Does the integral of the vis-viva equation have any meaning?

Question

If orbital speed of an elliptical orbit can be described by

$$ v(r) =\sqrt{GM\left(\frac{2}{r}-\frac{1}{a}\right)}, $$

then what would the meaning of its ($\mathrm{d}r$) integral be?

Answer 1

Given any $v(r)$, the integral over space can be transformed as follows, $$\int v\,\mathrm{d}r=\int v\,\frac{\mathrm{d}r}{\mathrm{d}t}\mathrm{d}t=\int v^2\,\mathrm{d}t.$$ Recalling that kinetic energy is $\sim mv^2$, then the right-most integral is proportional to the integral of kinetic energy over time, which yields an action. So the original integral is something like an action per unit mass, I suppose.
While the action is fairly common, I don't think the integral on the left-most side is a particularly common operation.

Answer 2

$$\oint v_r \mathrm dr,$$ i.e. the radial component of the velocity integrated around a radial orbit, is a useful quantity, sometimes called the radial action. For example, it is an adiabatic invariant, meaning that it is preserved under (among other things) a change to the gravitational potential, as long as the change happens over many orbits. Also, it is used in the construction of action-angle coordinates.

However, I can't think of a context where it makes sense to put the total speed $v\equiv |\vec v|$ (and not just $v_r$) as the integrand. Putting the total speed in the above integral would actually just yield 0, i.e. $$\oint |\vec v| \mathrm dr =0,$$ since the ascending and descending contributions would cancel.