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The Marsden-Weinstein theorem needs a Hamiltonian action, not merely an action by symplectomorphisms. Assume an equivariant moment map $\mu :P\to {\mathfrak{g}}^{\ast }$ exists, choose a coadjoint-fixed regular value $a$ (zero is the usual choice), and assume the action on $C={\mu }^{-1}(a)$ is free and proper. Regularity makes $C$ a submanifold of codimension $g=\dim G$. Equivariance and invariance of $a$ make $C$ invariant under $G$, and freeness and properness give a smooth quotient $P^{\prime }=C/G$. Thus

Let $\iota :C\hookrightarrow P$ and $\pi :C\to P^{\prime }$. For $v\in T_{p}C$ and $\xi \in \mathfrak{g}$,Moreover the conormal of $C$ is spanned by the independent $d{\mu }_{\xi }$, so nondegeneracy of $\omega$ gives $(T_{p}C{)}^{\omega }=\{{\xi }_P(p)\}$. The coadjoint-fixed value makes these orbit directions tangent to $C$. Hence the kernel of ${\iota }^{\ast }\omega$ is exactly the orbit tangent space. The restricted form is invariant and horizontal, so descends uniquely to a two-form ${\omega }^{\prime }$ withIt is closed because $\omega$ is closed. If a quotient tangent vector annihilates ${\omega }^{\prime }$, any lift lies in the kernel just identified and is vertical; the quotient vector is therefore zero. This proves nondegeneracy and completes symplectic reduction.

If the action is not free, singular strata or orbifold phenomena may occur. At a general noncentral regular value one quotients by its coadjoint stabilizer $G_a$, not by all of $G$; the regular dimension is $2n-g-\dim G_a$. These are genuine hypotheses behind the stated dimension formula.

Both proposed examples can be given explicitly. For symplectic reduction of an isotropic oscillator, take two modes withIts Hamiltonian flow is $z_j\mapsto e^{-it}{z}_j$, a circle action with moment map $H$. For $E>0$, $H^{-1}(E)$ is the sphere $S^3$ of radius $\sqrt{2E}$, and the action is free. The Hopf fibration identifies its quotient with ${\mathbb{CP}}^1\cong S^2$. On the chart $z_1\ne 0$, use $w=z_2/z_1$ and choose a section with $z_1=\sqrt{2E/(1+|w{|}^2)}$ real positive. Pulling back $\omega$ to this section givesWriting $w=x+iy$ gives $2Edx\wedge dy/(1+x^2+y^2{)}^2$, whose integral is $2\pi E$. This is the Fubini-Study form scaled by $E$ in the convention of area $2\pi$ at unit scale. The reduced dimension is $4-2=2$. At $E=0$, the circle fixes the origin and the smooth regular-level hypotheses fail.

For a unit-mass free particle in the plane, use the rotation circle action with angular-momentum moment map $\ell =x{p}_y-y{p}_x$. Choose ${\ell }_0\ne 0$ so the entire level avoids $r=0$ and the action is free. Polar canonical coordinates haveFixing $\ell ={\ell }_0$ and quotienting the rotation angle leavesThis planar rotational symplectic reduction replaces a planar free trajectory by radial motion with the centrifugal potential. A nonzero moment value is allowed because the circle is abelian and every value is coadjoint-fixed.