It’s Pi-Day! Time to celebrate maths!:D

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“The other planets dutifully followed the laws of planetary motion, but Uranus appeared to violate them. Breaking Kepler’s laws, Uranus moved too quickly for decades, then at the right speed, then too slowly. The observations weren’t easily dismissable, but their physical cause was unknown. An additional planet beyond Uranus, gravitationally tugging on it, offered a potential solution. Determining the mass, orbital parameters, and location of an unseen world presented incredible calculational challenges.”

On March 11, 1811, Urbain Le Verrier was born. As a mathematician of tremendous skill in France, he had only a passing initial interest in astronomy, until the 1840s, when the influential François Arago suggested that he take up the puzzle of Uranus’ orbit, which appeared to violate the laws of planetary motion. Le Verrier theorized that if there were an outer planet beyond Uranus with the right mass and orbital parameters, it could cause these observed orbital anomalies. On August 31, 1846, Le Verrier composed a letter detailing his predictions and sent it to the Berlin Observatory. On September 23, the letter arrived. That very night, the portion of the sky where Le Verrier claimed a new planet should be was clear, and less than one degree away from his location, there it was: the planet Neptune.

Anonymous asked:

Why

is it mr^2 omega and not some other weird formula that is conserved? Why not mr^3 omega or mr^2 omega^2 ?

This is a great question. And to be honest, there is no intuitive answer as to why it is defined this way or that.

Conservation laws can be understood better through the Lagrangian formulation of classical mechanics.

That’s the conservation of momentum for a free particle. It means that this quantity **mv **remains constant with time *(not m ^{2}v, not m^{2}v^{2 },just mv*).

And similarly for a rotating body, one can find that the quantity that remains constant wrt time is the angular momentum.

And that’s the best rationale using modern physics that can be provided for why Angular momentum takes the form that it does.

Any other form would just not be conserved. Sure, you can construct a Lagrangian that would give you the form that you need but that would not represent anything physical !

Hope that answers your question. Thanks for asking !

** If you have not heard about Lagrangian formulation of classical mechanics, the wiki article on Principle of Least action is a really good place to start..

The principle of Least/Stationary action remains central in modern physics and mathematics, being applied in thermodynamics, fluid mechanics, the theory of relativity, quantum mechanics, particle physics, and string theory.

If unit vectors always scared you for some reason, this neat little trick from * The story of i by Paul Nahin* involving complex numbers is bound to be a solace.

It allows you find the tangential and radial components of acceleration through simple differentiation. How about that!

Have a good one!

** r = r(t), θ = θ(t)