Question: why diff following some are missing and how to result in a*lambda(t) = -y(t)

int(y^2/2 - lambda*(u*diff(y,t)-diff(u,t)-a), t);

with(Physics):

diff(y(t)^2/2 - lambda(t)*(u(t)*diff(y(t),t)-diff(u(t),t)-a), u(t)); # u -> something

diff(lambda(t),t) disappear

should be

diff(lambda(t),t)+Physics[`*`](lambda(t), diff(y(t), t))

diff(y(t)^2/2 - lambda(t)*(u(t)*diff(y(t),t)-diff(u(t),t)-a), y(t)); # y -> something

u*diff(lambda(t),t) + lambda(t)*diff(u(t),t) disappear

should be u(t)*diff(lambda(t),t) + lambda(t)*diff(u(t),t) = y(t)

diff(y(t)^2/2 - lambda(t)*(u(t)*diff(y(t),t)-diff(u(t),t)-a), lambda(t)); # lambda -> something

-Physics[`*`](u(t), diff(y(t), t))+diff(u(t), t)+a

 which operation can obtain a*lambda(t) = -y(t)

i guess eliminate or annihilator, however eliminate is wrong and involutive manual do not mention use annihilator for differential equation case

eliminate({diff(lambda(t),t)+Physics[`*`](lambda(t), diff(y(t), t))

u(t)*diff(lambda(t),t) + lambda(t)*diff(u(t),t) = y(t)

-Physics[`*`](u(t), diff(y(t), t))+diff(u(t), t)+a = 0}

, u(t));