# Euler-Lagrange systems, autonomous or nonautonomous?

I was reading an article on Euler-Lagrange systems. It is stated there that since M(q) and C(q,q') depend on q, it is not autonomous. As a result, we cannot use LaSalle's theorem. I have uploaded that page of the article and highlighted the sentence. (ren.pdf)

Then, I read Spong's book on robotics, and he had used LaSalle's theorem. I am confused. (spong.pdf)

I did some research, and found out that non-autonomous means it should not explicitly depend on the independent variable. Isn't independent variable time in these systems? So, shouldn't they be considered autonomous?

• Autonomous systems are also called time invariant. They don't depend on the time variable. – CroCo Mar 27 '16 at 17:00
• Dear @CroCo, So you belive Euler-Lagrange systems are autonomous? – Has Mar 27 '16 at 21:15

If you structure the system dynamics equations to not depend on the independent variable, an Euler-Lagrange system is autonomous. If, however, you recast the equations to depend on the state variables (and their derivatives), as Ren did, it becomes non-autonomous. That clever manipulation allowed him to prove asymptotic stability for the consensus algorithm $\tau_i$. It might be possible to prove without taking that mathematical direction, but I cannot think of a better way to do it.