Let $q=\mathbf{k}\cdot \mathbf{n}$, $E=e^{iq(\tau -{\tau }_0)}$ and $T=h_S^{\prime }/(2k^2)$. The synchronous-to-Newtonian transformation with positive spatial perturbations gives $(h^{\prime }-h_S^{\prime })/6={\Phi }^{\prime }+{\Psi }^{\prime }+T^{\prime \prime }$. Since $[E(T^{\prime }-iqT){]}^{\prime }=E(T^{\prime \prime }+q^{2}T)$, the displayed identity follows by integration. The emission endpoint combines intrinsic and velocity shifts into the ordinary Sachs-Wolfe effect; the local observer endpoint is a monopole plus a dipole. Without anisotropic stress the remaining integral is the Integrated Sachs-Wolfe effect, $2\int E{\Phi }^{\prime }d\tau$. The angular term in the synchronous contraction is positive for this metric convention.

Facing a companion and facing the source of light are different relations. The Moon keeps its near side of the Moon approximately toward Earth, while the direction to the distant Sun is nearly fixed during the idealized orbit shown below. A fixed surface patch therefore turns toward, across and away from sunlight. The far side of the Moon has day and night too; its name does not mean permanent darkness. ^[ref].

If the Sun direction has fixed angle $\alpha$ and the marked surface patch has outward normal vector at angle $\theta$, its direct illumination is proportional to $\max (0,\cos (\theta -\alpha ))$. With synchronous rotation, $\theta =f+\pi$, so the patch goes through day and night even though it always faces Earth. In reality the Earth-Moon system advances around the Sun: the lunar spin and orbital cycle relative to distant stars is about $27.3$ Earth days, while the lunar phase cycle has a synodic period of about $29.5$ Earth days. These numbers describe the actual system, not the fixed-sunlight cartoon. ^[ref].

For a synchronously rotating exoplanet whose orbital companion is itself the illuminating star, the direction to the star turns with the orbit. On an ideal circular orbit, with perpendicular spin axis and no additional oscillations, this direction is fixed relative to the surface, giving persistent day and night hemispheres. Day-night heat redistribution then helps determine the nightside temperature; permanent lack of direct starlight does not imply zero temperature. Orbital eccentricity and departures from this ideal spin geometry modify that simple picture.

View a circular orbit along its perpendicular spin axis. Let $f$ be the orbital angle measured from a fixed direction, choosing an arbitrary zero for the circular limit of true anomaly, and let $\theta$ be the direction of an outward arrow painted on the body, measured from the same fixed direction. The direction from the body to its companion is $f+\pi$. Thus an inward-facing arrow satisfiesMore generally synchronous rotation keeps $\theta -f$ constant; its particular value selects which surface direction faces the companion. Over one orbital period $T$, both $f$ and the unwrapped $\theta$ increase by $2\pi$. This is the short geometric explanation for one turn per orbit. For a nonspinning body, $\dot{\theta }=0$ instead, and the companion direction sweeps through all surface longitudes. On an eccentric orbit, mean $1:1$ spin does not force $\theta -f$ to stay exactly constant: true anomaly advances nonuniformly, producing the longitudinal component of libration.

This group consists of time-one maps of Hamiltonian isotopies whose flows are defined throughout the specified time interval. With ${\iota }_{X_H}\omega =-dH$, composition of isotopies generated by $H_t,K_t$ has generator $H_t+K_t\circ {\varphi }_t^{-1}$, where ${\varphi }_t$ is the first isotopy. The inverse path has generator $-H_t\circ {\varphi }_t$. These formulas prove closure under composition and inversion. On a closed symplectic manifold, every smooth time-dependent Hamiltonian has a complete flow over a bounded time interval.

A subspace of a symplectic vector space is symplectic if the restricted symplectic form is nondegenerate. It has a symplectic orthogonal complement and the ambient space is the direct sum of those two subspaces.

A linear map $A:(V,{\omega }_V)\to (W,{\omega }_W)$ is symplectic when it pulls back the target symplectic form to the source form. Nondegenerate bilinear form makes such a map injective. In equal finite dimensions it is an isomorphism. The automorphisms form the symplectic group and preserve the symplectic orientation on the ambient space and on every symplectic subspace.

A rank-$n$ vector subbundle of a rank-$2n$ symplectic vector bundle is Lagrangian when each fibre is a Lagrangian subspace. If it exists, a compatible almost complex structure supplies a complementary Lagrangian subbundle by applying $J$. Existence is a global condition: the area-oriented tangent bundle of the two-sphere has no real line subbundle, by triviality of real line bundles over the sphere and the Hairy ball theorem.

For $H={\sum }_j(q_j^2+p_j^2)/2$ and $E>0$, the oscillator flow is a free circle action on the sphere of radius $\sqrt{2E}$. Its quotient is Complex projective space. For two modes, the reduced form in $w=z_2/z_1$ is $iEdw\wedge d\bar{w}/(1+|w{|}^2{)}^2$, of total area $2\pi E$. Reduction at zero energy is singular instead of a smooth quotient of the asserted dimension.

For a planar cotangent bundle, rotation has moment map $\ell =x{p}_y-y{p}_x$. At $\ell \ne 0$ its level is regular and the circle action is free. Polar canonical coordinates give $\omega =dr\wedge d{p}_r+d\theta \wedge d\ell$. Fixing $\ell$ and quotienting $\theta$ leaves $T^{\ast }{\mathbb{R}}_{>0}$ with the displayed form. The unit-mass free Hamiltonian becomes $p_r^2/2+{\ell }^2/(2r^2)$.

At a regular free circle group level of a moment map, the inverse image of a Lagrangian submanifold of the reduced space is a Lagrangian submanifold of the original space. Its restricted symplectic form is the pullback of the reduced form and hence vanishes. Its dimension is one more than that of the reduced Lagrangian, exactly half the ambient dimension. It is a circle bundle over the original Lagrangian.

For a line $⟨e⟩$, complete $(e,f)$ to a symplectic basis with complementary symplectic space $W$. The unipotent radical has elements $u(b,t)$ sending $e$ to $e$, $w$ to $w-B(w,b)e$ and $f$ to $f+b+te$. Its multiplication is $(b,t)(b^{\prime },t^{\prime })=(b+b^{\prime },t+t^{\prime }+B(b,b^{\prime }))$, giving order $q^{2m-1}$. Its commutator is central with parameter $2B(b,b^{\prime })$. It is Heisenberg-type in odd characteristic and abelian in characteristic two. The full stabilizer also scales $e,f$ inversely and acts symplectically on $W$.

With a Lagrangian subspace first in a paired symplectic basis, its stabilizer consists of $\left(\begin{array}{cc}A& AS\\ 0& A^{-\mathsf{T}}\end{array}\right)$ with $S$ symmetric. The additive symmetric-matrix unipotent radical has order $q^{m(m+1)/2}$; the Levi subgroup is $G{L}_m(q)$, acting by $S\mapsto AS{A}^{\mathsf{T}}$. Its index in $S{p}_{2m}(q)$ is ${\prod }_{i=1}^m(q^i+1)$, the number of Lagrangian subspaces.

The scalar center has order $\gcd (2,q-1)$. The quotient is simple except at $(m,q)=(1,2),(1,3),(2,2)$. The last exception is $S{p}_4(2)\cong S_6$.

This map preserves the alternating form and has inverse $T_{v,-c}$. Transvections along a fixed line form an abelian normal subgroup of its stabilizer. They generate the symplectic group by aligning successive symplectic basis pairs.

For $q>3$, embedded perfect $S{L}_2(q)$ groups contain every symplectic transvection. For $q=3,m\ge 2$, four coefficient-one transvections on the lines of an isotropic plane multiply to the identity; their abelianized class is annihilated by 3 and 4. For $q=2,m\ge 3$, seven transvections on an isotropic three-space multiply to the identity; their class is annihilated by 2 and 7. Thus all nonexceptional cases are perfect.

Let $n\ge 2$ be even and let $E\subseteq {\mathbb{F}}_2^n$ be the vector subspace of even-weight vectors. Its dot product is an alternating bilinear form, since $x\cdot x={\sum }_i{x}_i=0$ on $E$. The orthogonal complement of $E$ in ${\mathbb{F}}_2^n$ is spanned by the all-one vector $1$, which lies in $E$. Consequently the radical of a bilinear form on $E$ is $⟨1⟩$, andis a nondegenerate symplectic vector space of dimension $n-2$. The symmetric group acts by coordinate permutations, preserving this alternating bilinear form.

At $n=6$, this group action is faithful. A weight-two vector and its complement have weights two and four, so a permutation fixing its quotient class must fix that two-element subset itself. Fixing every two-element subset forces the identity. The resulting embedding $S_6\hookrightarrow S{p}_4(2)$ is an isomorphism, because the symplectic group over a finite field order formula gives $2^4(2^2-1)(2^4-1)=720=6!$.

The standard symplectic cylinder restricts one conjugate coordinate pair to a disk and leaves the remaining symplectic coordinates unrestricted. Its normalized symplectic capacity is $\pi R^2$, despite its infinite symplectic volume when $n>1$.

For $U^{T}JU=J$ with real orthogonal skew-symmetric $J$, the dual $\tilde{\rho }=J{\rho }^T{J}^{†}$ transforms as $\widetilde{U\rho U^{†}}=U\tilde{\rho }{U}^{†}$. Hence $I_J(\rho )$ is constant on the reachable unitary orbit of a density operator. Different values can obstruct density operator controllability even when the ordinary spectra agree. For a four-dimensional $J$ pairing indices $1,4$ and $2,3$, the two diagonal arrangements $(a,a,b,b)$ and $(a,b,b,a)$ have invariants $4ab$ and $2(a^2+b^2)$.

For a closed $2n$-dimensional symplectic manifold, a symplectic blowup of size $\epsilon =\pi R^2$ removes the symplectic volume of the radius-$R$ ball. Its exceptional divisor has lower dimension and contributes no ambient volume. The displayed volume loss shows that different size choices can give forms that are not related by a symplectomorphism.

For a symplectic blowup of a $2n$-dimensional symplectic manifold, the displayed formula gives its First Chern class. The exceptional divisor has normal bundle $\mathcal{O}(-1)$; the discrepancy coefficient is $n-1$. In dimension four it is one.