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I know that the Lagrangian $L$ is defined to be $T-V$, i.e. the difference between kinetic energy and potential energy. Also the Action $S$ is defined to be $\int Ldx$ and from this we can derive Newton's 2nd law of motion.

  1. If we get Newton's second law out, does it mean that the formulation is correct? Couldn't it be just a coincidence?

  2. Where do we derive these expressions for the Action and for the Lagrangian from?

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This site derives the principle of least action from Newton's laws. http://www.damtp.cam.ac.uk/user/tong/dynamics/two.pdf

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Newton's second law, $\mathbf{F}_{net}=\dot{\mathbf{p}}$, is the definition of force. Lagrangian and action are defined to be $ T-V $ and $\int L\: \mathrm {d} t$ (and not $\mathrm {d} x $) respectively. You don't derive anything from anything here (however we can talk about how $ T $ and $ V $ come about).

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    $\begingroup$ Newton's second law is not the definition of force. Force, mass, displacement, and time are defined elsewhere. Newton 2 is relationship between these quantities. The Lagrangian is $T-V$ only for mechanical systems. I wouldn't say that $T-V$ defines "Lagrangian". $\endgroup$ – garyp Apr 4 '14 at 13:30
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    $\begingroup$ I don't agree with you on the first point, though the second one is spot on. $\endgroup$ – user132181 Apr 4 '14 at 13:33
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    $\begingroup$ My first point is correct. Force is a dynamical quantity that is defined by how it is measured. A working definition of force is: the quantity that an ideal spring scale measures. $\endgroup$ – garyp Apr 4 '14 at 13:53
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    $\begingroup$ I should have finished the argument: Time is what a clock measures. Displacement is what a ruler measures. Mass is what a triple beam balance measures. Newton 2 written $\mathrm{d}^2 x/\mathrm{d}t^2 = F/m$ is a relationship among those four quantities, giving kinematical information (about motion) from dynamical inputs. $\endgroup$ – garyp Apr 4 '14 at 14:00
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    $\begingroup$ I'm not saying you can't measure mass, length and time. You can measure the mass $ m $ and the acceleration $\mathbf { a} $ of an object at time $ t $, multiply them together and say: the net force $\mathbf{F}_{net}$ on the object at time $ t $ is $ m\mathbf {a} $. It is completely consistent with mathematical formalisms. Also, I'm not the only one who thinks Newton's 2nd law is the definition of force. $\endgroup$ – user132181 Apr 4 '14 at 14:20

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