Do there exist any variational principles on the space of braids (or knots)? This is very speculative question and I do not know where to start looking up the literature, or if what I am looking for is even mathematically possible/meaningful.
Q: I am interested in finding out whether any variational or action principles have been derived anywhere in the physics or mathematical literature (e.g. in study of geometry of configuration spaces etc.) on the space of braids (or knots).
Explicitly, some example where a Lagrangian is a function of a braid representation, and it is minimized/extremized under some constraints. The resulting equation would be differential equation whose solution gives a trajectory in the space of braids.
 A: O'Hara introduced knot energies, and a Möbius invariant case was studied by Freedman-He-Wang. For prime knots, Zheng-Xu He subsequently showed that there exists a smooth minimizer (up to Möbius transformation), and now it is known that critical points are analytic. 
For braids, one can look at various variational principles by looking at metrics on Teichmuller space of the punctured sphere. The Teichmuller metric is a Finsler metric, with unique minimizing geodesics for pseudo-Anosov braids. The Weil-Petersson metric is a Riemannian metric which is incomplete, but for which pseudo-Anosovs also have unique minimizing representatives. 
The minimizing geodesic then gives a canonical braid representative up to conjugacy (by choosing the representatives to have fixed center of mass and constant moment). One can also find unique representatives for braids up to choosing basepoint surfaces (e.g. $n$ points lying on the $x$-axis in $\mathbb{R}^2$). 
For the 3-punctured sphere, the Teichmuller metric is the hyperbolic plane, and one can easily compute the geodesic. 
 
One can also look at solutions to the planar $n$-body problem to get braid realizations. Richard Montgomery first proved the existence of solutions to the 3-body problem.

