Clarification of classical field theory lecture notes by P. Deligne and D. Freed

2010-06-08T07:02:08Z

In section 1.1 under the subtitle system of classical particles with potential, the authors claim that "for a system of classical particles with rigid constraints, the configuration space is a Riemannian manifold X with Riemannian structure given by twice the kinetic energy."

I don't quite how the configuration space can be given by a Riemannian manifold, as it is more naturally viewed as a symplectic manifold and there appears to be no natural Riemannian structure on a symplectic manifold. Also the relation between the Riemannian structure and the kinetic energy also eludes me. The best interpretation I can think of is to impose a Riemannian structure on the cotangent bundle via Legendre transform, or the specification of a Lagrangian function. But this is not explcitly given.

Answer by Kevin Lin for Clarification of classical field theory lecture notes by P. Deligne and D. Freed

2010-06-08T07:40:47Z

Configuration space is, by definition, the position space of your particles. Phase space, on the other hand, is the space of pairs (position, momentum). The latter has a symplectic structure; the former has a Riemannian structure.

Regarding the relationship between kinetic energy and the Riemannian structure: You will recall from your high school physics class that kinetic energy is $\frac{1}{2} mv^2$. Of course the $v^2$ is really the dot product $v \cdot v$, in other words it's $g(v,v)$, where $g$ is the Riemannian metric and $v$ is a tangent vector. The $\frac{1}{2}$ explains the "twice the kinetic energy" part.

Clarification of classical field theory lecture notes by P. Deligne and D. Freed - MathOverflow

Clarification of classical field theory lecture notes by P. Deligne and D. Freed

Answer by Kevin Lin for Clarification of classical field theory lecture notes by P. Deligne and D. Freed