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AxiomOfChoice

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**Can someone please explain this remark in Landau-Lifshitz's *Mechanics* about fields?**

On pg. 15, it says that "individual components [of momentum] may be conserved even in the presence of

**an external field**."

What do they mean, exactly, by "external field?"

I know (or at least I

*think*I know) that a given component of linear momentum is conserved if the system is translational invariant in that direction (that is, the Lagrangian does not depend explicitly on that coordinate). But how am I supposed to meld this with the idea of a field?

Landau-Lifgarbagez goes on to say that "...in a uniform

**field**in the z-direction, the x and y components of momentum are conserved." I think I'd understand why this is if I had a clearer picture of what's meant by "field."

There is also an exercise at the end of this section that mentions the "

**field**of an infinite homogeneous plane." What does that mean? As best I can tell, he is talking about a particle that moves in a plane subject to a potential energy that does not depend on the coordinates parallel to the plane. For example, if the plane in question is the x/y plane, the potential U satisfies

[tex]

\frac{\partial U}{\partial x} = \frac{\partial U}{\partial y} = 0.

[/tex]

Am I right or barking up the wrong tree? Thanks.

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