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2 added 10 characters in body

I just wanted to expand two points of Greg's answer. Both are rather trivial additions, but took me a short while to understand, so I'm putting them here for completeness's sake and my own future reference.

First, here's a picture of Greg's idea of fixing the problem with Moser's trick. In his notation, we have volume forms $\mu_{\alpha,t} = \mu^t \mu_\alpha^{1-t}$. They are defined because the two volume form must have the same sign everywhere (since everything is orientation-preserving; note that because of this I doubt that this argument can be generalized to symplectic forms in higher dimensions).

The obvious, but wrong, solution (described in my comments to his answer) would be to apply Moser's theorem to $\mu_{\alpha,0}$ in this notation. This would correspond to flowing along the horizontal axes of the picture below. However, Greg's idea is to flow in a different direction: we fix $\alpha$ and vary $t$. Strictly speaking, we should also show that the resulting flow $\phi_{\alpha,1}$ will be smooth, but this should follow from the proof of Moser's theorem quite easily.

                                         Moser flow,
t        defines phi_alpha,t
(mu_{alpha,1} = mu)   1 ^          ^
|          .
|          .
|          .
|          .
(mu_{alpha,0} = mu_alpha)  0 ----------------> alpha
(phi_{alpha,0} = phi_alpha)


(Note that, for all t, $\mu_{0,t}=\mu_{1,t}=\mu$)

Secondly, Greg's answer implies that there is an isotopy $\psi_t: D^2 \to D^2$ such that each $\psi_t$ is volume preserving (or, equivalently, a symplectomorphism), $\psi_0$ is the identity, and $\psi_1=\psi$. Here's how we find the time-dependent Hamiltonian such that this is the Hamiltonian flow. Let Xt be the time-dependent vector field that is the derivative of the flow $\psi_t$. The fact our flow is volume-preserving is equivalent to the fact that the 1-form $\iota_{X_t} \omega$ is be closed for all $t$. Since we are on a disk, this form will also be exact. So, let $H_t$ be the function such that $dH_t = \iota_{X_t} \omega$. The Hamiltonian flow of the function $H_t$ is precisely $\psi_t$; this is immediate from the definitions.

1

I just wanted to expand two points of Greg's answer. Both are rather trivial, but took me a short while to understand, so I'm putting them here for completeness's sake and my own future reference.

First, here's a picture of Greg's idea of fixing the problem with Moser's trick. In his notation, we have volume forms $\mu_{\alpha,t} = \mu^t \mu_\alpha^{1-t}$. They are defined because the two volume form must have the same sign everywhere (since everything is orientation-preserving; note that because of this I doubt that this argument can be generalized to symplectic forms in higher dimensions).

The obvious, but wrong, solution (described in my comments to his answer) would be to apply Moser's theorem to $\mu_{\alpha,0}$ in this notation. This would correspond to flowing along the horizontal axes of the picture below. However, Greg's idea is to flow in a different direction: we fix $\alpha$ and vary $t$. Strictly speaking, we should also show that the resulting flow $\phi_{\alpha,1}$ will be smooth, but this should follow from the proof of Moser's theorem quite easily.

                                         Moser flow,
t        defines phi_alpha,t
(mu_{alpha,1} = mu)   1 ^          ^
|          .
|          .
|          .
|          .
(mu_{alpha,0} = mu_alpha)  0 ----------------> alpha
(phi_{alpha,0} = phi_alpha)


(Note that, for all t, $\mu_{0,t}=\mu_{1,t}=\mu$)

Secondly, Greg's answer implies that there is an isotopy $\psi_t: D^2 \to D^2$ such that each $\psi_t$ is volume preserving (or, equivalently, a symplectomorphism), $\psi_0$ is the identity, and $\psi_1=\psi$. Here's how we find the time-dependent Hamiltonian such that this is the Hamiltonian flow. Let Xt be the time-dependent vector field that is the derivative of the flow $\psi_t$. The fact our flow is volume-preserving is equivalent to the fact that the 1-form $\iota_{X_t} \omega$ is be closed for all $t$. Since we are on a disk, this form will also be exact. So, let $H_t$ be the function such that $dH_t = \iota_{X_t} \omega$. The Hamiltonian flow of the function $H_t$ is precisely $\psi_t$; this is immediate from the definitions.
The text passage is a detailed explanation of a mathematical concept, specifically an expansion on a proof or explanation provided by someone named Greg. It discusses volume forms, Moser's trick, and Hamiltonian flows, and includes a diagram to illustrate the concept of flowing in a different direction to fix a problem with Moser's trick.
no_question

the passage does not contain a mathematical question. it is an explanatory text that elaborates on a mathematical concept and provides additional details for understanding a specific proof or argument.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

2 added 10 characters in body

I just wanted to expand two points of Greg's answer. Both are rather trivial additions, but took me a short while to understand, so I'm putting them here for completeness's sake and my own future reference.

First, here's a picture of Greg's idea of fixing the problem with Moser's trick. In his notation, we have volume forms $\mu_{\alpha,t} = \mu^t \mu_\alpha^{1-t}$. They are defined because the two volume form must have the same sign everywhere (since everything is orientation-preserving; note that because of this I doubt that this argument can be generalized to symplectic forms in higher dimensions).

The obvious, but wrong, solution (described in my comments to his answer) would be to apply Moser's theorem to $\mu_{\alpha,0}$ in this notation. This would correspond to flowing along the horizontal axes of the picture below. However, Greg's idea is to flow in a different direction: we fix $\alpha$ and vary $t$. Strictly speaking, we should also show that the resulting flow $\phi_{\alpha,1}$ will be smooth, but this should follow from the proof of Moser's theorem quite easily.

                                         Moser flow,
t        defines phi_alpha,t
(mu_{alpha,1} = mu)   1 ^          ^
|          .
|          .
|          .
|          .
(mu_{alpha,0} = mu_alpha)  0 ----------------> alpha
(phi_{alpha,0} = phi_alpha)


(Note that, for all t, $\mu_{0,t}=\mu_{1,t}=\mu$)

Secondly, Greg's answer implies that there is an isotopy $\psi_t: D^2 \to D^2$ such that each $\psi_t$ is volume preserving (or, equivalently, a symplectomorphism), $\psi_0$ is the identity, and $\psi_1=\psi$. Here's how we find the time-dependent Hamiltonian such that this is the Hamiltonian flow. Let Xt be the time-dependent vector field that is the derivative of the flow $\psi_t$. The fact our flow is volume-preserving is equivalent to the fact that the 1-form $\iota_{X_t} \omega$ is be closed for all $t$. Since we are on a disk, this form will also be exact. So, let $H_t$ be the function such that $dH_t = \iota_{X_t} \omega$. The Hamiltonian flow of the function $H_t$ is precisely $\psi_t$; this is immediate from the definitions.

1

I just wanted to expand two points of Greg's answer. Both are rather trivial, but took me a short while to understand, so I'm putting them here for completeness's sake and my own future reference.

First, here's a picture of Greg's idea of fixing the problem with Moser's trick. In his notation, we have volume forms $\mu_{\alpha,t} = \mu^t \mu_\alpha^{1-t}$. They are defined because the two volume form must have the same sign everywhere (since everything is orientation-preserving; note that because of this I doubt that this argument can be generalized to symplectic forms in higher dimensions).

The obvious, but wrong, solution (described in my comments to his answer) would be to apply Moser's theorem to $\mu_{\alpha,0}$ in this notation. This would correspond to flowing along the horizontal axes of the picture below. However, Greg's idea is to flow in a different direction: we fix $\alpha$ and vary $t$. Strictly speaking, we should also show that the resulting flow $\phi_{\alpha,1}$ will be smooth, but this should follow from the proof of Moser's theorem quite easily.

                                         Moser flow,
t        defines phi_alpha,t
(mu_{alpha,1} = mu)   1 ^          ^
|          .
|          .
|          .
|          .
(mu_{alpha,0} = mu_alpha)  0 ----------------> alpha
(phi_{alpha,0} = phi_alpha)


(Note that, for all t, $\mu_{0,t}=\mu_{1,t}=\mu$)

Secondly, Greg's answer implies that there is an isotopy $\psi_t: D^2 \to D^2$ such that each $\psi_t$ is volume preserving (or, equivalently, a symplectomorphism), $\psi_0$ is the identity, and $\psi_1=\psi$. Here's how we find the time-dependent Hamiltonian such that this is the Hamiltonian flow. Let Xt be the time-dependent vector field that is the derivative of the flow $\psi_t$. The fact our flow is volume-preserving is equivalent to the fact that the 1-form $\iota_{X_t} \omega$ is be closed for all $t$. Since we are on a disk, this form will also be exact. So, let $H_t$ be the function such that $dH_t = \iota_{X_t} \omega$. The Hamiltonian flow of the function $H_t$ is precisely $\psi_t$; this is immediate from the definitions.
user
The text passage is a detailed explanation of a mathematical concept, specifically an expansion on a proof or explanation provided by someone named Greg. It discusses volume forms, Moser's trick, and Hamiltonian flows, and includes a diagram to illustrate the concept of flowing in a different direction to fix a problem with Moser's trick.

TYPE: NO_QUESTION

The passage does not contain a mathematical question. It is an explanatory text that elaborates on a mathematical concept and provides additional details for understanding a specific proof or argument.