Why Does Pi Appear in Various Mathematical Solutions?

[Naty1]

What is the physcial signifcance of pi's appearance in a mathematical solution, if any??

Pi, the circumference of a circle divided by diameter, shows up in all sorts of mathematical solutions..classical, quantum, sometimes in unexpected mathematical results.

As an example, Unruhs law is a simple equation giving the equivalent energy kT of a uniformly accelerating particle, as: kT = ha/ 2(pi)c (a quantum phenomena)

Source: Wikipedia, http://en.wikipedia.org/wiki/Bill_Unruh
I don't know how it was derived, so I don't know if pi was an input.

Another example is in simple harmonic motion

T= 2(pi) root(m/k) pi again! (a simple classical formulation)

In this case it likely arises as a result of formulation with trig functions and pi is used in the formulation and remains in the answer.


Is pi always in the output because it was an input...in other words does pi become an element of an answer only because it was an element of the original equation? Or can it emerge without being an input...and when it does what physcial significance can we attribute to a constant with a geometric origin??

 

[Crosson]

Pi appears because the geometry of space is an important part of physical processes. Factors of pi almost always come from doing integrations, it is not the case that they get arbitrarily put there.

can it emerge without being an input

Absolutely, for example the solution of the wave equation is derived from Newton's laws and it has pi in it for some cases of boundary conditions:

http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/wavsol.html#c2

The equation they refer to as "the one-dimensional wave equation" does not have pi in it, but its solutions do (how to find those solutions is too complicated to be explained, so this webpage just gives them to you).

 

[Tac-Tics]

Or can it emerge without being an input...

Pi is magical. But then again, lots of numbers are, so Pi isn't terribly special.

See http://en.wikipedia.org/wiki/List_of_formulas_involving_π for a long list of interesting places Pi can pop up.

For a good example of a proof where Pi seems to appear out of nowhere, check out the gaussian integral: http://en.wikipedia.org/wiki/Gaussian_integral You'll notice that we find circles in places you'd never expect!

Also, it can be demonstrated in number theory that the probability of two integers being relatively prime (having no common factors) is 6 over Pi squared =~ 60%.

Pi is mostly an artifact of mathematics, and so whenever you see Pi show up in a physical theory, it boils down to the mathematics of the theory. Euclidean space in particular is so full of Pi, it will make your head spin.

 

[Andy Resnick]

From:

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

"THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."

 

[HallsofIvy]

pi basically shows up in problems where you have some kind of circular or spherical symmetry. Since physics problems tend to be independent of a rotation of the coordinate system, they is a natural symmetry.

 

[Naty1]

pi basically shows up in problems where you have some kind of circular or spherical symmetry

.

Great observation!...I've been out of school and calculations for many years and forgot about some relationships I've used, like Euler's identity...


And the Wikipedia reference is fantatsic!

Fascinating!

I have to think about why some functions of (1 +/- x^2) involve pi?...I guess because x^2 +y^2 is a circle...

 

[Tac-Tics]

.

Great observation!...I've been out of school and calculations for many years and forgot about some relationships I've used, like Euler's identity...


And the Wikipedia reference is fantatsic!

Fascinating!

I have to think about why some functions of (1 +/- x^2) involve pi?...I guess because x^2 +y^2 is a circle...

Exactly. Pi, circles, e, and the pythagorean theorem are all best friends forever.

Really, check out the proof on the gaussian distribution. How do you calculate the area under the curve e^{-x^2}? Well, you take the integral to find the area: A = \int e^{-x^2} dx. Now, square both sides: A^2 = (\int e^{-x^2} dx)(\int e^{-y^2} dy) = \int e^{-(x^2 + y^2)} dx dy. Holy crap, x^2 + y^2! The rest of the proof involves using the squeeze theorem of calculus using well-known bounding functions, and the answer is \sqrt{\pi}! Spooky, spooky!

 

[atyy]

Exactly. Pi, circles, e, and the pythagorean theorem are all best friends forever.

Really, check out the proof on the gaussian distribution. How do you calculate the area under the curve e^{-x^2}? Well, you take the integral to find the area: A = \int e^{-x^2} dx. Now, square both sides: A^2 = (\int e^{-x^2} dx)(\int e^{-y^2} dy) = \int e^{-(x^2 + y^2)} dx dy. Holy crap, x^2 + y^2! The rest of the proof involves using the squeeze theorem of calculus using well-known bounding functions, and the answer is \sqrt{\pi}! Spooky, spooky!

A two dimensional Gaussian is a stack of circles!

 

[ValenceE]

Hello to all,


since all motion is helicoidal, implying a root of circular trajectories, then Pi is essential.





VE