Clarifying Symmetry in Le Bellac's Quantum Physics

[neelakash]

I was going through Le Bellac's Quantum Physics book.In the "symmetry" chapter 1st page(Classical physics), he makes the following comments a part of which look a bit weired to me...Each statement starts with "Invariance of the potential energy".Do you think this is meaningful?


*Invariance of the potential energy under time-translations implies conservation of mechanical energy E = K + V , the sum of the kinetic energy K and the potential energy V .
*Invariance of the potential energy under spatial translations parallel to a vector n implies conservation of the momentum component p_n .
*Invariance of the potential energy under rotations about an axis n implies conservation of the component j_n of the angular momentum.

 

[dextercioby]

If he's using the Newton formulation, then he must be tacitly or explicitely assuming that:

$$\vec{F} = -\nabla V$$.

 

[xepma]

The kinetic energy usually is already invariant under the symmetries he mentiones. Therefore, if the potential energy is invariant under those symmetries, then the whole system is invariant as well.

This is a famous result called Noether's Theorem. The theorem states that if a system is related invariant with respect to a specific symmetry, then there is a specific quantity which is conserved.

E.g.
Time translation invariance <-> Conservation of (mechanical) energy
Space translation <-> Conservation of momentum
Rotation symmetry <-> Conservation of angular momentum

This is quite a statement. If you consider the collission of two particles, for instance. Suppose the force between the two particles is invariant under space and time translations. Then the theorem implies that collissions will always obey conservation of energy and momentum. And I don't even have to specify what kind of force we are dealing with!

So yes: such statements are very meaningful. But you will learn to appreciate them later on ;-)

 

[neelakash]

Yes,I learned about Noether's theorem last semester...Under symmetry transformations of time translation, space translation or rotation, total mechanical energy,linear momnentum and angular momentum are conserved respectively.

My question is does the conservation of total energy,linear momentum and angular momentum imply conservation of potential energy in classical physics?I cannot see conservation of total energy/linear momentum or angular momentum would necessarily mean conservation of potential energy...

 

[neelakash]

Yea...the thing is other way around,actually.There is no need to invoke Noether's theorem and its conserved current.The author puts like this: conservation of the Lagrangian [$$\delta\ L=0$$]under space translation,time translation and space rotation gives rise to conservation of linear momentum,energy and the angular momentum.

However,whether $$\delta\ L=0$$ or not is decided by the potential function V(r1-r2)...Hence,if the potential energy is conserved,we must have $$\delta\ L=0$$ w.r.t. appropriate variables and the corresponding quantities are conserved.