Origin and demonstration of Newton's second law

[fab13]

At high school, we saw without demonstration the fundamental principle of dynamics (2th Newton's law), i.e :

$$\sum \vec{F}=m \vec{a}\,\,\,\,\,\,\,\,\,eq(1)$$

after, at university, we saw another expression of this 2th Newton's law :

$$\sum \vec{F}= \dfrac{d\vec{p}}{dt}\,\,\,\,\, \,\,\,eq(2)$$ with $\vec{p} = m\vec{v}$ the momentum.

From an historical point of view, which one was defined by Newton, eq(1) or (eq2) ?

The eq(2) allows to deduce the famous equation $E=mc^{2}$ by considering $dp=d(mv)=dm\,v +m\,dv$ but I think that Newton could not have access to $dm$ and so defined rather $m=\text{constant}$, didn't he ?

Does the origin of eq(1) and eq(2) come from physical experiments performed by Newton ?

Secondly, we can proove eq(2) thanks to Euler-Lagrange equation, taking $p_{i}$ the i-th momentum :

$$\dfrac{d}{dt}\bigg(\dfrac{ \partial L}{\partial \dot{q}_{i}}\bigg)=\dfrac{d\,p _{i}}{dt}=\sum F$$

Are there other ways to get the eq(2) ?

Finally, one told me that Newton has used geometric demonstration : Anyone could give me a link on these geometric prooves ?

Thanks for your help

 

[DrStupid]

From an historical point of view, which one was defined by Newton, eq(1) or (eq2) ?

The second.

Does the origin of eq(1) and eq(2) come from physical experiments performed by Newton ?

We will never know. He didn't explain how he derived this equation.