# Vargas' book about Differential Geometry

• Geometry
kiuhnm
I'm learning Differential Geometry (DG) on my own (I need it for robotics). I realized that there are many approaches to DG and one is Cartan's, which is presented in Vargas's book. I think that book is highly opinionated, but I don't know if that's a good or bad thing. Does anyone of you know that book or the approach described in it? More importantly, should I start from this book or should I first learn the popular way (e.g. from Boothby's book)?

Gold Member
(I need it for robotics)
May I ask how differential geometry is needed for robotics?

kiuhnm
I'm interested in locomotion and the modern formulation of mechanics relies on Differential Geometry. I come from Reinforcement Learning and want to know more about mechanics and dynamics to see if I can combine the two approaches.

Anyway, I don't like Vargas's book. It's almost unreadable because the author refers to concepts before defining or explaining them. I'm reading Victor Guillemin's notes right now and I like them.

Demystifier
Gold Member
If you want to apply mechanics to something practical (such as making a robot), then I would suggest you to avoid modern geometrical approaches to mechanics. A good old physical/engineering presentation of mechanics should be much more useful for practical purposes. Or perhaps you are doing general abstract theory of robotics without attempting to actually make a robot?

May I ask how differential geometry is needed for robotics?

There could be a few ways- motion of a jointed arm (for example), involves coordinate transformations. Trying to move a tool along a prescribed path is another. I've seen differential geometry appear in gear theory as well:

https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19900010277.pdf

Demystifier
kiuhnm
May I ask how differential geometry is needed for robotics?

I'm focusing on underactuated robotics which is related to nonholonomic mechanics. I've been ignoring the dynamics part so far because I've been using Reinforcement Learning, which is basically an approach where an agent figures out on its own how to maximize a reward signal. If the reward signal indicates how fast an agent is running, then the agent will learn how to run (with mixed results because of bad local minima).
That approach is very promising but I think it's not enough so I want to learn about dynamics and see if dynamics and mechanics can inform the RL algorithms in a useful way.
Here's the most recent and complete book I could find about nonholonomic mechanics: https://www.amazon.com/dp/1493930168/?tag=pfamazon01-20
It looks pretty advanced so I need to do some background study. Goldstein's book is a classic but I think that José & Saletan or Fasano & Marmi are better suited for my purpose, being more mathematical.

Gold Member
Goldstein's book is a classic but I think that José & Saletan or Fasano & Marmi are better suited for my purpose, being more mathematical.
In my experience (I am a theoretical physicist) "more mathematical" usually means more abstract, and hence less applicable. Abstraction is good for understanding the big picture, but not for solving specific practical problems. Nevertheless, you certainly know better what is better suited for you.

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kiuhnm
In my experience (I am a theoretical physicist) "more mathematical" usually means more abstract, and hence less applicable. Abstraction is good for understanding the big picture, but not for solving specific practical problems. Nevertheless, you certainly know better what is better suited for you.

I think one should learn enough math to be able to read and understand most papers and books relevant to its field. For instance I had to learn some measure theory just to be able to read some overly mathematical papers and that certainly paid off.