# Connectedness & Canonical Transformations

• kakarukeys
In summary, connectedness in the context of canonical transformations refers to the idea that a set of points in phase space remains connected under the transformation. Canonical transformations preserve the Hamiltonian by preserving the Poisson bracket structure, ensuring that the transformed equations of motion have the same form as the original equations. A symplectic transformation preserves the symplectic structure, while a canonical transformation also preserves the Hamiltonian. Liouville's theorem states that the volume of a region in phase space remains constant under a Hamiltonian flow, so canonical transformations cannot change the phase space volume. Canonical transformations are related to the principle of least action through the use of generating functions, which connect the mathematical formalism with the physical principle.
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?

Isn't a canonical transformation continuous?

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.

## 1. What is connectedness in the context of canonical transformations?

Connectedness refers to the idea that a set of points in phase space remains connected under a canonical transformation. This means that the transformation does not result in any "jumps" or discontinuities in the physical system.

## 2. How do canonical transformations preserve the Hamiltonian?

Canonical transformations preserve the Hamiltonian by preserving the Poisson bracket structure. This means that the transformed equations of motion will have the same form as the original equations of motion, and thus the Hamiltonian will be conserved.

## 3. What is the difference between a symplectic and a canonical transformation?

A symplectic transformation is a transformation that preserves the symplectic structure of a system, while a canonical transformation is a special type of symplectic transformation that also preserves the Hamiltonian. All canonical transformations are symplectic, but not all symplectic transformations are canonical.

## 4. Can canonical transformations change the phase space volume?

No, canonical transformations cannot change the phase space volume. This is a consequence of Liouville's theorem, which states that the volume of a region in phase space remains constant under a Hamiltonian flow.

## 5. How are canonical transformations related to the principle of least action?

Canonical transformations are related to the principle of least action through the concept of a generating function. The generating function is a mathematical tool used to find canonical transformations that correspond to physical systems with a given Hamiltonian. The principle of least action is a physical principle that dictates the path taken by a system in order to minimize the action, which is a function of the system's position and velocity. The generating function allows us to connect the mathematical formalism of canonical transformations with the physical principle of least action.

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