How can we prove that an ODE does not have a first integral (i.e., a constant of motion that is conserved along the trajectories)? For example, is it true that the pendulum system of ODEs $$\dot x_1 = x_2$$ $$\dot x_2 = -a\:\sin(x_1) – b\:x_2$$ does not have a first integral?
My intuition says that for ODE systems which represent a physical system with loss, the first integral (equivalently, the energy function) does not exist.
Best Answer
In general, a first integral representing some form of energy does exist; however, it might not be constant on trajectories. That's the case for your system, it's dissipative so you don't have a first integral of motion, in the sense of a conserved quantity.
For systems of the form $$\dot{x}_1=x_2$$ $$\dot{x}_2=F(x_1)$$ you can in general find a first integral of motion that represents some form of energy being conserved on the system's trajectories. However, when $$\dot{x}_2=F(x_1,x_2)\,,$$ as in your case, then the system is dissipative and there is no first integral of motion; you can try to integrate the equation, but the resulting quantity will not be maintained constant on trajectories. Hope this helps you in your question!