Category: math

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On the definition of  Angular MomentumAnonymous asked:Why is it…

On the definition of  Angular Momentum

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.

What makes the definition special?

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

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That’s the conservation of momentum for a free particle. It means that this quantity mv remains constant with time (not m2v, not m2v2 ,just mv).

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

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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.

Finding n roots of unity

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This leads to Quantization of θ  because there are only specific values that become possible for  θ  when we impose that after n rotations it has to return back to its same starting point. **

And the following are those values:

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The problem of finding the values of θ for given values of n is more generally known as the N roots of unity.  

We will leave it as an exercise, but the following animation plots all the possible the values of θ for integer values of n=2,3,4,.. from the above equation that we found:

                            Source

And those are the roots of the equation z^n  = z i.e if you start at these points on the unit circle and make n rotations you will get back to the same point that you started with.

Have a good one!

** A more physical way to think about this is matching boundary conditions.
For more insight on how boundary conditions lead to quantization, take a
look at this  post.

*** Finding out the n roots of unity (Video)

Never forget De Moivre’s formulaIf you ever find yourself…

Never forget De Moivre’s formula

If you ever find yourself trying to remember any one of those basic trigonometric formulas, the ideal starting point is the De Moivre’s formula:

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Above we have given you some examples of identities that can be easily derived using this formula. In fact, most trigonometric relations can be obtained from this formula after performing some basic algebra.

Try to obtain your favorite identity using this method and let us know how that went. Have a good one!

If unit vectors always scared you for some reason, this neat…

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)

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Introducing FYP’s book club

We are starting a new segment on the blog where we recommend one or two books in Math or Physics that everyone can read.

And this month it is : A Mathematician’s Apology by G.H.Hardy which is available for free to download here and here

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You are absolutely welcome to share your comments and reviews here once you are done. Also, if you would like us to check a book out, do let us know too!

Have a good one!