# What is the time-integral of motion for first order differential equations?

For a second order differential equation (many physical systems) in one variable, I know "procedures" to compute the energy. Given $$q''(t)=f(q(t),q'(t)),\ \ q(0)=q_0,\ \ q'(0)=v_0,$$ if we're lucky we can read off the related Lagrangian $L$, introduce $p=\frac{\partial L}{\partial q}$, do a Legendre transform and we got the Hamiltonian function $H(q,p)$ for which $\frac{\text d}{\text dt}H(q(t),p(t))=0$ for solutions of the differential equation.

We can be more exact and give all the conditions for Noethers theorem to hold and the result is that along the flow $X(t)=\langle q(t),p(t)\rangle$ in phase space given by a solution with initial conditions $X(0)$, the function $H(q,p)$ always takes the same values. It defines surfaces in phase space indexed by initial conditions $X(0)$.

I wonder how to view this for a priori first order systems $$q'(t)=f(q(t)),\ \ q(0)=q_0,$$ where I think this must be even simpler. E.g. I thing some functional $$q(t)\mapsto q(t)-\text{an integral over}\ f(q(t))\ \text{something},$$ should exists which will be constant for solutions of the equation, i.e. only depend on $q_0$. For each $f$, the functional dependence of this "energy" on "$q_0$" will be different.

However, I can't seem to find a general relation. What's the theory behind this, is there an energy'ish time integral of motion? What is the functional dependence on the intial condition, for a suitable constant of motion for for first order systems. And what would be the interpretation, given that we speak of a situation with only one initial condition?

If it is that case that the system is too restricted so that there is no meaningful geometrical interpretation, then let's think about a system of first order differential equations. This is like the one we generated from phase space, except that it doesn't really come from a second order situation and so the intial conditions aren't really e.g. intial position and velocity. I'm pretty sure there are situation where one considers such a directly generated flow (the equation might be more complicated than exponential flow $\dot x\propto x$), but I don't recall any talk about the time-constant of motion in these systems, or how to interpret it.

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$q'= \frac{\partial H}{\partial p}\qquad p'= -\frac{\partial H}{\partial q}$
$\frac{\partial H}{\partial p} = f(q),\quad p'=??$