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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 $F=ma$.
It is not enough to integrate the equations of motion in order to obtain the conservation laws. Newton derived conservation of momentum components in the absence of external forces from the third law of motion already in Principia. The idea to integrate forces to obtain changes in momentum and kinetic energy (called "vis viva" at the time) first appears in Boscovich's De Viribus Vivis (Rome, 1745), a fifty page contribution in Latin to the then raging vis viva controversy, see What was the vis viva controversy, including its philosophical aspects? for details. According
According to Iltis's d'Alembert and the Vis Viva Controversy:
"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.
D'Alembert included similar interpretations into the second edition of his Traite de Dynamique (1758), a milestone monograph on mechanics after Newton's Principia, which was widely read. It is unclear if he was aware of Boscovich's work. However, it was only Lagrange in Mecanique Analytique (1788), who introduced the notion of potential energy (the name was proposed by Rankine only in 1853), and showed that for potential forces the sum of kinetic and potential energies is an "integral of motion".
