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\begin{center}\vspace{1in}{\bf MATH 209, Last HW,  WINTER 2020}\\
  Mostly,   basic symplectic geometry.  \end{center}

Below $S$ denotes a symplectic manifold with symplectic form  $\omega$.
  Darboux coordinates on $S$ are 
coordinates $q^i, p_i$ such that $\omega = \Sigma dq^i \wedge dp_i$.  
The canonical one form $\theta$ on $S = T^*Q$ is given in coordinates by $\Sigma p_i dq^i$, 
 has a  nice intrinsic definition, and satisfies $\omega = - d \theta$. 

\vskip .2cm

a) Define the `cotangent lift' of a diffeo $Q \to Q$
by $T^* \phi (q, p) = (\phi(q), (d \phi_q)  ^{-1 *} p)$
for $p \in T^* _q Q$.    Show that the cotangent lift of any diffeo 
preserves the canonical one-form.
\vskip .2cm
b)  Find a symplectic transformation of $T^* \R$ which is not the
cotangent lift of a diffeo of $\R$.
\vskip .2cm
c).    The Hamiltonian vector field associated to a function 
$H$ on a symplectic manifold $S$,  also called its symplectic gradient,
is defined by
$$dH =  \omega (X_H, \cdot)$$.

Show that in Darboux coordinates, the ODE  $\dot \gamma = X_H (\gamma)$
reads
$$ \ddot q^i = \dd{H}{p_i}$$
$$\ddot p_i = -\dd{H}{q^i}$$

This system of ODEs is  called ``Hamilton's equations'' for the Hamiltonian $H$. 
\vskip .2cm
d) Verify that the flow of a Hamiltonian vector field preserves the symplectic form:
$$L_{X_H} \omega = 0$$.

\vskip .2cm

POISSON BRACKETS

e)     The Poisson bracket of two functions on a symplectic manifold
is defined by $\{f, g \} = \omega(X_f,  X_g)$.  Verify that
the map $$f, g \mapsto \{ f,  g \}$$ gives the vector space of smooth functions
on $S$ the structure of a Lie algebra.    Verify that also
$$\{f , g h \} = g \{f, h \} + h \{ f, g \}$$ for any three smooth functions $f, g, h$.

\vskip .2cm
g) Compute the Poisson bracket $\{f, g\}$ in Darboux coordinates.

\vskip .2cm
h)  With $X_H$ as above, verify that $X_H [f]: = df(X_H) = \{f, H \}$.
Conclude that Hamilton's equations is equivalent to the evolution equation
$\frac{df}{dt} = \{f , H \}$ valid for all smooth functions $f$. 
\vskip .2cm

i)  Verify that the map $H \mapsto X_H$ is either a Lie-algebra homomorphism
or anti-homomorphism, between the Lie algebra of functions on $S$ under Poisson
bracket, and the Lie algebra of vector fields which preserve the symplectic form,
under Lie bracket of vector fields.

\vskip .2cm

CO-TANGENT SPECIALS

j)  For $X$ a vector field on $Q$ write $P_X: T^* Q \to \R$
for the same vector field, but viewed via duality as a   fiber-linear function on $T^* Q$.
Thus:
$$P_X (q, p) = p(X(q)).$$

Verify that the Hamiltonian flow of the Hamiltonian vector field for $P_X$ is the cotangent lift of the
flow of $X$ on $Q$. 

k) Verify that $\{P_X, P_Y \} = - P_{[X, Y]} $ for vector fields $X, Y$ on $Q$. 

Verify that $$\{P_X, \pi^* f \} = \pi^* X[f]$$ where $\pi: T^*Q \to Q$ is the projection and $f$ an arbitrary smooth function on $Q$.

Verify that $\{ \pi^* f, \pi^* g \} = 0$

\vskip .2cm

FUBINI-STUDY; 

L)  As we have seen in lectures ,  $\CP^n = \bP({\mathcal H})$ is a symplectic manifold
whose symplectic form, called the Kahler form,  is induced by the symplectic form on  $\cH$ which 
is the imaginary part of the Hermitian inner product.    Here $\cH$ is a Hilbert space of dimension $n+1$. 

L1)     For $A: \cH \to \cH$
a self-adjoint operator, verify that $H([\psi]) = \langle \psi, A \psi \rangle/\langle \psi, \psi \rangle $
is a smooth real valued function on $\CP^n$. 

L2) Use the spectral theorem to relate the eigenspaces of $A$ to the fixed points of the
Hamiltonian vector field $X_H$.

L3) For the case $dim(\cH) = 2$ and $A$ having two distinct eigenvalues
show that the flow of $X_H$ corresponds to rotation of the sphere $S^2 = \CP^1$.
Find a formula relating   the period  of this  rotation to the difference of the two  eigenvalues.
 
 
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