Integration of equation of motion to obtain invariant angular momentum Studying the equations of motion in a central-symmetric potential, we obtain:
$ m ( r \ddot \theta + 2 \dot r \dot \theta ) = 0 \tag{1} $
From that my book concludes that the angular momentum ($m r^2 \dot \theta$) is constant:
$ \frac{d}{d t} (m r^2 \dot \theta ) = 0 \tag{2} $
This is true, as differentiation shows:
$$ \frac{d}{d t} (m r^2 \dot \theta ) = m ( r^2 \ddot \theta + 2 r \dot r \dot \theta ) $$
This is equal to (1) after division by $r$. How could I have discovered that starting from (1) itself? Is there any way to see that multiplication by $r$ makes the integral more easily solvable?
Clarification: What I'm after is the integration, not the conversation law. How do I know that I should multiply by $r$ when I obtain an integration such as $ \int r \ddot \theta + 2 \dot r \dot \theta dt $ ? Is there any guiding principle or do you just play around with the equation and hope for the best?
 A: There is no univeral way to find conserved quantities of a given physical system. The most common way they are found is by looking at symmetries of the system. In this case, noticing that the potential is central-symmetric leads us to the suspicion that angular momentum might be conserved. 
I cannot provide you with a definite answer about how people used to treat these systems before the Langrangian formalism. Note however that not all systems exhibit conserved quantities (although systems in exercises tend to have them, because having conserved quantities often goes hand in hand with analytic solvability). 
Systems with "enough" conserved quantities are called integrable and nowadays there are quite a few different interpretations of the word "enough", depending on the context of the model. The most well-known form of integrability is that in the sense of Liouville, which is the form that is probably most useful here. Although there is a general method of constructing Noether charges, finding all the conserved quantities is most often nothing more than clever guesswork. 
