113 lines
6.5 KiB
TeX
113 lines
6.5 KiB
TeX
|
\relax
|
||
|
|
\providecommand\hyper@newdestlabel[2]{}
|
||
|
|
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
|
||
|
|
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
|
||
|
|
\global\let\oldcontentsline\contentsline
|
||
|
|
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
|
||
|
|
\global\let\oldnewlabel\newlabel
|
||
|
|
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
|
||
|
|
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
|
||
|
|
\AtEndDocument{\ifx\hyper@anchor\@undefined
|
||
|
|
\let\contentsline\oldcontentsline
|
||
|
|
\let\newlabel\oldnewlabel
|
||
|
|
\fi}
|
||
|
|
\fi}
|
||
|
|
\global\let\hyper@last\relax
|
||
|
|
\gdef\HyperFirstAtBeginDocument#1{#1}
|
||
|
|
\providecommand\HyField@AuxAddToFields[1]{}
|
||
|
|
\providecommand\HyField@AuxAddToCoFields[2]{}
|
||
|
|
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}\protected@file@percent }
|
||
|
|
\@writefile{toc}{\contentsline {section}{\numberline {2}Some basic background material}{1}{section.2}\protected@file@percent }
|
||
|
|
\newlabel{def:manifold}{{2.1}{1}{}{definition.2.1}{}}
|
||
|
|
\newlabel{def:manifold@cref}{{[definition][1][2]2.1}{[1][1][]1}}
|
||
|
|
\newlabel{def:tangent}{{2.2}{2}{}{definition.2.2}{}}
|
||
|
|
\newlabel{def:tangent@cref}{{[definition][2][2]2.2}{[1][1][]2}}
|
||
|
|
\newlabel{def:vector_field}{{2.3}{2}{}{definition.2.3}{}}
|
||
|
|
\newlabel{def:vector_field@cref}{{[definition][3][2]2.3}{[1][2][]2}}
|
||
|
|
\newlabel{def:lie_algebra}{{2.4}{2}{}{definition.2.4}{}}
|
||
|
|
\newlabel{def:lie_algebra@cref}{{[definition][4][2]2.4}{[1][2][]2}}
|
||
|
|
\newlabel{prob:lie_alg}{{2.5}{2}{}{definition.2.5}{}}
|
||
|
|
\newlabel{prob:lie_alg@cref}{{[problem][5][2]2.5}{[1][2][]2}}
|
||
|
|
\newlabel{prob:lie_alg2}{{2.6}{3}{}{definition.2.6}{}}
|
||
|
|
\newlabel{prob:lie_alg2@cref}{{[problem][6][2]2.6}{[1][3][]3}}
|
||
|
|
\newlabel{def:adjoint}{{2.7}{4}{}{definition.2.7}{}}
|
||
|
|
\newlabel{def:adjoint@cref}{{[definition][7][2]2.7}{[1][4][]4}}
|
||
|
|
\newlabel{def:lie_group}{{2.8}{4}{}{definition.2.8}{}}
|
||
|
|
\newlabel{def:lie_group@cref}{{[definition][8][2]2.8}{[1][4][]4}}
|
||
|
|
\newlabel{prob:lie_group}{{2.9}{4}{}{definition.2.9}{}}
|
||
|
|
\newlabel{prob:lie_group@cref}{{[problem][9][2]2.9}{[1][4][]4}}
|
||
|
|
\newlabel{def:alg_of_group}{{2.10}{5}{}{definition.2.10}{}}
|
||
|
|
\newlabel{def:alg_of_group@cref}{{[definition][10][2]2.10}{[1][5][]5}}
|
||
|
|
\newlabel{prob:lie_group2}{{2.11}{5}{}{definition.2.11}{}}
|
||
|
|
\newlabel{prob:lie_group2@cref}{{[problem][11][2]2.11}{[1][5][]5}}
|
||
|
|
\newlabel{def:action}{{2.12}{6}{}{definition.2.12}{}}
|
||
|
|
\newlabel{def:action@cref}{{[definition][12][2]2.12}{[1][6][]6}}
|
||
|
|
\newlabel{def:right}{{2.13}{6}{}{definition.2.13}{}}
|
||
|
|
\newlabel{def:right@cref}{{[definition][13][2]2.13}{[1][6][]6}}
|
||
|
|
\newlabel{def:exp}{{2.14}{6}{}{definition.2.14}{}}
|
||
|
|
\newlabel{def:exp@cref}{{[definition][14][2]2.14}{[1][6][]6}}
|
||
|
|
\newlabel{prob:exp}{{2.15}{7}{}{definition.2.15}{}}
|
||
|
|
\newlabel{prob:exp@cref}{{[problem][15][2]2.15}{[1][7][]7}}
|
||
|
|
\newlabel{lem:1}{{2.16}{8}{}{definition.2.16}{}}
|
||
|
|
\newlabel{lem:1@cref}{{[lemma][16][2]2.16}{[1][7][]8}}
|
||
|
|
\@writefile{toc}{\contentsline {section}{\numberline {3}Approximating differential equations on manifolds}{8}{section.3}\protected@file@percent }
|
||
|
|
\newlabel{eq:3_1}{{3.1}{8}{Approximating differential equations on manifolds}{equation.3.1}{}}
|
||
|
|
\newlabel{eq:3_1@cref}{{[equation][1][3]3.1}{[1][8][]8}}
|
||
|
|
\newlabel{prob:num1}{{3.1}{8}{}{definition.3.1}{}}
|
||
|
|
\newlabel{prob:num1@cref}{{[problem][1][3]3.1}{[1][8][]8}}
|
||
|
|
\newlabel{eq:3_3}{{3.3}{8}{}{equation.3.3}{}}
|
||
|
|
\newlabel{eq:3_3@cref}{{[equation][3][3]3.3}{[1][8][]8}}
|
||
|
|
\newlabel{eq:2_2}{{3.4}{9}{Approximating differential equations on manifolds}{equation.3.4}{}}
|
||
|
|
\newlabel{eq:2_2@cref}{{[equation][4][3]3.4}{[1][9][]9}}
|
||
|
|
\newlabel{rem:1}{{3.2}{10}{}{definition.3.2}{}}
|
||
|
|
\newlabel{rem:1@cref}{{[remark][2][3]3.2}{[1][9][]10}}
|
||
|
|
\newlabel{def:dexp}{{3.3}{10}{}{definition.3.3}{}}
|
||
|
|
\newlabel{def:dexp@cref}{{[definition][3][3]3.3}{[1][10][]10}}
|
||
|
|
\newlabel{lem:2}{{3.4}{10}{}{definition.3.4}{}}
|
||
|
|
\newlabel{lem:2@cref}{{[lemma][4][3]3.4}{[1][10][]10}}
|
||
|
|
\newlabel{lem:2a}{{3.5}{10}{}{definition.3.5}{}}
|
||
|
|
\newlabel{lem:2a@cref}{{[lemma][5][3]3.5}{[1][10][]10}}
|
||
|
|
\newlabel{rem:b}{{3.6}{10}{}{definition.3.6}{}}
|
||
|
|
\newlabel{rem:b@cref}{{[remark][6][3]3.6}{[1][10][]10}}
|
||
|
|
\newlabel{eq:3_8}{{3.8}{10}{}{equation.3.8}{}}
|
||
|
|
\newlabel{eq:3_8@cref}{{[equation][8][3]3.8}{[1][10][]10}}
|
||
|
|
\newlabel{lem:3}{{3.7}{11}{}{definition.3.7}{}}
|
||
|
|
\newlabel{lem:3@cref}{{[lemma][7][3]3.7}{[1][10][]11}}
|
||
|
|
\newlabel{lem:4}{{3.8}{11}{}{definition.3.8}{}}
|
||
|
|
\newlabel{lem:4@cref}{{[lemma][8][3]3.8}{[1][11][]11}}
|
||
|
|
\newlabel{rem:3}{{3.9}{12}{}{definition.3.9}{}}
|
||
|
|
\newlabel{rem:3@cref}{{[remark][9][3]3.9}{[1][12][]12}}
|
||
|
|
\newlabel{expand1}{{3.17}{12}{Approximating differential equations on manifolds}{equation.3.17}{}}
|
||
|
|
\newlabel{expand1@cref}{{[equation][17][3]3.17}{[1][12][]12}}
|
||
|
|
\newlabel{expand2}{{3.18}{12}{Approximating differential equations on manifolds}{equation.3.18}{}}
|
||
|
|
\newlabel{expand2@cref}{{[equation][18][3]3.18}{[1][12][]12}}
|
||
|
|
\newlabel{expand3}{{3.19}{12}{Approximating differential equations on manifolds}{equation.3.19}{}}
|
||
|
|
\newlabel{expand3@cref}{{[equation][19][3]3.19}{[1][12][]12}}
|
||
|
|
\newlabel{lem:5}{{3.10}{13}{}{definition.3.10}{}}
|
||
|
|
\newlabel{lem:5@cref}{{[lemma][10][3]3.10}{[1][13][]13}}
|
||
|
|
\newlabel{lem:5_ode}{{3.21}{13}{}{equation.3.21}{}}
|
||
|
|
\newlabel{lem:5_ode@cref}{{[equation][21][3]3.21}{[1][13][]13}}
|
||
|
|
\newlabel{eq:3_23}{{3.23}{13}{}{equation.3.23}{}}
|
||
|
|
\newlabel{eq:3_23@cref}{{[equation][23][3]3.23}{[1][13][]13}}
|
||
|
|
\newlabel{i1}{{{{(i)}}}{13}{}{Item.12}{}}
|
||
|
|
\newlabel{i1@cref}{{[enumi][1][]{{(i)}}}{[1][13][]13}}
|
||
|
|
\newlabel{i2}{{{{(ii)}}}{13}{}{Item.13}{}}
|
||
|
|
\newlabel{i2@cref}{{[enumi][2][]{{(ii)}}}{[1][13][]13}}
|
||
|
|
\newlabel{eq:3_25}{{3.25}{13}{}{equation.3.25}{}}
|
||
|
|
\newlabel{eq:3_25@cref}{{[equation][25][3]3.25}{[1][13][]13}}
|
||
|
|
\newlabel{prob:1}{{3.11}{14}{}{definition.3.11}{}}
|
||
|
|
\newlabel{prob:1@cref}{{[problem][11][3]3.11}{[1][14][]14}}
|
||
|
|
\newlabel{prob:2}{{3.12}{15}{}{definition.3.12}{}}
|
||
|
|
\newlabel{prob:2@cref}{{[problem][12][3]3.12}{[1][15][]15}}
|
||
|
|
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}A first attempt at a numerical approximation}{16}{subsection.3.1}\protected@file@percent }
|
||
|
|
\newlabel{eq:3_26}{{3.26}{16}{A first attempt at a numerical approximation}{equation.3.26}{}}
|
||
|
|
\newlabel{eq:3_26@cref}{{[equation][26][3]3.26}{[1][16][]16}}
|
||
|
|
\newlabel{eq:3_27}{{3.27}{16}{A first attempt at a numerical approximation}{equation.3.27}{}}
|
||
|
|
\newlabel{eq:3_27@cref}{{[equation][27][3]3.27}{[1][16][]16}}
|
||
|
|
\newlabel{prob:3}{{3.13}{16}{}{definition.3.13}{}}
|
||
|
|
\newlabel{prob:3@cref}{{[problem][13][3]3.13}{[1][16][]16}}
|
||
|
|
\newlabel{prob:4}{{3.14}{17}{}{definition.3.14}{}}
|
||
|
|
\newlabel{prob:4@cref}{{[problem][14][3]3.14}{[1][17][]17}}
|
||
|
|
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Approximating the exponential map}{18}{subsection.3.2}\protected@file@percent }
|
||
|
|
\gdef \@abspage@last{18}
|