The Geodesic Problem on a Sphere | Calculus of Variations

[@tobology6781]

“An affront to your existence as a Chad” Golden.

[@whyiszeldagreen]

incredible

[@jaronfeld123]

@Kaikhosru Shapurji Sorabji Thanks for this I really needed it

[@MrCoffeypaul]

Indeed!

[@ethankorbin894]

I dont mean to be offtopic but does any of you know of a method to get back into an Instagram account? I was stupid forgot the login password. I would love any tips you can give me!

[@donaldkendall2645]

@Ethan Korbin instablaster :)

[@michaelzumpano7318]

This was sooo compact and clear! You included the right amount of detail. Best introduction to geodesics ever! Good job.

[@vaibhavyenare773]

KZbin chodke books padna chalu kr ussme bhalayi tera samjha kya , youtube se gahr ny chalta

[@sao8681]

I was going through this problem today but got stuck somewhere. I kept surfing the net for some help n watched two three videos from other sources but they were not helpful. At last, I clicked this video with less hope that it would help me but after watching for few minutes, I got pretty impressed with his explanation. I ended up hitting the like button n subscribing the channel. Indeed it has cleared all my doubts. Thank you so much 😊

[@reyhans3555]

I've been working on this for four hours from my professor's notes, and thanks to you, I've learned five minutes.

[@Astro-X]

10 XD

[@vaibhavyenare773]

Arey tu hamre modi sir ka notes padha rehta tujhe bss 12 gante lagte

[@Physixfixer1]

I looked at Geodesic from a different angle but now am with it. Thank-you sir

[@NikolasVassilev]

Most beautiful mathematics I've seen. Thank you.

[@skeletonrowdie1768]

this is calculus on steroids

[@Minnxlla]

* gets final exam review problem of finding the geodesic on a cylinder * Faculty of Khan: I got you

[@declanwk1]

thank you for all the useful detail in going through this calculation

[@78anurag]

Holy shit this was enlightening thanks. Always thought of solving of solving the geodesic problem but never came across a calculus of variations method.

[@PrettyMuchPhysics]

Concise and clear explanation! 👍 BTW, 7:30 great drawing!

[@vaibhavyenare773]

wotha inga drawing pakkava vanthe explaination mairu mari irriku taivoli

[@joshuawatt5582]

This man is a living legend

[@ashleybellasxo]

Hi just wanted to say thanks! I’m interested in calculating the bearing (or some people called it azimuth) of a point B as viewed from a point A on a sphere. This is exactly the head start I needed.

[@roshanpanda1662]

Thankyou sir for such a brief derivation.....

[@miqaelgali3161]

more vids: about conservation laws derivated by calculus of variations and how to find if stationary function is min or max. for functional - would be awesome

[@sudhir2854]

Ah, finally i can complete my assignment.

[@shinkolan1985]

this channel is gold!

[@abdulkaderalsalhi557]

Thank you for your effort; a great journey in shortest time to bring a hard topic home... Good Luck.

[@clem2874]

man you are a life saver.. thanks a lot for this video....

[@vaibhavyenare773]

Bacha liya bhai tujhe shukar maan

[@eamon_concannon]

4:00 Slightly more accurately, we need k^2< sin^2(phi)

[@srujaniam9762]

Amazing explanation

[@ritvikverma606]

Can you please post the boundary conditions calculations as well?

[@milessodejana2754]

you can also find ds by simply drawing an infinitesimal path that has side rd\phi and rsin\phi d\theta and use the Pythagorean theorem to find ds, if it helps :>

[@krisbarc4927]

Very cool video ! Have you ever tried to solve the geodesic problem on the Poincaré half-plane ? It is also a nice one in my opinion :)

[@FacultyofKhan]

Looks like a more complicated problem than what I'm used to ahahah. But once I get through Differential Geometry + the metric tensor + the geodesic equation, it seems like an exotic example I could take a crack at. Thank you for the suggestion and kind words!

[@frankdimeglio8216]

@@FacultyofKhan Greetings. Reply please. ON THE ESSENTIAL AND NECESSARY RELATION OF E=MC2 AS F=MA (IN BALANCE), AS ELECTROMAGNETISM/energy is clearly gravity AS WHAT IS NECESSARILY POSSIBLE/POTENTIAL AND ACTUAL IN BALANCE: Energy has/involves GRAVITY, AND ENERGY has/involves inertia/INERTIAL RESISTANCE. Gravity AND ELECTROMAGNETISM/energy are linked AND BALANCED opposites, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity ! Gravity is ELECTROMAGNETISM/energy. The stars AND PLANETS are POINTS in the night sky. ACCORDINGLY, the Moon AND THE PLANETS move AWAY very, very, very slightly in comparison to WHAT IS THE SUN !! SO, carefully consider what is THE EYE !! Finally, think about what is the speed of light (c). E=MC2 is CLEARLY F=ma (ON BALANCE), AS ELECTROMAGNETISM/energy is gravity ! Magnificent. It ALL does CLEARLY make perfect sense ON BALANCE !!! This does explain the fourth dimension, including the term c4 therewith !!! "Mass"/ENERGY involves BALANCED inertia/INERTIAL RESISTANCE consistent with/as what is BALANCED electromagnetic/gravitational force/ENERGY, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. (Gravity IS ELECTROMAGNETISM/energy.) Gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE, AS E=MC2 IS F=ma, AS ELECTROMAGNETISM/energy is gravity ! Accordingly, the rotation of WHAT IS THE MOON matches it's revolution. So, notice that what is THE MOON is basically dead or inert ON BALANCE !!! GREAT. INDEED, objects fall at the SAME RATE (neglecting air resistance, of course); AS ELECTROMAGNETISM/energy is gravity; AS E=MC2 IS F=ma ON BALANCE !!! The sky is BLUE, AND what is THE EARTH is ALSO BLUE !! (Notice what is THE EYE !!) ELECTROMAGNETISM/energy is gravity. E=MC2 is CLEARLY proven to be F=ma ON BALANCE, AS gravity IS ELECTROMAGNETISM/energy !!! Great. Carefully consider what is the Sun. (Very importantly, outer "space" involves full inertia; AND it is fully invisible AND black.) Consider what is BALANCED observer dependent/related experience ! Carefully consider the man (INCLUDING what is THE EYE) who actually IS in outer "space". NOW, think about TIME; AS the stars AND PLANETS are POINTS in the night sky. INSTANTANEITY is therefore FUNDAMENTAL to what is the FULL and proper UNDERSTANDING of physics/physical experience. So, here we have established what can basically and sensibly be understood as constituting a one dimensional relation. A white dwarf star is about the size of the Earth, AND it is ALSO (ON BALANCE) the PROJECTED form or fate (IN TIME) of the Sun !!! Great !!! Stellar clustering ALSO proves (ON BALANCE) that E=MC2 IS F=ma, AS ELECTROMAGNETISM/energy is gravity. Against what is outer "space", why would the Sun maintain what is its "preferential" existence ? The answer is THE EYE/the observer. INDEED, notice that THE EYE is invisible AND VISIBLE IN BALANCE; AS E=MC2 IS F=ma; AS gravity IS ELECTROMAGNETISM/energy ! Now, ON BALANCE, consider what is the speed of light (c). A galaxy is basically FLAT. SO, think about a TWO dimensional surface OR SPACE (ON BALANCE) as well. HALF of the galaxies are "dead" or inert, AS ELECTROMAGNETISM/energy is gravity; AS E=MC2 IS F=ma !! Carefully consider what is THE EYE !! TIME dilation ULTIMATELY proves (ON BALANCE) that E=MC2 IS F=ma, AS ELECTROMAGNETISM/energy is gravity. INDEED, TIME is NECESSARILY possible/potential AND actual IN BALANCE; AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. Gravity IS ELECTROMAGNETISM/energy. Great. E=MC2 IS F=ma. This NECESSARILY represents, INVOLVES, AND DESCRIBES what is possible/potential AND actual IN BALANCE, AS ELECTROMAGNETISM/energy is gravity. Gravity IS ELECTROMAGNETISM energy. Excellent !!! BALANCE and completeness go hand in hand !!! GRAVITATIONAL force/ENERGY IS proportional to (or BALANCED with/as) inertia/INERTIAL RESISTANCE, AS E=MC2 IS F=ma ON BALANCE; AS ELECTROMAGNETISM/energy is gravity. Gravity IS ELECTROMAGNETISM/energy. Consider WHAT IS THE MAN who IS standing on what is THE EARTH/ground. Touch AND feeling BLEND, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. Think about TIME !!! NOW, the stars AND PLANETS do REMAIN AS what are POINTS in the night sky. TIME dilation ULTIMATELY proves ON BALANCE that ELECTROMAGNETISM/energy is gravity, AS E=MC2 IS F=ma. This explains the cosmological redshift AND the "black holes". Gravity IS ELECTROMAGNETISM/energy. It ALL CLEARLY makes perfect sense, AS BALANCE AND completeness go hand in hand. Gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE, AS E=MC2 is CLEARLY F=ma ON BALANCE, AS ELECTROMAGNETISM/energy is gravity. The fact that the rotation of WHAT IS THE MOON matches it's revolution is not a meaningless coincidence, AS the fact that both the Sun AND the Moon are the SAME SIZE in the sky is not some sort of a meaningless coincidence. Notice the match with the size of what is THE EYE. Magnificent !!! It ALL CLEARLY makes perfect sense, AS BALANCE AND completeness go hand in hand. E=MC2 IS CLEARLY F=ma ON BALANCE, AS gravity AND ELECTROMAGNETISM/energy are linked AND BALANCED opposites; AS ELECTROMAGNETISM/energy is gravity ON BALANCE !!! Gravity IS ELECTROMAGNETISM/energy. Great !!! Carefully consider what are the POINTS in the night sky ON BALANCE !!! Think about the necessity (or essential nature) of TIME, AS E=MC2 IS clearly F=ma ON BALANCE !!! Great. It is proven. ("Mass"/ENERGY IS GRAVITY. ELECTROMAGNETISM/energy is gravity. E=MC2 IS F=ma.) Consider what is THE SUN AND what is THE EARTH/ground in DIRECT comparison, AS the stars AND PLANETS are POINTS in the night sky !!! GREAT !!! E=MC2 IS F=ma !!!! By Frank DiMeglio

[@yugeshkeluskar]

7:04 or can we *insert vsause music*

[@Peter_1986]

That's what I was thinking. xD

[@jacobvandijk6525]

It's even better when you speed up the video ;-)

[@andresyesidmorenovilla7888]

Integral table comment just broke me😂😂😂😂😂. Excellent video series btw👍

[@soccerbels7947]

Hahah fr lol

[@ZanielWalker]

AS A CHAD! love it!

[@alessandrotripoli814]

Regarding 5:34, I understand the trig sub, but on an integration table I am finding it to be arcsin(u/a)+c not arccos. Could you help explain why? Also should the second trig sub be called u? The last one is already using u and the integral is with respect to du so I am also a bit confused here. Thanks

[@thakurbabu1]

arcsin(x) = pi/2 - arccos(x). Notice the negative sign outside the integral, the pi/2 gets absorbed in the integration constant

[@FeanorPhys]

@@thakurbabu1 Thank you for the explanation. I had the same doubt.

[@maximdewildt219]

A question relating to the problem: if we find the second variation of this curve it is easy to show it is greater than zero, and therefore solns to euler-lagrange equations are minima. If instead we parametrise the curve as Phi(theta) why doesn't the result about second variation before imply that the second variation of the functional with Phi(theta) is also greater than zero?

[@rubenoliveira5305]

After seeing this video I realized how dumb I am, oh god.

[@fairpoor]

Hahahaha

[@Geert.Van.Boxelaer]

I was going to write this exact same comment :D Like, how do I calculate the great circle distances of a 3V tetrahedral geodesic projection, with a radius of say, 2,2m? How hard can it be ...

[@yarooborkowski5999]

Could You make some videos in the future about conservation laws derivated by calculus of variations? I mean Noether's theorem in classical mechanics and later in relativity. Best regards

[@FacultyofKhan]

Good suggestion! I plan to work on that later!

[@yarooborkowski5999]

@@FacultyofKhan Cool! I am waiting patiently for it. All the best for You.

[@josephanderson7237]

Great job, Great circle. 😊

[@solfeinberg437]

Maybe I need some pre-requisites but this is impossible to follow. Perhaps you have other videos, but I was hoping for something that would get me to the great circle solution for distances between points (on the surface of the earth) and illustrate the calculus of variations aspect of the problem, given a familiarity with calculus (and trigonometry / geometry).

[@sandeeptiwari5189]

I was stuck on the final integration, thanks

[@JJZB9]

Perfect explanation!

[@vaibhavyenare773]

Yede Kya samjha tujhe fukat ka pateli ( English Translation - What did you understand from this ?)

[@garyjenkins]

Interesting! Doesn't this give two solutions, where the minimum extrema is the shorter segment of the great circle passing through A and B, and the maximum extrema is the longer segment of the great circle through A and B? Pretty cool!

[@benjaminciotti3462]

Great video! Do you speed up the writing?

[@jameswilson8270]

Great video!

[@toseeornot2see]

Hi guys, I was hoping to see christoffel symbols (relativity), but this was a deep dive into calculus.

[@eamon_concannon]

We can also show that the geodesic on a sphere is part of a great circle by replacing Φ with π - Φ in θ(Φ) = arc cos (βcotθ) and showing that a point (x,y,z) on geodesic becomes (-x,-y,-z) which is on the same geodesic . WLOG we can set θ_0 = 0.

[@blzKrg]

Can anyone explain how he calculated the value of theta using substitution? I tried substituting the values he said but i am not getting anywhere

[@soumyasamratmandal1437]

Great work bro !

[@JS-gr9mz]

I think there’s a small mistake, because he lost the R in the partial derivative of F with respect to theta‘ in 3:20

[@darkaleksboy1548]

Noticed that too, but since it is a constant, maybe we can put it outside the integral and derivation steps?

[@florian-ij8rx]

At 9:07, why can you apply arccos? I don't see why \theta - \theta_0 is \in [0,\pi]... Furthermore, how to find an equation if \sin (\phi) = 0? Thanks you very much for your answer!

[@Skibidi-9z]

Great video.. Very well explained.. It helped me alot

[@humblehmathgeo]

You are amazing! Thank you sooo much!

[@MrChicken1joe]

Hey, thanks for the content. I dont get how you used the chain rule to express dx, dy, dz as d(theta) and d(phi). What does it mean if you just write dx instead of dx/dt? What is the inner an what ist the outer function here? Why do you just add the components of the partial derivatives dx etc.? And you used different "d", why? Whats the difference?

[@leif1075]

Why would anyone think of making a cotangent substitution when you only have sine in the equation.? Or Cosine ..can you please add a justification for that?

[@TheBigWazowski]

Dang coulda used this a couple months ago for my mechanics hw. Great video tho

[@FacultyofKhan]

Thank you! Hopefully you still learned something useful for your future assignments!

[@michaelgiglia2158]

Great video. I'm relatively new to differential geometry but I'm slowly trudging through everything. How would one go about getting the geodesic for a submanifold in R3 that doesn't have continuous curvature like the sphere. For example the "soft cube": x^4 + y^4 + z^4 = 1. I want to use this also be able to generate the geodesic between two points on this manifold (and subsequently be able to get the exponential map and logarithmic map). Also, what if I have a manifold that is smooth, but cannot be defined via a single expression? For example imagine a cube that has it's edges and verticies all curved such that they remain C-infinity smooth. Is it possible to also generate a geodesic for this? Or would this require a shooting method of sorts to solve for the geodesic?

[@بابکعطارها]

Hi you have a mistake in the integral. Integral of du/sqrt(a^2 -u^2) is equal to arcsin not arccos which you have written.

[@freddyfozzyfilms2688]

you are my hero

[@yarooborkowski5999]

Perfect. Thanks

[@karachiabubakari6092]

good introduction

[@prasannacs5358]

A geodesic does make sense if we take a 2 dimensional or a 3 dimensional Cartesian space. But if you map this to more than 3 dimension where do we even get to define a point in a space of more than 3 dimensions? Does it even make sense? ex;- a point in 4 dimensional space? Does this mean Geodesic or the shortest path has no definition beyond 3 dimensions? That is exactly where it becomes more counter intuitive and I believe calculating geodesic may make more sense.

[@parastooalimanish8526]

!That's amazing Thanks.can someone say what is app that he right with it and explain this video pls???

[@cesarmvbr]

Awesome!

[@miqaelgali3161]

plane equation through the origin is same as great circle equation ? in the great circle equation some point you should have R=constant term

[@FacultyofKhan]

I did set rho (the radial distance) to a constant R (the radius of the sphere) at about 1:09.

[@miqaelgali3161]

@@FacultyofKhan yes, but in the end Great circle equation should have some R parameter because it is a great circle of sphere. and if you take plain equation Ax+Bx+cx=0 - (is this a great circle equation ? - no) and put there spherical coordinates it would be same plain not the circle - am I wrong ? so what you are saying (I think) that plain equation in spherical coordinates and great circle has same equation ? - I don't really get that. x^2 +y^2+z^2=R^2 ax+bx+cx=0 this should be a great circle equation. because circle is one dimensional object. I just don't get that part of vid. and correct me if I am wrong

[@ayandandapat6920]

@@FacultyofKhan sir, why you are assuming great circle equation with a plane Equation just bcoz its circumference intersects the plane? ( Ref. 7:50)

[@cristianodejesus.9539]

@@FacultyofKhan How to calculate the equatorial diameter of the earth?

[@tripp8833]

0:56 much easier to just use line elements from cartesian to spherical

[@Rex-xj4dj]

I kinda just followed along until the integration of which theta was equal to, ain't no way I am gonna not be lazy and let you do the work for me lol

[@mktsp2]

Nice video but why don t I write any formulas? A couple of them wouldn’t t hurt!

[@kingplunger1]

At 3:20 you missed a factor R in the partial derivative and you might want to talk am little slower, but other than that, good video

[@nativesg3644]

@8.37 May I ask what the trigonometric identities used are?

[@FacultyofKhan]

Basically a variation of cos(a-b) = cos(a)cos(b) + sin(a)sin(b). See here for more (page 3): www.mathcentre.ac.uk/resources/uploaded/mc-ty-rcostheta-alpha-2009-1.pdf

[@chymoney1]

Is this Kahn academy for harder maths?

[@FacultyofKhan]

I make university-level videos on Maths and Physics, so yeah basically.

[@maxwellsequation4887]

@@FacultyofKhan Lol, great!!

[@ElFazo]

Great vidéo thanks

[@mukundyadav6913]

at 2:15, why was our variable of integration "phi" and not "theta"?

[@sidereal6296]

I love you dude

[@YossiSirote]

Why is the integral from phi A to phi B? Why are we integrating only wrt phi?

[@abusufian5391]

whenever you are taking the square root, you are taking only the positive root. why?

[@thevegg3275]

Is there an easier way to find a geodesic on a sphere? Place two points anywhere on a sphere. Using these any-two points you can define a great circle. Useing this great circle, define a cross-section. Viewing this cross-section you will see the two points between which you can trace a curve (following the outer edge of the sphere). And using the length of a curve formula you will determine the shortest distance and straightest path between these two points. No need for a metric tensor or Christoffel symbols. Thoughts?

[@KurtSansom]

Why isn’t the integral a double integral in Theta and phi?

[@lujason677]

I love this

[@adamboussif8035]

can someone explain why d(theta)/d(phi) = (thetha)' isn't theta a function of time ? or does it actually depend on phi ?

[@avamohammed7716]

please can you give me the book name from which you got information

[@lighthomefood3812]

Thank you😊

[@user-jj5se2is1w]

Hi, I didn't really understand what the final equation means, like, how do you find the distance between two points on a sphere using that equation?

[@RJ-fw6sk]

They lie on both the sphere and plane..So he just solved both of them

[@MrCoffeypaul]

Is R constant in a Geodesic?

[@FacultyofKhan]

For a geodesic on a sphere, yes, R would be a constant. For example, a geodesic on Earth (roughly a sphere) would have an R equal to the radius of Earth.

[@sagarghule7029]

Sir how to prove geodesic is complete on surface s in R^(n+1)

[@SMECHOULAN]

I thought this video was going to be the equation of the geodesic for general relativity... is there a connection?

[@tonmoydeka7319]

precise

[@carlossanchezaguilera1688]

When you use the substitution, what happen whit csc^2 φ?

[@maddog1918]

wow only 1:30 in to the video and I realize I need to go back and start at middle school math

[@souslicer]

wait, L doesnt have any theta prime but has theta. so df/dtheta should not be 0?

[@adityauppal8287]

dtheta/dphi is theta prime.

[@mullachv]

If the points A and B are on the same hemisphere (say, north), the geodesic is not part of the great-circle - is it?

[@abrahamtan5766]

You can always have a plane passing through the origin of a sphere to intersect with any two points on the surface of the sphere So any two points can be connected by a great circle

[@FacultyofKhan]

Agreed with Abraham. If you recall from Geometry, 3 points determine a unique plane, so 2 points on the sphere (even same hemisphere) + one at the center would determine a unique plane whose intersection with the sphere's surface would give the great circle.

[@hendrixgryspeerdt2085]

I know he you’re supposed to use sin^2 + cos^2 = 1, but does anyone know how you simplify the huge mess of sqrt(dx^2 + dy^2 + dz^2) into the formula for ds at 2:10 ?

[@aswathik4709]

ds^2 = dx^2 + dy^2 + dz^2, it is the distance between 2 points in the 3D cartesian plane. so, you have to square the dx, dy, dz term and add them together some of the terms will cancel out and some others can be reduced using the trig identity to get the equation, and finally if we take the common terms out and take the square root you will get the equation shown there

[@jerrydurkan8793]

all your derivatives are mixed up when you differentiate dx with respect to theta you get rcos(theta)cos(phi) theta ??

[@sukranochani5764]

thankyou sir

[@liyaelizabethantony5675]

isnt it z = r cos theta?

[@ingenuityreloaded2.089]

Cylinder is a good person.

[@Eduardo-hv1yh]

Can someone please help me understand the step at 3:47 when both sides have θ'^2 goes to only the left hand side having it?

[@nkhr2]

just distribute the K^2 on the right and then gather the θ'^2 on the left

[@kaustuvregmi1469]

5:06 it doesn't results the same. √u^2 + k^2 shows up in denominator 😳

[@yangvazquez]

5:55 where did minus sign go?

[@maxwellsequation4887]

The derivative of arcos(x/k)=1/√(k^2-x^2)

[@jerrydurkan8793]

bro the z coordinate is Z= Rcos(theta) not Rcos(phi)

[@adscft7597]

I think absolute of K should be less than 1

[@gibson7392]

"Theeeeeta"

[@solfeinberg437]

0:58 z = r cos theta not r cos phi.

[@solfeinberg437]

Oh, maybe you're okay - I was thinking theta was south from z axis.

[@Haighdr02]

you have all your thetas and phis mixed up... wtf

[@admiralhyperspace0015]

4:28. How in God's name you know that this substitution would work? Please anyone share with me the art of u substitution.

[@Peter_1986]

It is probably some sort of trigonometric substitution that is based on labelled sides of a triangle. You actually made me curious about that, so I will take a look at it myself, and then I will tell you about it if I figure it out.

[@beitexue2490]

hardcore