Geometry Differential Geometry: Book on its applications?

Click For Summary
SUMMARY

The discussion centers on the search for books that explore the applications of differential geometry beyond relativity. Participants recommend several resources, including "Geometric Mechanics" and "Applied Differential Geometry" by Burke, which cover applications in thermodynamics, Hamiltonian mechanics, and electromagnetism. Other notable mentions include "Discrete Differential Geometry" by Crane and works by Bruce J. West on fractional calculus. The conversation also touches on the relevance of differential geometry in fields like control theory and information theory.

PREREQUISITES
  • Familiarity with differential forms and differential geometry
  • Understanding of classical mechanics and its geometric applications
  • Knowledge of control theory and its mathematical foundations
  • Basic concepts of information geometry and stochastic geometry
NEXT STEPS
  • Research "Geometric Mechanics" for insights on differential geometry in classical mechanics
  • Explore "Discrete Differential Geometry: An Applied Introduction" by Keenan Crane
  • Investigate applications of differential geometry in control theory
  • Study Bruce J. West's works on fractional calculus for practical applications
USEFUL FOR

Mathematicians, physicists, and engineers interested in the practical applications of differential geometry in various fields, including mechanics, electromagnetism, and advanced mathematical theories.

s00mb
Messages
33
Reaction score
10
Hi, I'm already familiar with differential forms and differential geometry ( I used multiple books on differential geometry and I love the dover book that is written by Guggenheimer. Also used one by an Ian Thorpe), and was wondering if anyone knew a good book on it's applications. Preferably not just in the realm of relativity. I used to study a lot of pure mathematics topics and now I'm leaning towards applications and I've noticed that there is very little on the subject that I can find, which I think is a shame because it is my favorite subject (Fractional calculus is cool too, I even found some papers on fractional differential forms and geometry). Any suggestions? I'm flexible on this; it doesn't have to be a dedicated differential geometry book but I'd say if it has a few good chapters on applications that would be neat. I'm not too familiar on the subject of aerodynamics, does anyone know if that subject uses it? Thanks for your help! -James
 
Physics news on Phys.org
  • Like
Likes mpresic3, Abhishek11235, FourEyedRaven and 1 other person
Looks very promising, thank you!
 
  • Like
Likes Demystifier
Possibly useful [...these are on my to do list... someday]:

Hirani, Anil Nirmal (2003) Discrete exterior calculus. Dissertation (Ph.D.), California Institute of Technology. https://resolver.caltech.edu/CaltechETD:etd-05202003-095403

Crane, Keenan. Discrete Differential Geometry: An Applied Introduction
https://www.cs.cmu.edu/~kmcrane/Projects/DDG/
https://www.cs.cmu.edu/~kmcrane/Projects/DDG/paper.pdf

Bossavit, Alain. Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements
https://www.amazon.com/gp/product/0123885604/?tag=pfamazon01-20
related:
https://www.researchgate.net/publication/254470625_On_the_geometry_of_electromagnetism
https://www.researchgate.net/publication/200018385_Differential_Geometry_for_the_student_of_numerical_methods_in_Electromagnetism
https://www.researchgate.net/publication/242462763_Computational_electromagnetism_and_geometry_Building_a_finite-dimensional_Maxwell's_house
Daverz said:
Burke's Applied Differential Geometry

https://www.amazon.com/dp/0521269296/?tag=pfamazon01-20
The errata for the Burke's book is at
http://www.ucolick.org/~burke/forms/errata.ps
linked from http://www.ucolick.org/~burke/class/adg.html
 
  • Like
Likes Daverz and s00mb
Thanks for the additional references. I've started the homological algebra and have previously read some of Kranes paper I like them a lot. I haven't seen that before. I had a different book on discrete differential geometry but it was very jumbled with different topics kind of piled one on top of each other with no seeming attention to its order. I find the discrete stuff very interesting. I wonder if there is a discrete analog of hyperbolic geometry or if you can construct such a thing using Krane's stuff? I'll definitely read that thank you.
 
I do not know if this is still of interest to you

Differential geometry applied to classical mechanics is sometimes known as geometric mechanics.
https://www.amazon.com/dp/0387968903/?tag=pfamazon01-20
https://www.amazon.com/dp/038798643X/?tag=pfamazon01-20
https://www.amazon.com/dp/0821844385/?tag=pfamazon01-20
https://www.amazon.com/dp/0387967907/?tag=pfamazon01-20
https://www.amazon.com/dp/0521636361/?tag=pfamazon01-20
https://www.amazon.com/dp/0199212902/?tag=pfamazon01-20
https://www.amazon.com/dp/184816775X/?tag=pfamazon01-20
https://www.amazon.com/dp/1848167784/?tag=pfamazon01-20
https://www.amazon.com/dp/3319569511/?tag=pfamazon01-20

Control Theory
https://www.amazon.com/dp/1493930168/?tag=pfamazon01-20

Solid Mechanics
https://www.amazon.com/dp/9814616036/?tag=pfamazon01-20

Differential geometry applied to information theory is known as information geometry.
https://www.amazon.com/dp/B01BDF0CQM/?tag=pfamazon01-20
https://math.ucr.edu/home/baez/information/

Bruce J. West has written several books on fractional calculus. I know that he cares about applications.
 
Last edited:
  • Informative
Likes andresB
caz said:
I do not know if this is still of interest to you

Differential geometry applied to classical mechanics is sometimes known as geometric mechanics.
https://www.amazon.com/dp/0387968903/?tag=pfamazon01-20
https://www.amazon.com/dp/038798643X/?tag=pfamazon01-20
https://www.amazon.com/dp/0821844385/?tag=pfamazon01-20
https://www.amazon.com/dp/0387967907/?tag=pfamazon01-20
https://www.amazon.com/dp/0521636361/?tag=pfamazon01-20
https://www.amazon.com/dp/0199212902/?tag=pfamazon01-20
https://www.amazon.com/dp/184816775X/?tag=pfamazon01-20
https://www.amazon.com/dp/1848167784/?tag=pfamazon01-20
https://www.amazon.com/dp/3319569511/?tag=pfamazon01-20

Control Theory
https://www.amazon.com/dp/1493930168/?tag=pfamazon01-20

Solid Mechanics
https://www.amazon.com/dp/9814616036/?tag=pfamazon01-20

Differential geometry applied to information theory is known as information geometry.
https://www.amazon.com/dp/B01BDF0CQM/?tag=pfamazon01-20
https://math.ucr.edu/home/baez/information/

Bruce J. West has written several books on fractional calculus. I know that he cares about applications.

Yes I am still interested, which one would apply the most of geometric arguments in your opinion(in regard to the dynamics books)? I've seen stuff on information geometry, personally I prefer stochastic geometry over that. I am a little biased though because I have more experience with topics relating to the latter though.
 
s00mb said:
Yes I am still interested, which one would apply the most of geometric arguments in your opinion(in regard to the dynamics books)? I've seen stuff on information geometry, personally I prefer stochastic geometry over that. I am a little biased though because I have more experience with topics relating to the latter though.

I’ve gotten interested in advanced mechanics over COVID, so you are seeing a list of interesting things that I have found. I apologize for not having read and absorbed them all so that I can give a good review 😜
 
Last edited:
  • #10
caz said:
I’ve gotten interested in advanced mechanics over COVID, so you are seeing a list of interesting things that I have found. I apologize for not having read and absorbed them all so that I can give a good review 😜
That's no problem, the one about dynamics on manifolds seems to be the one for me to look at. I like manifold theory too, I imagine they'd apply some tensors or Riemann geometry in it.
 
  • #12
Check out "Global Calculus" by S. Ramanan. It is not about applications, but contains material/approach that is not generally discussed in books on differential geometry.
 
  • #13
love_42 said:
contains material/approach that is not generally discussed in books on differential geometry
Such as?
 
  • #14
Demystifier said:
Such as?
Sheaves, exact sequences, cohomology.
 
  • Informative
  • Like
Likes vanhees71 and Demystifier

Similar threads

Geometry Confusion about Differential Geometry Books
  • · Replies 11 ·
Replies
11
Views
3K
Geometry Geometrical books (differential geometry, tensors, variational mech.)
  • · Replies 2 ·
Replies
2
Views
3K
Suggestions for HS geometry book (proofs)
  • · Replies 14 ·
Replies
14
Views
2K
Geometry Differential Geometry Book
  • · Replies 10 ·
Replies
10
Views
3K
Geometry I would like suggestions regarding reading about geometry and manifolds
  • · Replies 16 ·
Replies
16
Views
3K
Geometry Book Recommendations in Differential Geometry
  • · Replies 19 ·
Replies
19
Views
4K
Geometry Vargas' book about Differential Geometry
  • · Replies 7 ·
Replies
7
Views
5K
Motivating definitions from differential geometry
  • · Replies 4 ·
Replies
4
Views
2K
Applied Resources for general vector differential equations?
  • · Replies 6 ·
Replies
6
Views
2K
Geometry Book on Differential Geometry/Topology with applications
  • · Replies 11 ·
Replies
11
Views
4K