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

[AxiomOfChoice]

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

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

 

[DeShark]

By external field, I presume they mean something like a gravitational field. A field is a term, which I (as a physicist) have always just thought of as the allocation of a value of some sort, i.e. number, vector, matrix, tensor, etc. to each point in a space. So we get a scalar field, vector field, matrix field and tensor field.

For example, the gravitation field can be given as a vector field where each vector represents the acceleration of an object of unit mass at that position. To find the actual acceleration, you must multiply the vector at that position by the reciprocal mass of the object.

Alternatively, it can be given by a scalar field, known as the gravitational potential. The force is given by the gradient of the vector field.

By external field, they mean a field whose origin is not related to the presence of other objects in the "system". So if we have two charged object, they will create an electric field between each other, but there will also usually be an externally applied gravitational field.

So if the Lagrangian of both the internally and externally applied field is independent of a particular co-ordinate, you may say that that component of momentum is conserved. I believe this is what the book is trying to say. When they mention the uniform field in the z-direction, I believe the authors were thinking about gravity and how it will not cause a change of momentum in the x and y directions.

The field of an infinite homogeneous plane means the value allocated to each point of an infinite homogeneous plane. In this case, U is taken to be the scalar potential (field). Thus the rate of change of the potential in the x-direction is equal to the rate of change of the potential in the y-direction which equals zero. So the force in all directions equals zero. This example doesn't sound very interesting to me though =)

 

[astrokreng]

In this context, an "external field" refers to any force or potential that acts on a system from outside of it. This could include gravitational, electromagnetic, or any other type of field that can exert a force on objects within the system.

In classical mechanics, the conservation of momentum holds true for a system that is isolated from external forces. However, when an external field is present, the individual components of momentum may still be conserved even if the total momentum of the system is not. This is because the external field can exert a force on the system, causing a change in its momentum, but the individual components may still remain constant.

The example given by Landau-Lifshitz about a uniform field in the z-direction illustrates this concept. In this case, the external field is the uniform force acting in the z-direction, and the x and y components of momentum are conserved because the system is translational invariant in those directions.

As for the exercise about the "field of an infinite homogeneous plane," you are correct in your understanding that it refers to a particle moving in a plane with a potential energy that does not depend on the coordinates parallel to the plane. This type of potential energy can be described by the equation you provided, where the partial derivatives with respect to x and y are both equal to 0.

I hope this helps clarify the concept of external fields and how they relate to the conservation of momentum in classical mechanics.