Question on conservation of angular momentum

[Tac-Tics]

Is conservation of angular momentum a direct consequence of conservation of linear momentum? It seems like it is. Here is a short proof I derived.

Assume that we have a particle moving in space with constant (conserved) momenum p. The particle's position is given by x(t) = x0 + t/m * p. The angular momentum of the system is given by L(t) = r(t) x p = x0 x p + t/m * p x p. By antisymmetry, the second term, p x p is zero, so L(t) = x0 x p, which is constant. Therefore, angular momentum is conserved.

Assuming my proof is correct for this case, does this proof work in a general case?It seems that the reverse would also be true... that conservation of angular momentum implies the conservation of angular momentum.

 

[Bob_for_short]

For a free particle indeed, the energy (there is only a kinetic part) and the angular momentum conservation laws follow from p=const.

In general case, in a central potential, for example, the linear momentum is not conserved, but the angular one as well as the total energy are conserved. In fact, the conservation laws follow from the Newton equation and in general case they are independent. In Lagrangian language they correspond to different symmetries.

 

[maverick_starstrider]

Noether's Theorem states that the conservation of angular momentum is nothing more than the restatement of the homogeneity of "angle" (for lack of a better term) in our universe. Just as conservation of linear momentum is due to the homogeneity of space.

 

[Bob_for_short]

Actually for a system of N interacting particles there are 2N-1 conservation laws but only few of them (7) are additive in particles. The others are non-linear combinations of particle variables.

 

[rcgldr]

Is conservation of angular momentum a direct consequence of conservation of linear momentum?

You also need to take into account any work done by moving an object "inwards" or "outwards" which changes the speed of the rotating object, but preserves angular momentum.

In the thread linked to below, the example was a puck (actually a point mass, as the angular momentum of the puck itself was ignored in this thread) sliding on a fictionless surface attached to a string. The initial problem is one about the string wrapping around a post, which exerts a torque on the post, where angular momentum is conserved only if you take into account the angular momentum of whatever the post is attached to (typically the earth). In this case, the linear speed of the puck remains constant, but it's angular momentum decreases as it spirals inwards (or increases if it spirals outwards).

The later part of the thread deals with a string being pulled through a frictionless hole at the end of a infinitely small diameter pipe, allowing the string to be pulled inwards or to be relaxed and pulled outwards by the centrifugal reaction force of the puck. In this case, angular momentum is conserved, and the speed of the puck varies inversely with the radius. If the string is pulled inwards to reduce the radius by 1/2, then the pucks linear speed is doubled and the tension increases by a factor of 8. Angular momentum, related to mass x speed x radius is preserved mass x (speed x 2) x (radius / 2).

https://www.physicsforums.com/showthread.php?t=185178

The math for the string through a pipe (hole) case was covered in post 34 from that thread:

https://www.physicsforums.com/showthread.php?p=1436456&postcount=34

The last paragraph in post 34, about the string wrapping around the post is wrong. In this case, the tension in the string is perpendicular to the path of the puck, so there is no "forwards" or "backwards" force, and the speed of the puck remains constant, but the angular momentum of the puck decreases as it spirals inwards (the torque on the post would be increasing the angular momentum of whatever it's attached to).

The math showing that the string wrapping around the post case (involute of circle) results in a line perpendicular to the path of the puck was shown in post #32:

https://www.physicsforums.com/showthread.php?p=1435942&postcount=32

So getting back to the classic example of a ice skater spinning (friction free) while holding a pair of weights, the skater does work by pulling the weights inwards, so the linear speed of the weights is increased, and angular momentum is preserved.

 

[russ_watters]

Is conservation of angular momentum a direct consequence of conservation of linear momentum?

It seems that the reverse would also be true... that conservation of angular momentum implies the conservation of angular momentum.

They are to different forms/applications of the same concept: conservation of momentum.