%First integrals of the Toda lattice;
%Program name: hamin.our;
linelength 72;
off nat$
out haminl$

operator h, q, p, qt, pt;
depend h, q(j), p(j);

%This is the Hamilton function;
H := (p(1)^2 + p(2)^2 + p(3)^2)/2 +
E^(q(1)-q(2)) + E^(q(2)-q(3)) + E^(q(3)-q(1));

for j := 1:3 do
<<pt(j) := -df(h,q(j)); qt(j) := df(h,p(j))>>;

%Examle 1: Ansatz for a first integral;
I1 := p(1) + p(2) + p(3);
R1 := for l := 1:3 sum (pt(l)*df(I1,p(l)) + qt(l)*df(I1,q(l)));
If R1=0 then write "I1 is a first integral"
else write "I1 is not a first integral";

%Example 2: Ansatz for a first integral;
I2 := p(1)*p(2)*p(3) - p(1)*E^(q(2)-q(3)) - p(2)*E^(q(3)-q(1))
-p(3)*E^(q(1)-q(2));
R2 := for l := 1:3 sum (pt(l)*df(I2,p(l)) + qt(l)*df(I2,q(l)));
If R2=0 then write "I2 is a first integral"
else write "I2 is not a first integral";

write ";end"$
shut haminl$
on nat$