Landau Mechanics Chapter 2 Problem 1: Direction of Potential Energy?

[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?

 

[Rick16]

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?

I understand the case with a single potential energy field, or perhaps I should rather say force field if I want to speak of the direction of the field. I can orient my coordinate axes such that the force points along one of the axes. The momentum of a particle moving in this field would then only be affected by the field along this coordinate axis and would be conserved in the plane of the other coordinate axes. But in the present case we have two potential energies, which do not necessarily have their gradients pointing in the same direction. Why would the boundary between the half-spaces be perpendicular to both forces?

 

[Ibix]

You don't seem to be reading the problem correctly. It states that the potential is $U_1$ in half of space and $U_2$ in the other half. Can you state where the force due to this potential is non zero?

 

[Matterwave]

It seems your confusion is not reading that the U1 and U2 are constants.

 

[GiorgioPastore]

A possible schematization for this problem is the x-dependent three-dimensional potential function
$$
U(x)=U_1+(U_2-U_1)\frac{\left(1+\tanh\left( \frac{x}{w} \right)\right)}{2}.
$$
Over an interval controlled by the width $w$, $U(x)$ goes from values very close to $U_1$ at negative values of $x$, to values very close to $U_2$ at large values of $x$. In the limit of $w \rightarrow 0$, $U(x)=U_1$ for negative $x$ and $U(x)=U_2$ for positive $x$.

For any positive width $w$, the force has only the $x$ component $F_x(x)$, always the same sign (negative if $U_2>U_1$ and positive if $U_1>U_2$, and it is concentrated in a region of width of a few $w$. Moreover, in the limit $w \rightarrow 0$, $F_x(x)=(U_2-U_1)\delta(x)$, where $\delta(x)$ is the Dirac delta.

I hope that such an example could help to clarify and solve the problem. Of course, there is nothing special about the continuous family of functions I have chosen. Every other family converging towards a step-function equal to $U_1$ in the negative $x$ half-space and to $U_2$ in the positive $x$ half-space will be equivalent.