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Complex variables and classical mechanics

  1. Jun 22, 2015 #1
    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!
  2. jcsd
  3. Jun 22, 2015 #2
    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.
  4. Jun 22, 2015 #3


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    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.
  5. Jun 23, 2015 #4


    Staff: Mentor

    I reccomend 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]


    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: Jun 23, 2015
  6. Jun 23, 2015 #5
    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!
  7. Jun 23, 2015 #6
    Silly me, I forgot a version of classical mechanics known as Koopman- von Neuman, it uses extensively the complex numbers.
  8. Jun 24, 2015 #7
    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.
  9. Jun 24, 2015 #8
    Quaternions rather than complex numbers, perhaps?
  10. Jun 24, 2015 #9
    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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