Why do objects experience acceleration due to gravity?

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Objects accelerate due to gravity based on the context of the discussion. In basic physics, this acceleration is explained by Newton's law, where acceleration (a) equals the force of gravity (F) divided by mass (m). In more advanced discussions, such as general relativity, objects accelerate because spacetime itself moves past them, with the Earth's surface accelerating upwards at approximately 10 m/s² relative to spacetime. This dual perspective highlights the difference between classical mechanics and modern physics in understanding gravitational acceleration.
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Why do objects accelerate due to gravity?
 
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It depends on the context. If you are talking first semester physics or something like that, then the acceleration is due to the force of gravity according to Newton's law: a = F/m. If you are talking more sophisticated as in general relativity, then objects accelerate because spacetime moves past them. The surface of the Earth accelerates upwards at a rate of about 10 m/s<sup>2</sup> with respect to spacetime.
 
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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