Conservation and Symmetry

Delta²

We say that conservation of linear momentum follows from the translational symmetry while conservation of angular momentum from directional (rotational) symmetry. Can anyone explain what exactly do we mean by these kind of symmetries and how they imply conservation of certain quantities?

zeromodz

As the universe gets bigger, it increases in entropy and asymmetry. Linear momentum appears to always be conserved, always traveling from one place to another. Angular momentum isn't really conserved like linear momentum is because you can completely get rid of angular momentum with some external torque.

Cleonis

Let me first make the statements more precise. It goes back to a theorem about the mathematics of theories of motion. That is, the theorem says something about the math of the theory, not necessarily about the physics.

There is a correspondence: when there is a symmetry in the theory then there is a correspondong conserved entity in that theory. For instance, given the definition of kinetic energy there is a correspondence between conservation of kinetic energy and symmetry with respect to time translation.

So if you're curious whether some attempt at formulating a new theory will lead to a theory that implies conservation of energy it suffices to figure out whether the theory is symmetrical with respect to time translation.

Our theories of motion have in common that for any system going through its motions (for example the solar system) the orientation in space is not a factor; there is no dependence on orientation; there is symmetry under shifts of orientation. According to the theorem there must be a corresponding conserved entity, and as we know that's angular momentum.

In any derivation of conservation of angular momentum from the laws of motion the independence on orientation in space is part of it. For example, there is Newton's derivation of Kepler's law of areas from the laws of motion. Among the elements used there is the fact that the same reasoning applies for all orientations in space.

Doc Al

Why do you allow external torques but not external forces?

zeromodz

I say torque because the force has to be perpendicular for angular momentum to be lost. If I were to apply a force directly parallel to the earth relative to the sun, the angular momentum is still conserved because sin90 = 1. Its very important to understand that it must be external torque which is:

t = FRsin = Iα

Delta²

In Lagrangian mechanics why the symmetry is always mentioned as invariance of Lagrangian with respect to time or space and why we always take small perturbations of the time and space variables and consider Langrangian to be invariant for these small changes? (what about big changes???)

From Wikipedia Noether's theorem it states that the theorem doesnt hold for systems that can not be modeled with a Lagrangian. Isnt it possible that these systems can be modelled with some other way which still has some symmetry with time or space?

DrGreg

I think Doc Al's point was if you say "you can completely get rid of angular momentum with some external torque", why don't you also say "you can completely get rid of linear momentum with some external force"?

Doc Al

Exactly. (Thanks, DrGreg. )