Importance of differential geometry in physics?

[cytochrome]

How important is differential geometry in physics? Can someone give me some applicable fields?

 

[DiracPool]

General relativity and finding the shortest distance to Grandma's house over the hills and far away.

 

[kweba]

...and finding the shortest distance to Grandma's house over the hills and far away.

Hahaha this is funny, I like your humor. Like! (sorry, there's no like button, but you get the gist.)

Are you referring to Geodesics (shortest path between two points)?

 

[DiracPool]

Are you referring to Geodesics (shortest path between two points)?

Yes, that's why Reimann developed his models originally, it was for land survey work. Little did he know what would become of it.

 

[kweba]

How important is differential geometry in physics? Can someone give me some applicable fields?

Like DiracPool said, Differential Geometry is the key mathematics in Einstein's General theory of Relativity.

It is used to derive Einstein's field equations to describe the curvature of Spacetime in the presence of a body of mass and energy. This curvature of Spacetime results to the phenomenon we know as gravity, including how the planets move in orbit around the Sun, among others. The mathematics (Differental Geometry) used in Einstein's field equations also predicted the existence of black holes, including concepts of the Event Horizon and Singularity as key and important regions in black holes. (I think it was the Schwarzschild solution to Einstein's equations who pioneered this prediction/discovery, being the first pioneering solution to Einstein's field equations.)

To learn and read more about the mathematics of General Relativity, and how Differential Geometry is extensively used in the theory and other areas of physics, here's some links:

http://en.wikipedia.org/wiki/Mathematics_of_general_relativity[/PLAIN]
(Granting the wikipedia article shows and discusses high level mathematics, so it might be hard to really get a grasp on them. I myself tried to read it, but lacking knowledge of tensors and differential geometry in general, makes it very difficult to understand)

http://en.wikipedia.org/wiki/Differential_geometry#Applications

Here's also the Wikipedia page of Differential Geometry

To quote the Wiki, under the Applications section:

In physics, three uses will be mentioned:

1. Differential geometry is the language in which Einstein's general theory of relativity is expressed. According to the theory, the universe is a smooth manifold equipped with a pseudo-Riemannian metric, which describes the curvature of space-time. Understanding this curvature is essential for the positioning of satellites into orbit around the earth. Differential geometry is also indispensable in the study of gravitational lensing and black holes.
   
2. Differential forms are used in the study of electromagnetism.
   
3. Differential geometry has applications to both Lagrangian mechanics and Hamiltonian mechanics. Symplectic manifolds in particular can be used to study Hamiltonian systems.

 

[kweba]

Yes, that's why Reimann developed his models originally, it was for land survey work. Little did he know what would become of it.

Ohh really? I didn't know that. I thought Riemann developed his models just for pure mathematical pursuits? Was it for real-world applications? And his works were primarily based from the earlier works of Gauss, right? (Wikipedia says that Riemann studied under Gauss, with Gauss being his doctoral advisor.) It was said that Gauss discovered/developed non-Euclidean Geometries but did not publish it.

By land survey, you mean: http://en.wikipedia.org/wiki/Surveying ? I must admit, it's the first time I heard about such a thing, atleast as a scientific field kind of sense. What is it about?

 

[Andy Resnick]

How important is differential geometry in physics? Can someone give me some applicable fields?

Other than GR, it's essential in continuum mechanics (stress-strain), both in the bulk and at deformable interfaces.

 

[WannabeNewton]

Are you referring to Geodesics (shortest path between two points)?

Hate to be technical but they aren't necessarily the shortest paths. They extremize but don't necessarily minimize. Other than what has already been mentioned, differential geometry is very important in more advanced formulations of classical mechanics.

 

[DiracPool]

I thought Riemann developed his models just for pure mathematical pursuits? Was it for real-world applications? And his works were primarily based from the earlier works of Gauss, right?

I don't know, maybe it was Gauss... One of those two. Check out this nifty video.

http://www.youtube.com/watch?v=pUHu6tw_1Mc&list=PLF56602BAC693237E&index=8

 

[kweba]

0Hate to be technical but they aren't necessarily the shortest paths. They extremize but don't necessarily minimize.[/QUOTE]

Oh thank you for correcting me, and clearing it out!

Other than what has already been mentioned, differential geometry is very important in more advanced formulations of classical mechanics.

Which are Lagrangian mechanics and Hamiltonian mechanics, right? Oh cool it says that in Wikipedia:

To quote the Wiki page of Differential Geometry, under the Applications section, the third item under Physics:

Differential geometry has applications to both Lagrangian mechanics and Hamiltonian mechanics. Symplectic manifolds in particular can be used to study Hamiltonian systems.

I did not know this. I thought since Lagrangian mechanics and Hamiltonian mechanics are still considered to be "classical" in Physics, I thought the mathematics utilized in these formulations are limited to Calculus/Analysis. I considered Differential Geometry to be relatively modern and advanced, so yeah. But then again, I still haven't studied even an undergraduate physics/mathematics course (Still not yet in University/College), so obviously I'm wrong to assume. :)

 

[Timo]

To answer the first question: It's pretty unimportant compared to e.g. differentiation and integration in R^n, linear algebra or differential equations.

 

[kloptok]

Differential geometry is also needed if one wants to understand the geometric setting of gauge theories which are formulated using a mathematical object called fiber bundles. In short, it is good to study differential geometry if you want to be a theoretical physicist.

 

[kweba]

I don't know, maybe it was Gauss...

Apparently:

Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it. This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory. Research on these geometries led to, among other things, Einstein's theory of general relativity, which describes the universe as non-Euclidean. His friend Farkas Wolfgang Bolyai with whom Gauss had sworn "brotherhood and the banner of truth" as a student, had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry. Bolyai's son, János Bolyai, discovered non-Euclidean geometry in 1829; his work was published in 1832. After seeing it, Gauss wrote to Farkas Bolyai: "To praise it would amount to praising myself. For the entire content of the work... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years."

http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#Middle_years_.281799.E2.80.931830.29

Riemann, in turn, pioneered and develop his own branch of differential geometry, Riemannian Geometry, which is the key specific mathematics from Differential Geometry used in GR.

To quote http://en.wikipedia.org/wiki/Riemannian_geometry:

http://en.wikipedia.org/wiki/Riemannian_geometry]Riemannian[/PLAIN] geometry originated with the vision of Bernhard Riemann expressed in his inaugurational lecture Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (English: On the hypotheses on which geometry is based)...Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them...It enabled Einstein's general relativity theory...

One of those two. Check out this nifty video.

http://www.youtube.com/watch?v=pUHu6tw_1Mc&list=PLF56602BAC693237E&index=8

This is a cool video, thanks!

 

[Sankaku]

Yes, that's why Reimann developed his models originally, it was for land survey work.

[citation needed]

There is no way this is believable without some kind of reference.