Landau Mechanics Chapter 2 Problem 1

[Rick16]

TL;DR

orientation of potential energy

Problem Statement

A particle of mass m moving with velocity v1 leaves a half-space in which its potential energy is a constant U1 and enters another in which its potential energy is a different constant U2.
Determine the change in the direction of motion of the particle.

Beginning of Landau's Solution

The potential energy is independent of the coordinates whose axes are parallel to the plane separating the half-spaces. The component of momentum in that plane is therefore conserved...
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The rest of the solution is straightforward, if only I understood the beginning. Why would the potential energy have this specific direction? The problem statement does not mention anything about it.

 

[Ibix]

It doesn't say that the potential has a direction (which wouldn't make sense for a scalar anyway), it says that if you pick coordinates with one axis parallel to the boundary, the potential's value does not depend on that coordinate. What does that tell you about any force it generates?