Applications of group theory to classical mechanics

Question

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?

Answer 1

(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).