Translation of Marsden-Weinstein-Meyer into classical mechanics language The Marsden-Weinstein-Meyer theorem is expressed in a too general way to be understood by a mean square physicist, as me. However, if we limit the scope to a Hamiltonian mechanics, it should be possible to express it in the Hamiltonian language, at least to some extent.
So my question is what the M.W.M. theorem says on the following problem. I have a Hamiltonian $H$, and I know that it is in involution with another function $J$, i.e. $\{H,J\}=0$. 
1) What are the additional hypothesis that must be met by $H$ and $J$, so that the M.W.M. theorem can be applied?
2) Under those hypothesis, what is the thesis?
But: since the problem is expressed in simple and practical terms, also the answer must be absolutely practical. Derivatives, Poisson parenthesis, solutions of differential equations are allowed; everything that can be followed by a physicists is allowed; but no Lie group, no symplectomorphism, no coadjoint action is allowed in the answer. Where this is not possible, examples should be given.
Literature references are also welcome!
Edit: I already asked related questions, which are however different from this one. Here, I'm asking what are exactly the hypothesis and the thesis of MWM limited to a specific situation. Not a general explanation of reductions. In particular, I would like to know: what are the hypothesis on $J$, if the result is globally valid, and what is the relation with Poincarè reduction.
 A: 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 $h(p_1,q_1)=p_1^2+q_1^2+J_0^2$. 
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.  
A: I’m not sure if this is what you are looking for, but you can think of this as fixing the values of conserved quantities. You have a function on phase space which commutes with the
Hamiltonian using the Poisson bracket. Therefore, the value of this function is conserved. Pick a value, and you can describe the equations of motion using one fewer variables. 
As a simple example, take the cotangent bundle to $S^1 \times \mathbb{R}$ with the usual Hamiltonian. Then, $J$ is the momentum along the circle, which is conserved. Fix that to the value $j$. Then the motion of a particle can be described solely using the phase space $T^*\mathbb{R}$ with the Hamiltonian $\frac{p^2 + j^2}{2m}$. This is arises from the full phase space as the space $J^{-1}(j)/U(1)$, where the quotient arises because we have fully described the motion in the circle direction. This is precisely the quotient space in the theorem. 
