= Solution
Let $\mu:P\to\mathbb R$ be the <moment map> for the <circle group>, assume $c$ is a <regular value>, and assume the circle action on $N=\mu^{-1}(c)$ is free. The action is proper because the circle is compact. Then $N$ has dimension $2n-1$ and $P_c=N/SO(2)$ is a smooth <manifold> of dimension $2n-2$.
If $\xi_P$ generates the action, $\omega(\xi_P,v)=d\mu(v)=0$ for every $v\in TN$. In fact $(TN)^\omega$ is exactly the line spanned by $\xi_P$, since $TN=\ker d\mu$ is a hyperplane and $\omega$ is nondegenerate. Therefore the kernel of the restricted two-form is precisely the orbit direction. The restriction is invariant and horizontal, so it descends to a unique nondegenerate closed two-form $\omega_c$ with
$$
\boxed{\pi^*\omega_c=\iota^*\omega.}
$$
This proves the <Marsden-Weinstein theorem> in the circle case. If the <Hamiltonian> $H$ is invariant, then $\{\mu,H\}=0$: its <Hamiltonian flow> stays on $N$, projects to the quotient and is generated there by the descended function $H_c$. This is a reduced <Hamiltonian system> in $2n-2$ local variables. At a critical level or a nonfree orbit the quotient can be singular, so the smooth dimension assertion needs the stated hypotheses.
For a mass-$m$ particle in the plane with <central force> potential $V(r)$, work on $r>0$ and write
$$
p_xdx+p_ydy=p_rdr+\ell d\theta,\quad
p_r=\frac{xp_x+yp_y}{r},\quad \ell=xp_y-yp_x.
$$
Exterior differentiation gives $\omega=dr\wedge dp_r+d\theta\wedge d\ell$. Rotation translates $\theta$ and has <moment map> $\ell$. Fix $\ell=\ell_0$ and quotient out $\theta$ to obtain the <planar rotational symplectic reduction>
$$
\boxed{\omega_{\mathrm{red}}=dr\wedge dp_r,\qquad
H_{\mathrm{red}}=\frac{p_r^2}{2m}+\frac{\ell_0^2}{2mr^2}+V(r).}
$$
The reduced equations are $\dot r=p_r/m$ and $\dot p_r=\ell_0^2/(mr^3)-V'(r)$. The centrifugal term arises from the conserved angular momentum; it is not an extra force imposed by hand. Reconstruct the angle from $\dot\theta=\ell_0/(mr^2)$. For $\ell_0\ne0$ the origin cannot lie on the level; for $\ell_0=0$ the same local reduction holds away from the origin, while the full phase-space origin is a singular fixed orbit.
Back to article page
