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This blog is primarily for me to blog my responses to books that I'm reading. Sometimes I blog about other stuff too, though.

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Monday, July 18, 2016

Newton: "Axioms, or Laws of Motion"

After stating the three laws of motion, Newton starts into corollaries.  The first two corollaries involve the parallelogram law of forces, resolution of forces into components, and an example with tension in ropes.  The next two corollaries (3-5) are more interesting, involving conservation of momentum (though not stated in that language; to Newton momentum was "quantity of motion") and the conservation of center of mass velocity.

Corollaries 5 and 6 show that the relative motion of a system of interacting bodies is unaffected if we move to a different inertial reference frame, or to a non-inertial reference frame at constant acceleration.

The Scholium shows us what a genius he is.  He cites prior work by Huygens, Wren, and Wallis on collisions, but notes that they only worked out elastic collisions.  He works out an example with an inelastic collision of two hard objects on pendula, shows how to estimate the effect of dissipative forces in a self-consistent manner, and then describes an experiment that he did to test his calculations (involving a 10 foot pendulum and errors no larger than 3 inches for the maximum height reached by objects, i.e. 2.5% error).

Having showed his skill as an experimental, he uses symmetry arguments and the impossibility of perpetual motion to demonstrate the validity of the Third Law for attractive forces, including a thought experiment on the stability of the earth as a self-gravitating object.  Since the Third Law can be shown (in the Lagrangian formalism) to arise from translational invariance, I like his use of symmetry arguments here.  It may be the earliest precursor of Noether's Theorem.

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