Here is a quotation that I found on Wolfram website:
"A first integral associated with the independent variable t exist if f is independent of t and does not contain any second or higher derivatives of the coordinates."
Since we have a=f(q,u,t) as a firt integral, it will not have a...
Here I have a translation from French to English of the original paper by Poisson about his brackets. I cannot understand why the function a=f(q,u,t) doesn't have a second order derivative (in q or u). The problem is on the top of the third page (second .JPG) after he took the time derivative...