Complex variables and classical mechanics

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Discussion Overview

The discussion explores the role and applications of complex variables and complex analysis within the context of classical mechanics. Participants examine various instances where these mathematical tools may be relevant, including theoretical and practical applications.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants mention the use of complex variables in the action-angle variable for the Kepler problem as discussed in the book by Jose and Saletan.
  • Others highlight the utility of complex variables in electrical engineering, particularly for analyzing alternating current, and in fluid mechanics for potential flow and solutions to the Laplace equation.
  • One participant references the historical resistance to complex numbers in science, suggesting their eventual acceptance due to their usefulness.
  • A participant notes the Koopman-von Neumann formulation of classical mechanics, which extensively employs complex numbers.
  • Another participant shares an anecdote about encountering a problem in an elementary school textbook that required complex numbers for a proper solution, while also questioning the potential use of quaternions instead.
  • One participant discusses a specific case involving the imaginary unit \(i\) and its role in equations, suggesting it can sometimes be excluded to yield correct answers.

Areas of Agreement / Disagreement

Participants express varying degrees of agreement on the usefulness of complex variables in classical mechanics, but no consensus is reached on specific applications or the extent of their relevance. Multiple competing views and examples are presented without resolution.

Contextual Notes

Some claims rely on specific definitions or contexts, such as the distinction between complex numbers and quaternions, which may not be universally applicable. Additionally, the discussion includes anecdotal evidence that may not generalize across all areas of classical mechanics.

Segala
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Dear all,
I'd like to know what is the place/use of complex variables (and complex analysis) in classical mechanics. By the way, is there any?

Thanks for your help. Best regards!
 
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In Jose and Saletan book there are integrals that they solve using complex variable theory, namely the action-angle variable for the Kepler problem (not the easiest way to do it though)

I don't really reacall other instance where complex analysis enter classical mechanics.
 
Complex variables are useful in electrical engineering for analyzing alternating current. They are also used in studying potential flow in fluid mechanics and for analyzing solutions to things like the Laplace equation, which finds application in fluid and solid mechanics. They are also quite useful in analyzing periodic phenomena using Fourier transforms.
 
I recommend an entertaining book that explains how scientists for centuries tried to avoid complex, but finally gave in because it is so useful in many ways.

An Imaginary Tale: The Story of [the Square Root of Minus One]https://www.amazon.com/dp/0691146004/?tag=pfamazon01-20

I'll never forget what Leonard Susskind once said, "Physicists are not interested in what is true. They are interested in what is useful."
 
Last edited:
Thanks for the answers. They start to convince myself of what I did suspect.

Dear anorlunda, thanks a lot for the book recommendation and, most of all, for the quote. Very true!

I am, by principle, interested in all physics and mathematics and, why not, engineering. However, graduate school awaits for me and time is a harsh mistress. I must optmize things the most I can and that won't go without sacrifices. Yes, Susskind is right... great news!

Best regards!
 
Silly me, I forgot a version of classical mechanics known as Koopman- von Neuman, it uses extensively the complex numbers.
 
Once, looking through elementary school textbook I have encountered a problem dealing with axis orientation. It was very strange, but without using complex unit this problem couldn't get proper solution.
 
mac_alleb said:
Once, looking through elementary school textbook I have encountered a problem dealing with axis orientation. It was very strange, but without using complex unit this problem couldn't get proper solution.

Quaternions rather than complex numbers, perhaps?
 
Not exactly complex numbers, rather trick with
I = Sqrt(-1). It appears in both equations and successfully excluded, giving correct answer.
 

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