First Order in Time Derivatives + Phase Space

[thatboi]

Hey all,
I am reading David Tong's notes on Chern-Simons: https://www.damtp.cam.ac.uk/user/tong/qhe/five.pdf
and he makes the following statement that doesn't make much sense to me: "Because the Chern-Simons theory is first order in time derivatives, these Wilson loops are really parameterising the phase space of solutions, rather than the configuration space."
What exactly does he mean by "solutions"? Also what is the relation between being first order in time derivative and these different spaces.

 

[A. Neumaier]

"Because the Chern-Simons theory is first order in time derivatives, these Wilson loops are really parameterising the phase space of solutions, rather than the configuration space."
What exactly does he mean by "solutions"? Also what is the relation between being first order in time derivative and these different spaces.

The phase space of a dynamical system defined by a stationary action principle is isomorphic to the space of initial conditions of the resulting system of differential equations. A differential equation for ##q(t)## that is first order in time needs the initial value for ##q## only, hence has the configuration space as phase space. But if the differential equation is of second order in time, one needs initial conditions on ##q## and its first time derivative, which after the conventional Legendre transform leads to a first order system in a Hamiltonian phase space that is ''twice as large'', whose points are labelled by ##(q,p)##.