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)

 

[paranormal]

Sure, I would be happy to provide some guidance on this problem. First, let's start by defining angular momentum. Angular momentum is a measure of an object's rotational motion and is defined as the product of the moment of inertia and the angular velocity. In other words, it is a measure of how much rotational motion an object has and how fast it is rotating.

Now, let's take a look at the problem from Landau Lifshitz. This problem is asking you to find the angular momentum of a rigid body rotating around a fixed axis. To solve this problem, you will need to use the formula for angular momentum, which is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

To find the moment of inertia, you will need to use the formula I = ∫r^2dm, where r is the distance from the axis of rotation to the mass element and dm is the mass element. This formula can be used to find the moment of inertia for any rigid body.

Once you have found the moment of inertia, you will need to find the angular velocity. This can be done by using the formula ω = Δθ/Δt, where Δθ is the change in angle and Δt is the change in time. This formula can be used to find the angular velocity for any rotating object.

Once you have both the moment of inertia and the angular velocity, you can plug them into the formula L = Iω to find the angular momentum of the rigid body.

I hope this helps to clarify the problem and gives you some guidance on how to solve it. Remember to always double check your calculations and units to ensure accuracy. Best of luck!