My understanding is that classical configuration space as you mentioned is usually thought of as including dimensions for velocities, or even better, momenta, in addition to positions. For even two particles in a one dimensional space, that is already four dimensions and you can’t graph it, so I can understand why you showed it the way you did. However you can show one particle, perhaps in a force field, which can be useful.
The advantage of including momentum is that a single point in configuration space has all the information needed to calculate its evolution forward (or backward) in time. A single point determines an entire trajectory (a line, or curve) in configuration space. That means that two nearby points determine two different trajectories, and in fact all of configuration space can be divided into non-intersecting trajectory lines. Only in this formulation is the Liouville Theorem true, about conservation of configuration space volumes. If you start with a certain volume in configuration space, and evolve it forward (or again, backward) in time, the volume doesn’t change. However, in most classical configurations, physics tends to be chaotic and the shape does change as you describe, developing “fingers” and “folds” and becoming very complex in structure, which leads on a crude scale to an apparent increase in the volume.
This is exactly right except that the space in which Liouville’s theorem holds is called phase space. Phase space is the cotangent bundle over configuration space; i.e., if the configuration space is an n-dimensional manifold M, then for every point in M there is a copy of an n-dimensional vector space. These n-vectors* represent momenta, and both a configuration and a momentum are necessary to uniquely specify a state of a classical (Hamiltonian) system.
* More precisely, they are one-forms—linear functions of n-vectors; i.e. they eat n-vectors and spit out scalars. One-forms are also called covariant vectors, whence the other kind are called contravariant. They are dual to each other (for a given n), and thus (contravariant) vectors can equivalently be considered linear functions of one-forms instead.
I think it would be nice if the post were edited to reflect this distinction. It wouldn’t take much effort; just a sentence inserted at the point where it switches from talking about configuration space to talking about phase space, and appropriate tweaks to a few subsequent sentences. The Wikipedia article Configuration space links here, by the way.
To clarify (seven years later), “configuration space” is the name physicists use for the space recording just the particle’s positions, and “phase space” is the name for the space recording their positions and momentums.
My understanding is that classical configuration space as you mentioned is usually thought of as including dimensions for velocities, or even better, momenta, in addition to positions. For even two particles in a one dimensional space, that is already four dimensions and you can’t graph it, so I can understand why you showed it the way you did. However you can show one particle, perhaps in a force field, which can be useful.
The advantage of including momentum is that a single point in configuration space has all the information needed to calculate its evolution forward (or backward) in time. A single point determines an entire trajectory (a line, or curve) in configuration space. That means that two nearby points determine two different trajectories, and in fact all of configuration space can be divided into non-intersecting trajectory lines. Only in this formulation is the Liouville Theorem true, about conservation of configuration space volumes. If you start with a certain volume in configuration space, and evolve it forward (or again, backward) in time, the volume doesn’t change. However, in most classical configurations, physics tends to be chaotic and the shape does change as you describe, developing “fingers” and “folds” and becoming very complex in structure, which leads on a crude scale to an apparent increase in the volume.
This is exactly right except that the space in which Liouville’s theorem holds is called phase space. Phase space is the cotangent bundle over configuration space; i.e., if the configuration space is an n-dimensional manifold M, then for every point in M there is a copy of an n-dimensional vector space. These n-vectors* represent momenta, and both a configuration and a momentum are necessary to uniquely specify a state of a classical (Hamiltonian) system.
* More precisely, they are one-forms—linear functions of n-vectors; i.e. they eat n-vectors and spit out scalars. One-forms are also called covariant vectors, whence the other kind are called contravariant. They are dual to each other (for a given n), and thus (contravariant) vectors can equivalently be considered linear functions of one-forms instead.
I think it would be nice if the post were edited to reflect this distinction. It wouldn’t take much effort; just a sentence inserted at the point where it switches from talking about configuration space to talking about phase space, and appropriate tweaks to a few subsequent sentences. The Wikipedia article Configuration space links here, by the way.
To clarify (seven years later), “configuration space” is the name physicists use for the space recording just the particle’s positions, and “phase space” is the name for the space recording their positions and momentums.