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The first example where we cannot use global Darboux coordinates to work this out is the harmonic oscillator $J=(1/2)(p_1^2+q_1^2+\dots+p_n^2+q_n^2)$ where $X=\mathbb{R}^{2n}$ with usual Darboux coordinates. The regular values $J_0$ of $J$ are any nonzero values. The level sets $X_{J_0}$ of $J$ are spheres: $J=J_0$ is a sphere of radius $\sqrt{2J_0}$. The quotient space $Y$ of flow lines is a complex projective space: if we take $z_j=p_j+\sqrt{-1}q_j$, then there is a unique flow line on $X_{J_0}$ for each complex line spanned by a vector $z=(z_1,\dots,z_n)$. The symplectic structure on the complex projective space is the famous Fubini--Study symplectic structure. See Arnold, Mathematical Methods of Classical Mechanics, p. 24, for this example with $n=2$, and Appendix 3 for the general construction of this symplectic structure.

In cases when your Lie group is 1-dimensional and simple connected, i.e. the real number line, i.e. when there is precisely one function $J$ as the moment map, i.e. the cases you want to know about, then MWM is Poincare reduction: change variables locally to get $J=p_n$, and then $H$ turns out not to depend on $q_n$, and $p_n$ is constant along flow of $H$, so on level sets of $p_n$, $H$ reduces to one fewer variable. I don't know a reference for this.

A regular point $x_0$ of a function $y=f(x)$ is a point at which at least one partial derivative $\partial f/\partial x_i$ is not zero. A regular value $y_0$ of a function $y=f(x)$ is a point so that every point $x_0$ at which $f(x_0)$ is equal to $y_0$ is a regular point. By a theorem of Sard, almost every value of a smooth function is a regular value.

Take a Hamiltonian function $H$ on a symplectic manifold $X$, i.e. a Hamiltonian system with Hamiltonian $H$. If we pick a regular value $J_0$ of $J$, then the level set $X_{J_0}\subset X$, i.e. the set of points where $J=J_0$, is a submanifold of $X$ invariant under the Hamiltonian flow of $J$. Suppose that the set of flow lines are parameterized by a smooth manifold $Y$ of dimension one less than the dimension of the level set. Let $\varphi\colon X_{J_0} \to Y$ be the map taking each point $x\in X_{J_0}$ to the flow line of $J$ through that point $x$. Then there is a function $h$ on $Y$, so that $H(x)=h(\varphi(x))$ for any point $x\in X_{J_0}$. (We say that $H$ descends down to $Y$, and write $h$ as $H$.) This $h$ is a Hamiltonian of a Hamiltonian system on that smooth manifold $Y$, for a natural symplectic structure. Simplest example: if $J=p_n$ in Darboux coordinates, i.e. $H$ is independent of $q_n$, then we can use coordinates $q_1,\dots,q_{n-1},p_1,\dots,p_{n-1}$ for that quotient manifold. In that case, we can write $H(p_1,\dots,p_n,q_1,\dots,q_n)$ as a function $h(p_1,\dots,p_{n-1},q_1,\dots,q_{n-1})$. (I am hoping that you are familiar with these $p$ and $q$ Darboux coordinates.)

A symplectic structure on $Y$ means that there is some way to set up a Hamiltonian system on $Y$, but a precise definition requires a familiarity with differential forms or some other mathematical structure which I can't perhaps give you. The point is then that $\varphi\colon X_{J_0} \to Y$ takes Hamiltonian paths of $H$ on $X$ (which, when they start on $X_{J_0}$, always stay on $X_{J_0}$) to Hamiltonian paths of $h$ on $Y$, for the associated Hamiltonian system with $h$ as Hamiltonian function.

It is true, as Michael says, that this essentially says that ignorable things are ignorable, i.e. that if you don't use $q_i$, you can skip using $p_i$ too. If you would like an easy way to see that, note that near any regular point of a function $J$, there are Darboux coordinates in which $J=p_n$. So that reduce the local picture to the study of Hamiltonians $H$ for which $\{p_n,H\}=0$, and this you can work out trivially by hand.

Let me give an example, as the result is perhaps still not clear. If $H(p_1,q_1,p_2,q_2)=p_1^2+q_1^2+p_2^2$, and $J(p_1,q_1,p_2,q_2)=p_2$, then $h(p_1,q_1)=p_1^2+q_1^2+J_0^2$.

Looking back at this answer, it doesn't belong on mathoverflow, and I don't think the question does either.

In cases when your Lie group is 1-dimensional and simple connected, i.e. the real number line, i.e. when there is precisely one function $J$ as the moment map, i.e. the cases you want to know about, then MWM is essentially Poincare reduction: locally change variables to get $J=p_n$, and then $H$ turns out not to depend on $q_n$, and $p_n$ is constant along flow of $H$, so on level sets of $p_n$, $H$ reduces to one fewer variable. However, the MWM story is not purely local. MWM requires, even here, global hypotheses, and gives a global conclusion. The function $J$ perhaps cannot be globally made into $p_n$, since Darboux coordinates are only local. However, if the flow lines of $J$ on a level set of $J$ can be parameterized by a smooth manifold, we can make a global statement as below. I don't know a reference for this.

A regular point $x_0$ of a function $y=f(x)$ is a point at which at least one partial derivative $\partial f/\partial x_i$ is not zero. A regular value $y_0$ of a function $y=f(x)$ is a point so that every point $x_0$ at which $f(x_0)$ is equal to $y_0$ is a regular point. By a theorem of Sard, almost every value of a smooth function is a regular value.

Take a Hamiltonian function $H$ on a symplectic manifold $X$, i.e. a Hamiltonian system with Hamiltonian $H$. If we pick a regular value $J_0$ of $J$, then the level set $X_{J_0}\subset X$, i.e. the set of points where $J=J_0$, is a submanifold of $X$ invariant under the Hamiltonian flow of $J$. Suppose that the set of flow lines are parameterized by a smooth manifold $Y$ of dimension one less than the dimension of the level set. Let $\varphi\colon X_{J_0} \to Y$ be the map taking each point $x\in X_{J_0}$ to the flow line of $J$ through that point $x$. Then there is a function $h$ on $Y$, so that $H(x)=h(\varphi(x))$ for any point $x\in X_{J_0}$. (We say that $H$ descends down to $Y$, and write $h$ as $H$.) This $h$ is a Hamiltonian of a Hamiltonian system on that smooth manifold $Y$, for a natural symplectic structure. 

Simplest example: if $J=p_n$ in global Darboux coordinates, i.e. $H$ is independent of $q_n$, then we can use coordinates $q_1,\dots,q_{n-1},p_1,\dots,p_{n-1}$ for that quotient manifold. In that case, we can write $H(p_1,\dots,p_n,q_1,\dots,q_n)$ as a function $h(p_1,\dots,p_{n-1},q_1,\dots,q_{n-1})$.

A symplectic structure on $Y$ means that there is some way to set up a Hamiltonian system on $Y$, but a precise definition requires a familiarity with differential forms or some other mathematical structure which I can't perhaps give you. The point is then that $\varphi\colon X_{J_0} \to Y$ takes Hamiltonian paths of $H$ on $X$ (which, when they start on $X_{J_0}$, always stay on $X_{J_0}$) to Hamiltonian paths of $h$ on $Y$, for the associated Hamiltonian system with $h$ as Hamiltonian function.

It is true, as Michael says, that this essentially says that ignorable things are ignorable, i.e. that if you don't use $q_i$, you can skip using $p_i$ too. If you would like an easy way to see that, note that near any regular point of a function $J$, there are Darboux coordinates in which $J=p_n$. So that reduce the local picture to the study of Hamiltonians $H$ for which $\{p_n,H\}=0$, and this you can work out trivially by hand.

Let me give an example, as the result is perhaps still not clear. If $H(p_1,q_1,p_2,q_2)=p_1^2+q_1^2+p_2^2$, and $J(p_1,q_1,p_2,q_2)=p_2$, then $h(p_1,q_1)=p_1^2+q_1^2+J_0^2$.

A regular point $x_0$ of a function $y=f(x)$ is a point at which at least one partial derivative $\partial f/\partial x_i$ is not zero. A regular value $y_0$ of a function $y=f(x)$ is a point so that every point $x_0$ at which $f(x_0)$ is equal to $y_0$ is a regular point. By a theorem of Sard, almost every value of a smooth function is a regular value.

Take a Hamiltonian function $H$ on a symplectic manifold $X$, i.e. a Hamiltonian system with Hamiltonian $H$. If we pick a regular value $J_0$ of $J$, then the level set $X_{J_0}\subset X$, i.e. the set of points where $J=J_0$, is a submanifold of $X$ invariant under the Hamiltonian flow of $J$. Suppose that the set of flow lines are parameterized by a smooth manifold $Y$ of dimension one less than the dimension of the level set. Let $\varphi\colon X_{J_0} \to Y$ be the map taking each point $x\in X_{J_0}$ to the flow line of $J$ through that point $x$. Then there is a function $h$ on $Y$, so that $H(x)=h(\varphi(x))$ for any point $x\in X_{J_0}$. (We say that $H$ descends down to $Y$, and write $h$ as $H$.) This $h$ is a Hamiltonian of a Hamiltonian system on that smooth manifold $Y$, for a natural symplectic structure. Simplest example: if $J=p_n$ in Darboux coordinates, i.e. $H$ is independent of $q_n$, then we can use coordinates $q_1,\dots,q_{n-1},p_1,\dots,p_{n-1}$ for that quotient manifold. In that case, we can write $H(p_1,\dots,p_n,q_1,\dots,q_n)$ as a function $h(p_1,\dots,p_{n-1},q_1,\dots,q_{n-1})$. (I am hoping that you are familiar with these $p$ and $q$ Darboux coordinates.)

A symplectic structure on $Y$ means that there is some way to set up a Hamiltonian system on $Y$, but a precise definition requires a familiarity with differential forms or some other mathematical structure which I can't perhaps give you. The point is then that $\varphi\colon X_{J_0} \to Y$ takes Hamiltonian paths of $H$ on $X$ (which, when they start on $X_{J_0}$, always stay on $X_{J_0}$) to Hamiltonian paths of $h$ on $Y$, for the associated Hamiltonian system with $h$ as Hamiltonian function.

It is true, as Michael says, that this essentially says that ignorable things are ignorable, i.e. that if you don't use $q_i$, you can skip using $p_i$ too. If you would like an easy way to see that, note that near any regular point of a function $J$, there are Darboux coordinates in which $J=p_n$. So that reduce the local picture to the study of Hamiltonians $H$ for which $\{p_n,H\}=0$, and this you can work out trivially by hand.

Let me give an example, as the result is perhaps still not clear. If $H(p_1,q_1,p_2,q_2)=p_1^2+q_1^2+p_2^2$, and $J(p_1,q_1,p_2,q_2)=p_2$, then $h(p_1,q_1)=p_1^2+q_1^2+J_0^2$.

Looking back at this answer, it doesn't belong on mathoverflow, and I don't think the question does either.

In cases when your Lie group is 1-dimensional and simple connected, i.e. the real number line, i.e. when there is precisely one function $J$ as the moment map, i.e. the cases you want to know about, then MWM is Poincare reduction: change variables locally to get $J=p_n$, and then $H$ turns out not to depend on $q_n$, and $p_n$ is constant along flow of $H$, so on level sets of $p_n$, $H$ reduces to one fewer variable. I don't know a reference for this.

A regular point $x_0$ of a function $y=f(x)$ is a point at which at least one partial derivative $\partial f/\partial x_i$ is not zero. A regular value $y_0$ of a function $y=f(x)$ is a point so that every point $x_0$ at which $f(x_0)$ is equal to $y_0$ is a regular point. By a theorem of Sard, almost every value of a smooth function is a regular value.

Take a Hamiltonian function $H$ on a symplectic manifold $X$, i.e. a Hamiltonian system with Hamiltonian $H$. If we pick a regular value $J_0$ of $J$, then the level set $X_{J_0}\subset X$, i.e. the set of points where $J=J_0$, is a submanifold of $X$ invariant under the Hamiltonian flow of $J$. Suppose that the set of flow lines are parameterized by a smooth manifold $Y$ of dimension one less than the dimension of the level set. Let $\varphi\colon X_{J_0} \to Y$ be the map taking each point $x\in X_{J_0}$ to the flow line of $J$ through that point $x$. Then there is a function $h$ on $Y$, so that $H(x)=h(\varphi(x))$ for any point $x\in X_{J_0}$. (We say that $H$ descends down to $Y$, and write $h$ as $H$.) This $h$ is a Hamiltonian of a Hamiltonian system on that smooth manifold $Y$, for a natural symplectic structure. Simplest example: if $J=p_n$ in Darboux coordinates, i.e. $H$ is independent of $q_n$, then we can use coordinates $q_1,\dots,q_{n-1},p_1,\dots,p_{n-1}$ for that quotient manifold. In that case, we can write $H(p_1,\dots,p_n,q_1,\dots,q_n)$ as a function $h(p_1,\dots,p_{n-1},q_1,\dots,q_{n-1})$. (I am hoping that you are familiar with these $p$ and $q$ Darboux coordinates.)

A symplectic structure on $Y$ means that there is some way to set up a Hamiltonian system on $Y$, but a precise definition requires a familiarity with differential forms or some other mathematical structure which I can't perhaps give you. The point is then that $\varphi\colon X_{J_0} \to Y$ takes Hamiltonian paths of $H$ on $X$ (which, when they start on $X_{J_0}$, always stay on $X_{J_0}$) to Hamiltonian paths of $h$ on $Y$, for the associated Hamiltonian system with $h$ as Hamiltonian function.

It is true, as Michael says, that this essentially says that ignorable things are ignorable, i.e. that if you don't use $q_i$, you can skip using $p_i$ too. If you would like an easy way to see that, note that near any regular point of a function $J$, there are Darboux coordinates in which $J=p_n$. So that reduce the local picture to the study of Hamiltonians $H$ for which $\{p_n,H\}=0$, and this you can work out trivially by hand.

Let me give an example, as the result is perhaps still not clear. If $H(p_1,q_1,p_2,q_2)=p_1^2+q_1^2+p_2^2$, and $J(p_1,q_1,p_2,q_2)=p_2$, then $h(p_1,q_1)=p_1^2+q_1^2+J_0^2$.

Looking back at this answer, it doesn't belong on mathoverflow, and I don't think the question does either.