What is the difference between configuration space and phase space?

Question

1. What is the difference between configuration space and phase space?

2. In particular, I notices that Lagrangians are defined over configuration space and Hamiltonians over phase space. Liouville's theorem is defined for phase spaces, so is there an equivalent conservation law for the configuration space?

3. In the process of modeling a physical system, when it is appropriate to use one instead of the other?

Answer 1

1) Configuration space is, in a sense, the possible "positions" of the mechanical system. The states of motion, eg. velocities/momenta are not part of the configuration space.

The configuration space (especially when constraits are in the picture) is modelled as some real, $n$-dimensional differential manifold, which I'll denote as $\mathcal{C}$.

The velocity phase space is the set of all "positions" and "velocities" together. If $\mathcal{C}$ is the configuration space, then the velocity phase space naturally has the structure of the tangent bundle over the configuration space, $T\mathcal{C}$. If a point $p\in\mathcal{C}$ is stated, then the elements of the fiber $T_p\mathcal{C}$ are the possible generalized velocities of the system in the "$p$ configuration".

Since the Lagrangian depends on positions and velocities, it is a scalar field on $T\mathcal{C}$.

The momentum phase space, which is usually called the phase space is the cotangent bundle $T^*\mathcal{C}$. In this case, the fiber ${T_{p}}^*\mathcal{C}$ is the set of all possible momenta of the system in the "$p$ configuration".

2) I don't think so, the Liouville-theorem heavily employs the fact that the momentum phase space is naturally a symplectic manifold. The velocity phase space is not, in general, a symplectic manifold. To be able to ascertain how the phase flow transforms a volume, you need to have a structure that defines volume, which in the momentum phase space, the symplectic form does, while in the velocity phase space, there is no such canonical structure.

3) Unfortunately I am not sure enough in this to post a definitive answer.

Answer 2

1. Point in configuration space represents configuration of the system, i.e. positions of the constituent particles. Point in phase space represents state of the system, i.e. positions and velocities of the constituent particles together.

2. No. Liouville's theorem has no simple analogue in the configuration space.

3. Depends on what is the task at hand and what are the preferences of the person working on it. Liouville's theorem plays some role in statistical physics, so Hamiltonian formalism there is used much more than the Lagrangian formalism. There are also cases where the Hamiltonian formalism is used in mechanics; in approximate astronomical calculations (perturbation theory), in problems involving small oscillations and others.

   There are cases, however, when it is cumbersome to construct Hamiltonian scheme from the already available Lagrangian scheme or Newton's equations of motion (finding the momenta and the Hamiltonian function is sometimes very tedious, and in some cases, impossible without further complications called constraints). If the task is to obtain some equations of motion, often the Lagrangian formalism or Newtonian theory is easier to use and sufficient.