Two points with zero velocity in some inertial frame move in a straight line

Question

I've been trying to solve a problem in Arnold's Mathematical Methods of Classical Mechanics in which I'm supposed to show that given a mechanical system of two points such that they have zero velocity in some inertial frame then the motion of the two points will stay on the line connecting them in the initial moment. I think I should use the invariance of the solution to Newton's equation under uniform translation in the direction of this line, but I don't see how.

Answer 1

Assume a frame where $\mathbf x_1(0)$ and $\mathbf x_2(0)$ lie on a common line $L$ through the origin, and where $\dot{\mathbf x}_1(0) = \dot{\mathbf x}_2(0) = \mathbf 0$. (This is possible by the problem description and translation of the origin.) Let $\mathbf u$ be a constant nonzero vector parallel to $L$. Then we can write

$$\mathbf x_i(0) = y_i^0 \mathbf u, \quad \dot{\mathbf x}_i(0) = 0 \, \mathbf u \quad (i = 1,2)$$

for some (unique) numbers $y_1^0, y_2^0$.

We know from examples 1 and 2 that invariance under time and space translation implies

$$\ddot{\mathbf x}_i = \mathbf f_i(\mathbf x_1 - \mathbf x_2, \dot{\mathbf x}_1 - \dot{\mathbf x}_2).$$

Moreover, if $\mathbf v$ and $\mathbf w$ are parallel vectors in $\mathbb R^3$, then a rotation $G$ about $\mathbf v$ leaves both invariant and, by example 3,

$$\mathbf f_i(\mathbf v, \mathbf w) = \mathbf f_i(G \mathbf v, G\mathbf w) = G \mathbf f_i(\mathbf v, \mathbf w),$$

implying that $\mathbf f_i(\mathbf v, \mathbf w)$ will also be parallel to $\mathbf v$. In particular,

$$\mathbf f_i(x \, \mathbf u, y \, \mathbf u) \parallel \mathbf u, \quad \forall (x,y) \in \mathbb R^2.$$

Hence define new functions $f_i: \mathbb R^2 \to \mathbb R$ given by

$$f_i(x,y) \mathbf u = \mathbf f_i(x \, \mathbf u, y \, \mathbf u)$$

and consider the one-dimensional initial value problem

$$\ddot y_i = f_i(y_1 - y_2, \dot y_1 - \dot y_2), \quad y_i(0) = y_i^0, \quad \dot y_i(0) = 0.$$

This problem has a unique solution (by the assumptions used in the chapter) on $I$ (where $I$ is the time interval considered in the problem) and by letting

$$\mathbf x_i(t) = y_i(t) \mathbf u$$

we obtain a solution for the original (three-dimensional) problem. This solution is also unique and it is clearly confined to the line $L$.