# What is the difference between configuration space and phase space?

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?

• Lagrangians are defined over the tangent bundle to the configuration space. The phase space is just the cotangent bundle to the configuration space. The bridge between the two is the natural symplectic structure, and the relation between the Lagrangian and Hamiltonian is a Legendre transform. May 23, 2015 at 18:35
• May 23, 2015 at 18:42
• @Phoenix87 that's spot on, should post as an answer.
– zzz
May 23, 2015 at 20:47
• Take a look at this. Oct 20, 2016 at 22:39

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.

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.