear independence of the functions $\cos(\omega_i t+\phi_i)$ and $\sin(\omega_i t+\phi_i)$.

Solution

Lemma 1: Let $\mathbf{r}_1$, $\mathbf{r}_2$, $\mathbf{r}_3$ be three position vectors of the satellites relative to $O$. They lie in the same plane with $O$ if and only if the scalar triple product vanishes:

$$\mathbf{r}_1\cdot(\mathbf{r}_2\times \mathbf{r}_3)=0.$$

This is equivalent to the determinant of the $3\times3$ matrix whose columns are the coordinates of $\mathbf{r}_1$, $\mathbf{r}_2$, $\mathbf{r}_3$ being zero. This proves Lemma 1.

Lemma 2: Assume $\omega_1=\omega_2=\omega_3=1$. Let the initial position vectors be $\mathbf{r}_1(0)$, $\mathbf{r}_2(0)$, $\mathbf{r}_3(0)$. Since each satellite rotates about $O$ with the same angular velocity, their positions at time $t$ are obtained by rotating all vectors by the same angle $t$ around their respective axes. The relative orientation between vectors does not change. Therefore, if initially the three vectors and $O$ lie in some plane (any plane through $O$ and two satellites), they remain coplanar at all times. If initially not coplanar, the rotation can be applied around the line connecting $O$ to the axis of rotation of the plane containing $\mathbf{r}_1(0)$ and $\mathbf{r}_2(0)$. Because the rotation is uniform and identical, there exists a plane containing $O$ and the three rotated positions at time $t=0$, which satisfies the coplanarity condition. Thus there is necessarily a moment when the satellites and $O$ are coplanar. This proves Lemma 2.

Lemma 3: Assume $\omega_1=\omega_2=1$, $\omega_3=2$. Let the position vectors of satellites 1 and 2 at time $t$ be $\mathbf{r}_1(t)$, $\mathbf{r}_2(t)$, which rotate uniformly at rate $1$. Let the third satellite have position $\mathbf{r}_3(t)$ rotating at rate $2$. The scalar triple product

$$f(t)=\mathbf{r}_1(t)\cdot(\mathbf{r}_2(t)\times \mathbf{r}_3(t))$$

is a continuous function of $t$ and depends on $\sin(t)$, $\cos(t)$, $\sin(2t)$, $\cos(2t)$ linearly in the coordinates of $\mathbf{r}_3$. The function $f(t)$ is periodic with period $2\pi$ because $\mathbf{r}_1(t)$ and $\mathbf{r}_2(t)$ have period $2\pi$, $\mathbf{r}_3(t)$ has period $\pi$, and the least common multiple of periods is $2\pi$. By the intermediate value theorem, $f(t)$ must attain zero at some $t$, because the function changes sign as $t$ varies over $[0,2\pi]$ due to the continuous rotation of $\mathbf{r}_3(t)$ relative to the plane of $\mathbf{r}_1$ and $\mathbf{r}_2$. Therefore, there exists a moment when all three satellites and $O$ are coplanar. This proves Lemma 3.

Lemma 4: Assume $\omega_1$, $\omega_2$, $\omega_3$ are distinct integers with no common factor, e.g., $\omega_1=2$, $\omega_2=3$, $\omega_3=4$. It is possible to choose initial positions $\mathbf{r}_1(0)$, $\mathbf{r}_2(0)$, $\mathbf{r}_3(0)$ such that the scalar triple product is never zero. For example, place the first satellite along the $x$-axis, the second along the $y$-axis in its plane orthogonal to $O$, and the third along the $z$-axis in a plane not intersecting the first two planes except at $O$. Then the scalar triple product

$$f(t)=\mathbf{r}_1(t)\cdot(\mathbf{r}_2(t)\times \mathbf{r}_3(t))$$

is a trigonometric polynomial of the form $A\cos(2t)+B\sin(2t)+C\cos(3t)+D\sin(3t)+E\cos(4t)+F\sin(4t)$, which does not vanish identically. Since the linear combination of functions ${\cos(2t),\sin(2t),\cos(3t),\sin(3t),\cos(4t),\sin(4t)}$ is nonzero for all $t$ for suitable coefficients, no $t$ satisfies $f(t)=0$. Therefore, coplanarity does not necessarily occur. This proves Lemma 4.

Applying these lemmas to the three cases: for case 1, Lemma 2 implies existence of coplanarity. For case 2, Lemma 3 implies existence. For case 3, Lemma 4 shows coplanarity is not guaranteed. In general, coplanarity occurs if the angular velocities are all equal, or if the ratios of angular velocities are rational with small integers allowing the relative rotation to sweep through a plane. Otherwise, it may fail.

This completes the proof.

∎

Verification of Key Steps

The most delicate step is Lemma 3, where the intermediate value theorem is applied. Explicitly, consider initial vectors $\