Connectedness & Canonical Transformations

[kakarukeys]

This is a question I found no answer from books.

Is Connectedness and Simply-connectedness preserved by Canonical Transformations?

If an area in phase space is connected (simply connected), will it still connected (simply connected) in the new phase space of new canonical variables?

 

[Palindrom]

Isn't a canonical transformation continuous?

 

[GSXR750]

Connectedness and simply-connectedness are both topological properties that describe the structure of a space. In the context of canonical transformations, these properties refer to the structure of phase space.

A canonical transformation is a change of variables in phase space that preserves the Hamiltonian equations of motion. This means that the trajectories of a system in the original phase space will be transformed into new trajectories in the new phase space, but the dynamics of the system will remain the same.

Connectedness refers to the property of a space where any two points can be connected by a continuous path. Simply-connectedness is a stronger condition, where any closed path in the space can be continuously deformed into a point. These properties are important in physics as they determine the behavior of physical systems, such as the existence of closed orbits.

In general, canonical transformations do not preserve connectedness or simply-connectedness. This is because a change of variables can introduce new boundaries or holes in the phase space, thus changing its topological structure. For example, a canonical transformation that involves a nonlinear transformation of coordinates can result in a phase space with new boundaries or holes.

However, there are certain special cases where connectedness and simply-connectedness are preserved by canonical transformations. One such case is when the transformation is a linear transformation of coordinates. In this case, the phase space will remain connected and simply-connected.

In summary, the preservation of connectedness and simply-connectedness by canonical transformations depends on the specific transformation being applied. In general, these properties are not preserved, but there are special cases where they are.