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looking for completion of pendulum (constrained motion)

equations using multiindex notation (1 independent variable [t] and 3 dependent variables [x,y,F])

fricas
jb:=IJB('t,'u,'p,1,5);
Type: Type
fricas
jbe:=JBE jb;
Type: Type
fricas
de := JDE(jb, jbe);
Type: Type
fricas
ck := CKP(jb, jbe);
Type: Type
fricas
eq1:jbe := m*P(4,[1]) + U(1)/L*U(3)

\label{eq1}{{L \  m \ {p_{1}^{4}}}+{{u^{1}}\ {u^{3}}}}\over L(1)
Type: JetBundleExpression?(IndexedJetBundle?(t,u,p,1,5))
fricas
eq2:jbe := m*P(5,[1]) + U(2)/L*U(3) + m*g

\label{eq2}{{L \  m \ {p_{1}^{5}}}+{{u^{2}}\ {u^{3}}}+{L \  g \  m}}\over L(2)
Type: JetBundleExpression?(IndexedJetBundle?(t,u,p,1,5))
fricas
eq3:jbe := U(1)^2 + U(2)^2 - L^2

\label{eq3}{{u^{2}}^{2}}+{{u^{1}}^{2}}-{{L}^{2}}(3)
Type: JetBundleExpression?(IndexedJetBundle?(t,u,p,1,5))
fricas
eq4:jbe := P(1,[1]) - U(4)

\label{eq4}-{u^{4}}+{p_{1}^{1}}(4)
Type: JetBundleExpression?(IndexedJetBundle?(t,u,p,1,5))
fricas
eq5:jbe := P(2,[1]) - U(5)

\label{eq5}-{u^{5}}+{p_{1}^{2}}(5)
Type: JetBundleExpression?(IndexedJetBundle?(t,u,p,1,5))
fricas
printSys([eq1,eq2,eq3,eq4,eq5])$de

\label{eq6}\begin{array}{c}
{\ }
\
{{{{L \  m \ {p_{1}^{4}}}+{{u^{1}}\ {u^{3}}}}\over L}= 0}
\
{\ }
\
{{{{L \  m \ {p_{1}^{5}}}+{{u^{2}}\ {u^{3}}}+{L \  g \  m}}\over L}= 0}
\
{\ }
\
{{{{u^{2}}^{2}}+{{u^{1}}^{2}}-{{L}^{2}}}= 0}
\
{\ }
\
{{-{u^{4}}+{p_{1}^{1}}}= 0}
\
{\ }
\
{{-{u^{5}}+{p_{1}^{2}}}= 0}
\
(6)
fricas
pendulum:de := makeSystem [eq1,eq2,eq3,eq4,eq5]

\label{eq7}\begin{array}{c}
{\ }
\
{{{L \  m \ {p_{1}^{4}}}+{{u^{1}}\ {u^{3}}}}= 0}
\
{\ }
\
{{{L \  m \ {p_{1}^{5}}}+{{u^{2}}\ {u^{3}}}+{L \  g \  m}}= 0}
\
{\ }
\
{{-{u^{4}}+{p_{1}^{1}}}= 0}
\
{\ }
\
{{-{u^{5}}+{p_{1}^{2}}}= 0}
\
{\ }
\
{{{{u^{2}}^{2}}+{{u^{1}}^{2}}-{{L}^{2}}}= 0}
\
(7)
Type: JetDifferentialEquation?(IndexedJetBundle?(t,u,p,1,5),JetBundleExpression?(IndexedJetBundle?(t,u,p,1,5)))
fricas
setOutMode(14)$ck

\label{eq8}0(8)
fricas
setRedMode(1)$ck

\label{eq9}0(9)
fricas
complete(pendulum)$ck

\label{eq10}\ (10)

\label{eq11}\mbox{\rm <em> </em> <em> </em> <em> </em> <em> </em> <em> </em> <em> </em> <em> </em> <em> \hbox{\axiomType{Final}\ } \hbox{\axiomType{Result}\ } </em> <em> </em> <em> </em> <em> </em> <em> </em> <em> </em> <em> </em> <em> </em>}(11)

\label{eq12}\ (12)

\label{eq13}\mbox{\rm \hbox{\axiomType{Equation}\ }}{R_{1}}\mbox{\rm involutive !}(13)

\label{eq14}\mbox{\rm \hbox{\axiomType{System}\ } without prolonged equations.\hbox{\axiomType{Dimension}\ } :}7(14)

\label{eq15}\begin{array}{c}
{\ }
\
{{{L \  m \ {p_{1}^{5}}}+{{u^{2}}\ {u^{3}}}+{L \  g \  m}}= 0}
\
{\ }
\
{{{L \  m \ {p_{1}^{4}}}+{{u^{1}}\ {u^{3}}}}= 0}
\
{\ }
\
{{-{u^{5}}+{p_{1}^{2}}}= 0}
\
{\ }
\
{{-{u^{4}}+{p_{1}^{1}}}= 0}
\
{\ }
\
{{{{u^{2}}^{2}}+{{u^{1}}^{2}}-{{L}^{2}}}= 0}
\
(15)

\label{eq16}\ (16)

\label{eq17}\mbox{\rm \hbox{\axiomType{Cartan}\ } characters :}{4}(17)
Type: Void




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