# Who was first to integrate Newton's equations of motion to derive the conservation laws for mechanical energy and momentum?

I'm wondering who is the first person in the history who came up with an idea to integrate Newton's $F=ma$ to obtain the law of mechanical energy conservation? And When did it happen?

Also I have the same questions for the law of momentum conservation.

I know that these laws were known by experience and intuitively formulated before someone realized that these are integrated from $F=ma$. But I want to know who found the relationship between the laws and $F=ma$.

"Employing both the ancient scholastic categories and the new mathematical methods of his time, Boscovich discussed the graphical representation of a pressure applied through a time and a force applied over a distance... Without taking a position on the definition of force, Boscovich measured the velocity acquired as a ratio composed of the pressure and its duration. A geometrical image is generated by the line representing the pressure with time as the second dimension of the diagram. The pressure is thus a function of time. Interpreting this in modern terminology, the momentum mv would be represented as the integral of these instantaneous pressures (or impulses) over a time, or $\int mdv = \int pdt$.
Boscovich suggested that, if the time coordinate is replaced by the space traversed and the pressure coordinate by the force which at any instant produces the velocity proportional to it, a second aspect of the phenomenon is represented... The new term 'force' must be interpreted as an entity proportional to the velocity engendered at any instant. If the pressure coordinate is changed to the force and the time coordinate to the space then the new geometrical image producing the velocity would be represented in modern notation as $\int Fds$. We would then interpret vis viva as $\int mvdv = \int Fds$ (where $ds = vdt$). Boscovich does not bring the mass into this analysis.