Feynman’s trick of parametric integration applied to Laplace Transforms

Parametric Integration is an Integration technique that was popularized by Richard Feynman but was known since Leibinz’s times. But this technique rarely gets discussed beyond a niche set of problems mostly in graduate school in the context of Contour Integration.

A while ago, having become obsessed with this technique I wrote this note on applying it to Laplace transform problems  and it is now public for everyone to take a look.

( Link to notes on Google Drive ) 

I would be open to your suggestions, comments and improvements on it as well. Cheers!

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 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 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)

Continued fractions are a fascinating way to represent never ending, never-repeating decimal expansions like pi or e.

As a fun exercise on this Friday evening, take out your smart phone and open the calculator app and try this :

What do you get? Do you recognize this number? If you did, then try answering why it mysteriously popped out of nowhere to represent this continued fraction.

Have fun!

Notes from a discussion about Hands

Recently, I found myself in the middle of a discussion with some close friends about the physics and maths that one can associate with our hands. Here are some notes from that discussion.

We apologize in advance if this post seems too factual and a bit disoriented, we couldn’t find a better way to share this information.

Hand Protractor

If you stretch out your hand, you will notice that the fingers on your hand can be calibrated to be used as a protractor like so:

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Count from 0 to 1023 **

Want to count till 1023 just on your hands? Well, just assign each finger a binary value and chug along:

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Also: Check out the binary dance by Vihart

Rock, paper, scissor, Lizard,Spock

A simple of rock,paper, scissor is by itself a lot of fun and is a playground for game theorists and mathematicians

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But adding new gestures to the game reduces the odds of getting a tie but increases the complexity of the game. One popular five-weapon expansion is “rock-paper-scissors-Spock-lizard”, invented by Sam Kass and Karen Bryla.

Remembering number of days of the month

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                                                 Source

The golden truth

This picture gets thrown everywhere on the Internet to illustrate the divine connection between hands and the golden ratio.

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BUT in a study using X-rays conducted at St. Luke’s Medical Center, Chicago, it was found that only 1 of 12 bone length ratios contained the ratio phi in the 95% confidence interval.

And it was concluded that:

The application of the Fibonacci sequence to the anatomy of the human
hand, although previously accepted, is a relationship that is not
supported mathematically.

Drip system on a farm land?

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                                          Source Video

What looks like a drip irrigation system on a farm land is actually a close up of your hands when sweating.

The reason why this visualization is fascinating is because modern techniques (like full field optical coherence tomography (FF-OCT) )in Forensics can extract sweat pores information from the fingerprint offering more information about the perpetrator crucial to the investigation.

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Snapping fingers and cracking knuckles

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When you snap your fingers, in order to get the satisfying full “snap” sound , the fast-moving second finger
must hit both the palm and a small portion of the third finger.

If it
hits only the palm and not the third(ring) finger, there will be a significant
reduction in the total ‘snap’ sound. Try it out!

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When you get tired of snapping your fingers, try cracking your knucles. The sharp sound that you hear is caused by the collapse of a cavitation bubble (see image above) in the sinuovial fluid present at the joint.

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Fingers, shadows and the transit of Venus

If you are wondering how on earth can all of these words be possibly related, then try answering this question:

When you bring two of your fingers closer in the back drop of a light source, long before your fingers actually touch, the edges magically seem to touch each other.

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Once you hypothesize a solution for this, head over here to understand what’s actually happening.


That’s about it! Hopefully not everything mentioned in this post seemed too trivial to you. Each topic mentioned in this post deserves a post on its own and is a lot of fun to explore.

We encourage to play around, have fun and have a great weekend ahead.

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** The structure of sign language is yet another thing that is absolutely fascinating to explore. Click here and here  to quench your thirst for knowledge

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!

Using Complex numbers in Classical Mechanics

When one is solving problems on the two dimensional plane and you are using polar coordinates, it is always a challenge to remember what the velocity/acceleration in the radial and angular directions (v_r , v_{\theta}, a_r, a_{\theta} ) are. Here’s one failsafe way using complex numbers that made things really easy :

z = re^{i \theta}

\dot{z} = \dot{r}e^{i \theta} + ir\dot{\theta}e^{i \theta} = (\dot{r} + ir\dot{\theta} ) e^{i \theta}

From the above expression, we can obtain v_r = \dot{r} and v_{\theta} = r\dot{\theta}

\ddot{z} =  (\ddot{r} + ir\ddot{\theta} + i\dot{r}\dot{\theta} ) e^{i \theta}   + (\dot{r} + ir\dot{\theta} )i \dot{\theta} e^{i \theta} 

\ddot{z} =  (\ddot{r} + ir\ddot{\theta} + i\dot{r}\dot{\theta}  + i  \dot{r} \dot{\theta} - r\dot{\theta}\dot{\theta} )e^{i \theta} 

\ddot{z} =  (\ddot{r} - r(\dot{\theta})^2+ i(r\ddot{\theta} + 2\dot{r}\dot{\theta} ) )e^{i \theta} 

From this we can obtain a_r = \ddot{r} - r(\dot{\theta})^2 and a_{\theta} = (r\ddot{\theta} + 2\dot{r}\dot{\theta}) with absolute ease.

Something that I realized only after a mechanics course in college was done and dusted but nevertheless a really cool and interesting place where complex numbers come in handy!