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Today, a friend and I solved a classical mechanics problem using group theory. The problem was the following:

Around a circumference, there are $N$ children evenly spaced. In the center, there is a tire. Each children pulls the tire by a rope with equal forces. Is the resultant force always zero?

My friend associated each force vector with a complex root of unity, and using the fact that the group of the $N$-roots of unity is cyclic, showed the identity: $$1 + \zeta + \zeta^2 + \cdots + \zeta^{n-1} = 0$$ which equals saying that the resultant force is zero. I considered the set $\Omega$ of all permutations of the children, with a group action $\pi\colon \mathbb{Z}/N\mathbb{Z}\times \Omega \to \Omega$ by cyclic permutations. I considered the "force" function $f\colon \Omega \to \mathbb{R}^2$ which associated each system with it's resultant force vector. I also considered the group action $\varphi\colon \mathbb{Z}/N\mathbb{Z} \times \mathbb{R}^2 \to \mathbb{R}^2$ by rotations of the plane. Then, I argumented the following identities for all $S \in \Omega, x \in \mathbb{Z}/N\mathbb{Z}$: $$f(\pi(x, S)) = \varphi(x, f(S))$$ $$f(\pi(x, S)) = f(S)$$ which implied $f(S)$ is a fixed point of $\varphi$, and thus is always the zero vector.

I never expected a application of group theory to classical mechanics (though I know about it's uses in crystallography). Are there other well know examples of this?

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    $\begingroup$ Ultimately, group theory is the study of symmetry. In many situations, the symmetry of a problem makes it easier to solve, or guarantees a certain outcome. In the easiest examples, one could intuitively figure out how to exploit that symmetry. However, in situations involving elaborate mathematical structures, group theory allows us to accomplish these ends systematically. $\endgroup$ – Ben Grossmann Aug 18 '16 at 13:42
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    $\begingroup$ I would expect that you could find some interesting examples in chemistry. $\endgroup$ – Ben Grossmann Aug 18 '16 at 13:51
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(I would leave a comment, but apparantly I'm not allowed to do that yet)

If you ask a physicist what group theory is, his answer will probably come out to be "the study of symmetry". Symmetry plays a huge role in modern physics and also in Classical Mechanics. The modern treatment of Classical Mechanics, involving symplectic and poisson manifolds, makes heavy use of groups.

One of the important points where group theory comes in is, that constants of motion (observable quantities of a system, that do not change with time) correspond to invariance of the equations of motion under the smooth action of a group. This is called Noether's theorem and remains very important beyond Classical Mechanics. Well-known conservation laws like energy, momentum and angular momentum correspond to the invariance of the system in question under time translation, space translation and rotation. The symmetry group of classical mechanics is called the Gallilei group. The symmetry group of relativistic physics is the Poincaré group. Modern physics is full of group theory (Lie groups are used most often): For example the Wightman axioms for quantum field theory require to specify an irreducible representation of the Poincaré group. The classification (due to Wigner) of (certain) irreducible unitary representations of the Poincaré group leads physicists to the classification of elementary particles according to their mass and spin.

If you want to find out more about applications of group theory to mechanics, I would recomend the books by V. Arnold (Mathematical Methods of Classical Mechanics) or Marsden and Ratiu (Introduction to Mechanics and Symmetry).

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    $\begingroup$ I really didn't expect such profound study of symmetry on classical mechanics. Though both of these books seems a bit unaccessible for a undergraduate, I think I'll benefit from a short, informal reading at the local library, just to get a feel of it (especially Mardsen's and Ratiu's book). Thanks! $\endgroup$ – Henrique Augusto Souza Aug 18 '16 at 14:11
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    $\begingroup$ The point re: "beyond classical mechanics" is an important one: Most of the applications I know of are not classical but quantum (either at the level of QM or QFT), though Noether's theorem is certainly relevant in classical mechanics. $\endgroup$ – Semiclassical Aug 18 '16 at 15:26

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