Topics in Geometric Mechanics: Week 9 Tutorial
$
%% not provided in align* environment
\def\newsavebox#1{}
\def\intertext#1{\text{#1}}
\def\let#1#2{}
\def\makeatletter{}
\def\makeatother{}
\newenvironment{theorem}{\textbf{Theorem.}\rm}{}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End:
%% sets
\def\R{\mathbf{R}}
\def\Z{\mathbf{Z}}
\def\C{\mathbf{C}}
\def\CP{\C{}P}
\def\Q{\mathbf{Q}}
\def\sphere#1{\mathbf{S}^{#1}}
\def\rp#1{\R P^{#1}}
\def\set#1{\left\{ #1 \right\}}
\def\st{\,\mathrm{s.t.}\,}
\def\extalg#1#2{\Lambda^{#1}(#2)}
\def\symalg#1#2{\mathsf{S}^{#1}(#2)}
\def\tenalg#1#2{\mathsf{T}^{#1}(#2)}
\def\hom#1#2{\mathrm{Hom}(#1;#2)}
\def\extproduct{\wedge}
\def\symmetricproduct{\cdot}
\def\tensorproduct{\otimes}
\def\image#1{\mathrm{Im}\,#1}
\def\kernel#1{\mathrm{Ker}\,#1}
\def\Trace[#1]#2{\mathrm{Tr}\ifx#1\empty\else{}_{#1}\fi\ifx#2\empty\else\,(#2)\fi}
\def\Trce#1{\mathrm{Tr}(#1)}
\def\orbit#1#2{{#1}\cdot{#2}}
\def\stab#1#2{\mathrm{stab}_{#1}(#2)}
\def\Span{\mathrm{span}}
\def\cinfty#1{C^{\infty}(#1)}
%% Lie groups
\def\GL#1{\mathrm{GL}(#1)}
\def\SL#1{\mathrm{SL}(#1)}
\def\Symp#1{\mathrm{Sp}(#1)}
\def\Orth#1{\mathrm{O}(#1)}
\def\SOrth#1{\mathrm{SO}(#1)}
\def\Unitary#1{\mathrm{U}(#1)}
\def\SUnitary#1{\mathrm{SU}(#1)}
\def\Torus#1{\mathbf{T}^{#1}}
\def\H#1{\mathbf{H}^{#1}}
\def\Diagonal#1{\mathbf{Diag}(#1)}
\def\diag#1{\mathbf{diag}\left(#1\right)}
\def\Liegp#1{\mathrm{#1}}
%% Lie algebras
\def\Mat#1#2{\mathrm{Mat}_{{#1} \times {#1}}(#2)}
\def\Matrect#1#2#3{\mathrm{Mat}_{{#1} \times {#2}}(#3)}
\def\gl#1{\mathfrak{gl}(#1)}
\def\spl#1{\mathfrak{sl}(#1)}
\def\symp#1{\mathfrak{sp}(#1)}
\def\orth#1{\mathfrak{o}(#1)}
\def\sorth#1{\mathfrak{so}(#1)}
\def\unitary#1{\mathfrak{u}(#1)}
\def\sunitary#1{\mathfrak{su}(#1)}
\def\torus#1{\mathfrak{t}^{#1}}
\def\liealg#1{\mathfrak{#1}}
\def\liebracket#1#2{[#1,#2]}
\def\Ad#1#2{\mathrm{Ad}_{#1}{#2}}
\def\ad#1#2{\mathrm{ad}_{#1}{#2}}
\def\coAd#1#2{\mathrm{Ad}^*_{#1}{#2}}
\def\coad#1#2{\mathrm{ad}^*_{#1}{#2}]}
\def\pb#1#2{\left\{{#1},{#2}\right\}}
%% operators
\def\D#1{\,\mathrm{d}{#1}\,}
\def\Dat#1#2{\,\mathrm{d}_{#1}{#2}\,}
\def\lieder#1#2{{\sf{L}}_{#1}{#2}}
\def\covder#1#2{\,\frac{D #2}{d{#1}}}
\def\liebracket#1#2{\left[#1,#2\right]}
\def\crossproduct{{\boldsymbol{\times}}}
\def\ip#1#2{\langle #1, #2 \rangle}
\def\interiorproduc#1#2{\iota_{#1}{#2}}
\def\norm#1{|#1|}
\def\lt{<}
\def\gt{>}
\def\ddt#1#2{\frac{d{#2}}{d{#1}}}
\def\dndtn#1#2#3{\frac{d^{#3}{#2}}{d{#1}^{#3}}}
\def\wint#1#2{\int\limits_{#1}^{#2}}
\def\didi#1#2{\frac{\partial{#1}}{\partial{#2}}}
\def\dindin#1#2#3{\frac{\partial^{#3}{#1}}{\partial{#2}^{#3}}}
\def\eulerlagrange#1#2#3#4{\ddt{#1}{}\didi{#4}{#3} - \didi{#4}{#2}}
\def\imagpart#1{\mathrm{Im}(#1)}
\def\realpart#1{\mathrm{Re}(#1)}
\def\equivclass#1{\left[#1\right]}
%% labels, etc.
\def\labelenumi{{\bf\Alph{enumi}.}}
\def\solutioncolor{blue}
\def\solutiontopsep{3mm}
\def\solutionbotsep{0mm}
\newenvironment{solution}{\vspace{\solutiontopsep}\noindent\textbf{Solution}. \textcolor{\solutioncolor}\bgroup}{\egroup\vspace{\solutionbotsep}}
%% redo quote environment
\def\quote{}
\def\endquote{}
%% redo enumerate environment
\makeatletter
\let\oldenumerate\enumerate
\let\oldendenumerate\endenumerate
\def\enumerate{\bgroup\oldenumerate\setlength{\itemsep}{3mm}\setlength{\topsep}{3mm}}
\def\endenumerate{\oldendenumerate\egroup}
\makeatother
\def\museincludegraphicsoptions{%
width=0.25\textwidth%
}
\def\museincludegraphics{%
\begingroup
\catcode`\|=0
\catcode`\\=12
\catcode`\#=12
\expandafter\includegraphics\expandafter[\museincludegraphicsoptions]
}
\def\musefigurewidth{!}
\def\musefigureheight{!}
\newsavebox{\mypicboc}%
\def\includefigure#1#2#3#4{%
\edef\fileending{#4}%
\def\svgend{svg}%
\def\pdftend{pdf_t}%
\def\pdftexend{pdf_tex}%
\def\bang{!}%
\begin{lrbox}{\mypicboc}%
\ifx\fileending\pdftend%
\input{#3.#4}%
\else%
\ifx\fileending\pdftexend
\input{#3.#4}%
\else%
\ifx\fileending\svgend%
\input{#3.#4}%
\fi\fi%
\includegraphics{#3.#4}%
\fi%
\end{lrbox}%
\edef\thiswidth{#1}\edef\thisheight{#2}%
\ifx\thiswidth\bang%
\ifx\thisheight\bang%
\resizebox{!}{!}{\usebox{\mypicboc}}%
\else%
\resizebox{!}{\thisheight}{\usebox{\mypicboc}}%
\fi%
\else%
\ifx\thisheight\bang%
\resizebox{\thiswidth}{!}{\usebox{\mypicboc}}%
\else%
\resizebox{\thiswidth}{\thisheight}{\usebox{\mypicboc}}%
\fi\fi}
$
-
Prove that the centre of the Poisson algebra of $\sorth{3}^*$ is non-trivial.
-
Let $H : T^* \R^n \to \R$ be given by $H(q, p) = \frac{1}{2} (\norm{q}^2 + \norm{p}^2)$.
-
Derive Hamilton's equations.
-
Show that, if we write $z = q + i p$, then $\phi_t(z) = \exp(-i t) z$ solves Hamilton's equations.
-
Let $S = H^{-1}(\frac{1}{2})$. $\Omega | S$ is degenerate. Show that $\kernel{\Omega|S}$ is spanned by $X_H$.
-
Let $C = S/\sim$ where $z \sim z'$ iff $z'$ lies on the orbit of $z$. Show that $C$ is a symplectic manifold of dimension $2n-2$ (it is known as complex projective space and usually denoted by $\CP^n$).
-
Show that $\CP^1$ with the symplectic form above is symplectomorphic to the unit sphere $\sphere{2}$ with its area form.
![[FSF Associate Member 10686]](http://static.fsf.org/nosvn/associate/fsf-10686.png)
