How Many Laws of Motion Did Newton Discover?

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Newton discovered three primary laws of motion, known as Newton's First, Second, and Third Laws. These laws describe the relationship between the motion of an object and the forces acting on it. Additionally, there are other laws attributed to Newton, such as the law of universal gravitation and the law of cooling, but they are not classified as laws of motion. The discussion highlights a common misconception regarding the number of laws Newton is credited with. Overall, the consensus is that there are three fundamental laws of motion.
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I curious about Newton have how many laws?
 
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1. Newton's 1st law.
2. Newton's 2nd law.
3. Newton's 3rd law.
4. Newton's law of universal gravitation.
5. Newton's law of cooling.
 
What i know is 3.
1.Newton's 1st law
2.Newton's 2nd law
and 3rd law.
 
I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...
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