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I was studying Central Force Problem and got stuck at a point. In the book Analytical Mechanics (Hand, Finch) they said that from the translation invariance and rotational invariance one can tell about the nature of the potential. It's not clear to me how is this done. Someone please explain

$Translational \ Invariance$ $$V(\vec r_1, \vec r_2)=V(\vec r_1-\vec r_2)$$ $Rotational \ Invariance$ $$V(\vec r_1-\vec r_2)=V(|\vec r_1-\vec r_2|)$$

Angular Momentum Is Conserved for Central Forces

If the potential energy of interaction between the two particles depended on the direction of $\vec r_1 — \vec r_2$, it would not be rotationally invariant. This could only happen if there were a preferred direction in space. Since we've already assumed a central force, which has no preferred direction by definition, we can assume rotational invariance.

As a counter example, if our particies had internal degrees of freedom, like a spin $\vec S$ or a magnetic moment, rotational invariance would not follow automatically from the assumption of no external forces. (We could have a term in the Lagrangian like $\vec S • \vec r$. This is invariant only if we simultaneously rotate both the spin and the coordinates together.) But we do not want to consider such complications here.

In this case we conclude: no $$\text {external forces} \iff \text {translational invariance}$$ $$\iff V (\vec r_1, \vec r_2 ) \approx V(\vec r_1 — \vec r_2 )$$ (4.30)

The center of mass frame is defined as the reference frame in which the velocity of the c.o.m. is zero.

plus only $$\text {central force} \iff \text {rotational invariance} $$ $$\iff V (|\vec r_1 - \vec r_2|) = V(\vec r_1 — \vec r_2 ) = V(r)$$(4.31)

The term "central force" thus means that the interaction between the particles doesn't depend on either their absolute spatial position or orientation but only on the distance between them.

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  • $\begingroup$ Assume these two are the only things in the universe. What happens when you change the positions of both by the same vector? $\endgroup$ – Raziman T V Jan 21 '17 at 8:10
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Consider a 2 particle system which has translation invariance i.e. if the system is allowed to move in space then the state of the system is invariant (does not change). Thus, potential can be defined using the new conjugate variable $q$ = $r_1$ - $r_2$ (this says that the dependence is only on the relative position of 1 and 2).

But again you have rotational invariance therefore you could rotate the sytem by $\pi$ and the potential has to be the same. Thus, the conjugate state variable is |$r_1$ - $r_2$|.

Now for the external force part. Just think what happens when a 3rd particle is introduced outside of the system. Is it still rotaionally and translationally invariant?

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In translational invariance you move the coordinates of all the particles by the same displacement vector d. Note now that if your potential is a function of r2 + r1 instead of r2-r1 then the new potential will change its value since its argument will now contain r2 + r1 + 2d after displacement. This will then violate the translational invariance. However if your potential is r2-r1 the vector d will cancel out. As regards the connection between external forces and translational invariance consider moving all the particles upward in earth gravity by the same amount. This has now changed the potential energy of the entire system. Thus no external force is a requirement of translational invariance which finally leads to conservation of total linear momentum. As regards rotational invariance you rotate all particles by the same angle. Now if your potential depends on the "vector" r2-r1 and not on its magnitude and if you rotate the system you now have a vector not the same as r2-r1 and thus your potential changes violating rotational invariance.

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  • $\begingroup$ As regards the connection between external forces and translational invariance consider moving all the particles upward in earth gravity by the same amount. This has now changed the potential energy of the entire system. Thus no external force is a requirement of translational invariance which finally leads to conservation of total linear momentum. $\endgroup$ – SAKhan Jan 21 '17 at 17:10

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