Angular Momentum, Problem from Landau Lifshitz

[Jimmy84]

This is problem 3 from section 9 of Mechanics, Landau Lifgarbagez.
I have been trying to understand the problem but I have no idea how to solve it.

Can someone give me a hand please? any comment or suggestion is very welcome.

Thanks for your time.

Best regards.

 

[facenian]

It is all about something physicists are very fond of,ie,symmetries. For instance in a) you can not notice any displacement parallel to the plane either can you notice any rotation about an axis which is perpendicular to the plane. You must check doing the formal derivation working with the Lagrangian of the system.

 

[rmadsanmartin]

Sorry for the bad english!

M is constant when the movement is parallel to the axis of simetry of the field
ie:

a) if the field is a plane xy--->symmetry z axis--->M_z=doesnt change

P is constant when the movement is in the "same field", in a) if the particle moves in any direction of x or y P is constant, the reason is because the vectors of the field are orientated in the direction of the axis of symetry (in case a) ), then P only change in that direction.

ie: b) the symetry is a cylinder, then Mz doesn't change in a Z-cylinder. But if you imagine the field, is like infinite cylinders, all parallel, then if you want the particle moves "in the same field", only the z-motion is the correct.

 

[facenian]

in case b) M_z=const. and P_z=const. I think a field compatible with cilindrical symetry mus be one that points in the radial direction perpendicular to z and it's magnitud depends only on the distance to the z axis.
However the key to this problem is understanding what kind o motion does not change the Lagrangian and this allows you to do it fomally(mathematically)