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1 point

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2nd Jun 2022

I am not surprised at all to here that. To be able to explain something so well means that you understand it extremely well. I used to tutor for calc, o-chem, and lab techniques like IR and NMR, and yeah I learned teaching wasn't for me either lol.

If you don't mind, could you provide some ideas on math studies for a particular goal? I've worked as a software developer/engineer, as well as some data research, but I'm looking to beef up my math and physics knowledge to move into more data/software/computer architecture engineering roles. I have studied math up to PDEs, some minor Linear Algebra, and a sprinkle of Quantum Mechanics relating to electron states for inorganic chemistry, but I want to be able to understand high level concepts like those found in this book:

The Variational Principles of Mechanics (Dover Books on Physics) https://www.amazon.com/dp/0486650677/ref=cm_sw_r_apan_i_705GMASH9R2FK9XY1SME?_encoding=UTF8&psc=1

Searching Reddit for that title is how I found your post to begin with lol. In your opinion, what type of math should I be studying to be able to understand the concepts in that book?

My idea was to refresh my Calc knowledge and then do:

Linear Algebra > ODE/PDE > Real/Complex/Functional Analysis as well as studying Proofs > Combinatorics > Newtonian Mechanics > Lagrangian Mechanics

What do you think of that? If you have any book reqs, that'd be awesome too!

1 point

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4th Dec 2021

Search for Analytical Mechanics.

Here one of my favorites:[Variational Principles of MEchanics]( https://www.amazon.co.uk/Variational-Principles-Mechanics-Dover-Physics/dp/0486650677/ref=asc\_df\_0486650677/?tag=googshopuk-21&linkCode=df0&hvadid=310867999190&hvpos=&hvnetw=g&hvrand=15627765740017970246&hvpone=&hvptw...)

1 point

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9th Sep 2019

Any suggestions on how to approach high-level physics without a formal math background?

I am an engineer with an academic concentration in signals processing and a minor in physics, so I do have a strong quantitative background. However, my training was heavily slanted towards ad-hoc problem solving rather than rigorous analysis, so I find myself lost as I tackle topics grounded in formal mathematics.

Specifically, I have been reading Lanczos' <em>The Variational Principles of Mechanics</em>, a popular analytical mechanics text, with great difficulty.

Is it worth reading a pure math book on differential geometry or something similar? How do most graduate students study advanced physics, when an undergraduate physics education doesn't use much math beyond basic PDEs?

2 points

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1st Dec 2020

For deeper readings I recommend Variational Principles in Classical Mechanics (pdf link) as well as the highly recommended Variational Principles of Mechanics by Lanczos (amazon link).

1 point

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3rd Sep 2021

Good sir Cornelius has written an excellent gem of a book that should be more well known:

https://www.amazon.com/Variational-Principles-Mechanics-Dover-Physics/dp/0486650677

It's been Doverized so can also be picked up inexpensively.

Some preliminary lectures on first order perturbation theory may also build intuition. I find the arguments particularly intuitive and concrete, particularly those used in physics approximations.

1 point

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9th Apr 2016

There are great derivations on Wikipedia and in a variety of variational mechanics texts. The Lanczos text in particular is very good, as is Landau and Lifshitz.

As /u/psisquared2, the calculus of variations method is basically an extension of the basic principle of calculus to functionals. In the derivation of the derivative, you approximate the slope of a line between the point f(x) and f(x+a), and then find the behavior as a approaches 0. Similarly, calculus of variations involves adding a perturbation factor εΦ(t) to the function q(t), where Φ(t) is a function itself. You can use the techniques described in the wikipedia page to find the value of q(t) such that any perturbation yields a greater value for the integral, just like optimization in differential calculus. Instead of setting the second derivative or gradient equal to zero, you can set the expression L(q + εΦ, q' + εΦ', t) - L(q, q', t) equal to zero (called the first variation), an apply some techniques of calculus to derive the Euler-Lagrange equations. This isn't the simplest subject, so feel free to ask questions.

1 point

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2nd Nov 2015

For more info, see this book by Lanczos.

1 point

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21st Aug 2015

Thanks. I mainly recommend the variational principles of mechanics by Lanczos for something more physics based (but rigorous for the most part).

For something that is just math (with some physics inspiration), I have found the book be Hans Sagan to be helpful.

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