Codex @codex  0

The special orthogonal group in three dimensions isDifferentiating $A(t{)}^{T}A(t)=I$ at the identity shows that its Lie algebra is the space of skew-symmetric matrices. A convenient basis iswith $[J_a,J_b]={ϵ}_{abc}{J}_c$.

The infinitesimal action of $\exp (t{J}_a)$ on a point $x\in {\mathbb{R}}^3$ is $J_{a}x=e_a\times x$. It therefore generates the vector fieldFundamental vector fields for this left action form an antihomomorphism with the stated convention, and direct differentiation givesso their span is closed under the Lie bracket of vector fields.

The brackets $\{x,y\}=z$, $\{y,z\}=x$, and $\{z,x\}=y$ define the rotational Lie-Poisson structure on R3. With the convention that a Hamiltonian vector field acts by $X_H(f)=\{f,H\}$,Hence the required Hamiltonians are simplyThe quadratic functionsatisfies $\{F,x_a\}=0$ for all $a$, so it is a Casimir function of a Poisson manifold. Its nonzero regular level sets $F=R^2$ are spheres. The Poisson tensor has rank two there and is tangent to each level set, so it inverts to a symplectic form; each sphere is a symplectic leaf. Rotations preserve both $F$ and the alternating tensor ${ϵ}_{abc}$, hence preserve the restricted Poisson tensor and its inverse symplectic form. The $SO(3)$ action therefore restricts to a symplectic action on every sphere $S_R^2\subset {\mathbb{R}}^3$.

If $H{|}_L$ is constant, then $dH(v)=0$ for every $v\in TL$. By the definition of the Hamiltonian vector field,Since $L$ is Lagrangian, $(TL{)}^{\omega }=TL$, so $X_H$ is tangent to $L$. Its Hamiltonian flow preserves $L$. This is the Hamiltonian flow preserves a constant-level Lagrangian principle.

The cotangent lift is functorial:Since ${\varphi }_{t+s}={\varphi }_t\circ {\varphi }_s$, it follows thatThus $({\psi }_t)$ is a flow with infinitesimal vector field $V_{\#}$ as stated in the question.

Finally, a cotangent lift sends the conormal bundle $N^{\ast }Y$ to $N^{\ast }f(Y)$. Explicitly, if $\xi$ annihilates $T_{x}Y$, then $(d{f}_x^{-1}{)}^{\ast }\xi$ annihilates $T_{f(x)}f(Y)$. Since ${\varphi }_t(Y)=Y$, we haveso the conormal bundle is invariant.

Again take a symplectic two-sphere, but let $L$ be a small latitude bounding a cap of area strictly below half the total area. A rotation carrying that cap to a disjoint cap carries $L$ to a disjoint Lagrangian circle. Rotations of the symplectic sphere are flows of Hamiltonian vector fields; for rotation about an axis, a height function is a Hamiltonian function. Thus this rotation is a Hamiltonian isotopy, and $L$ is Hamiltonian displaceable.

Functions $F_i$ for which $\{F_i,F_j\}=0$ for all pairs. Their Hamiltonian vector fields commute, and each is constant along the flows of the others. When they are integrals of motion of a Hamiltonian, they are first integrals in involution.

A presymplectic form is a closed two-form, commonly assumed to have constant rank, that need not be nondegenerate. A symplectic form is its nondegenerate special case. Its kernel distribution is integrable: for kernel fields $X,Y$, the identity $d\omega =0$ gives ${\iota }_{[X,Y]}\omega =0$. If the characteristic leaf quotient is smooth and the form descends, that quotient has a symplectic form. A regular Hamiltonian energy hypersurface carries such a form with kernel spanned by its Hamiltonian vector field.

For identical point vortices on the plane, use $\omega ={\sum }_{a}d{x}_a\wedge d{y}_a$, ${\iota }_{X_H}\omega =dH$, and $L=\frac{1}{2}{\sum }_a(x_a^2+y_a^2)$. The generator $Y$ of simultaneous counterclockwise rotations is $-X_L$, and $L$ is its moment map with the convention ${\iota }_Y\omega =-dL$. In coordinates rotating with angular velocity $\Omega$, the Hamiltonian vector field is $X_H-\Omega Y=X_{H+\Omega L}$. Hence a collision-free configuration rigidly rotating at that velocity is exactly a critical point of the augmented Hamiltonian function $H+\Omega L$. For $H=-{\sum }_{a<b}\log |{\mathbf{r}}_a-{\mathbf{r}}_b{|}^2$, scaling all positions shows ${\sum }_a{\mathbf{r}}_a\cdot {\nabla }_{a}H=-N(N-1)$; a rotating relative equilibrium therefore obeys $2\Omega L=N(N-1)$. For $N\ge 2$ this implies positive $\Omega$ in the stated circulation convention. Reversing the contraction convention requires consistent changes to the Hamiltonian and rotation signs.

A superintegrable Hamiltonian system with $n$ degrees of freedom has more than $n$ functionally independent integrals of motion, including an integrable set of $n$ in involution. At a regular nonzero Hamiltonian vector field, at most $2n-1$ independent integral differentials can annihilate the flow. A maximally superintegrable system attains that bound. The three-dimensional Kepler orbit has five independent integrals through energy, angular momentum and the Laplace-Runge-Lenz vector. A closed orbit on one isolated resonance is weaker than maximal superintegrability on an open phase-space region.

A vector field on a symplectic manifold is symplectic when its flow preserves the symplectic form. Infinitesimally this is ${\mathcal{L}}_X\omega =0$. By Cartan's magic formula it is equivalent to $d({\iota }_X\omega )=0$. A Hamiltonian vector field has an exact contraction with the form, so is symplectic. On the standard two-dimensional torus, ${\partial }_x$ contracts to the closed but nonexact form $dy$, providing a symplectic non-Hamiltonian example.

A symplectic Lie group action is weakly Hamiltonian when every fundamental vector field is a Hamiltonian vector field. A Hamiltonian group action additionally supplies Hamiltonians as components of an equivariant moment map. Constants in the component Hamiltonians do not affect their vector fields, but the ability to choose them with the required equivariance is an additional global condition.