Canonical transformation vs symplectomorphism

[lriuui0x0]

I have read that canonical transformation is basically a symplectomorphism which leaves the symplectic form invariant. My understanding is that the canonical transformation is a passive picture where we keep the point on the phase space fixed and change the coordinate chart, where symplectomorphism is the active picture where we map the point to a different point using the same chart.

My question is couldn't symplectomorphism map a point outside of a chart? How do we make sense of such transformation as a coordinate change?

 

[ergospherical]

It's true that canonical transformations $f$ preserve the symplectic 2-form $\omega$ in the sense that $f^* \omega = \omega$ (that's the defining feature of such a transformation!). What does the question mean?

My question is couldn't symplectomorphism map a point outside of a chart?

 

[andresB]

I have read that canonical transformation is basically a symplectomorphism which leaves the symplectic form invariant. My understanding is that the canonical transformation is a passive picture where we keep the point on the phase space fixed and change the coordinate chart, where symplectomorphism is the active picture where we map the point to a different point using the same chart.

Citation please?

Usually, what I've read is that a canonical transformation can be interpreted in a pasive way (you are changing the label of the points) or in an active way (you are moving the points). Both are canonical transformations.

The same way that a rotation can be understood by saying that the points are fixed and the coordinate axes are rotating, or by saying that the points are rotating and the axes are fixed. They are both a rotation (and rotations are CT, btw).

 

[lriuui0x0]

Usually, what I've read is that a canonical transformation can be interpreted in a pasive way (you are changing the label of the points) or in an active way (you are moving the points). Both are canonical transformations.

I'm just trying to understand in what sense symplectomorphism is canonical transformation. My understanding is the following. When you have a symplectomorphism, that's an active picture of the transformation which maps a point on the phase space to a different point. If we have a coordinate chart that contains both the old and the new point, we see the new point will have a new set of coordinate values. This is how I translate the symplectomorphism to coordinate language, which I think is the active canonical transformation you talked about.

Now what happens if this function maps a point outside of the range of a chart? If this is the case, there's no common chart that contains both the old point and the new point, how do I know about the coordinate changes?

 

[andresB]

Now what happens if this function maps a point outside of the range of a chart? If this is the case, there's no common chart that contains both the old point and the new point, how do I know about the coordinate changes?

It is not something I've encountered in practice. But in any case, your phase space should be equipped with an atlas. If your point goes outside a given chart, you just use a different chart (from the same atlas) to label it.

 

[lriuui0x0]

It is not something I've encountered in practice. But in any case, your phase space should be equipped with an atlas. If your point goes outside a given chart, you just use a different chart (from the same atlas) to label it.

Hmmm... I fee it's weird that the old & new coordinates are described by different charts, since they are completely not comparable that way. The transformation between the coordinates would not reflect the symplectomorphism itself, but also a random chart transition function.

So is the answer basically "you can always find a chart containing $x$ and $f(x)$ for the symplectomorphism $f$ that's worth considering"?

 

[andresB]

Hmmm... I fee it's weird that the old & new coordinates are described by different charts, since they are completely not comparable that way. The transformation between the coordinates would not reflect the symplectomorphism itself, but also a random chart transition function.

So is the answer basically "you can always find a chart containing $x$ and $f(x)$ for the symplectomorphism $f$ that's worth considering"?

Well, yes. Given x and f(x) there is always a chart that contains both (as long as the phase space is connected)

But I also don't understand your objections to using transition functions.