# Complex variables and classical mechanics

• Segala
In summary, complex analysis is used in classical mechanics to solve problems that are more difficult with quaternions.
Segala
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!

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 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]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.

## 1. What are complex variables and how are they used in classical mechanics?

Complex variables are numbers that have both a real and imaginary component. In classical mechanics, these variables are used to represent physical quantities in mathematical equations. They allow for more elegant and concise solutions to problems that involve motion and forces.

## 2. Can complex variables be applied to all areas of classical mechanics?

Yes, complex variables can be applied to all areas of classical mechanics, including kinematics, dynamics, and statics. They are particularly useful in solving problems involving oscillatory motion and resonance.

## 3. How do complex variables differ from real variables in classical mechanics?

Complex variables differ from real variables in that they have both a real and imaginary component, while real variables only have a real component. This allows for a more comprehensive representation of physical quantities and more efficient problem-solving techniques.

## 4. What are some common applications of complex variables in classical mechanics?

Some common applications of complex variables in classical mechanics include analyzing the motion of a simple harmonic oscillator, solving for the trajectory of a projectile, and determining the behavior of a vibrating string.

## 5. Are there any limitations to using complex variables in classical mechanics?

While complex variables are extremely useful in solving many problems in classical mechanics, there are some cases where they may not be applicable. For example, when dealing with systems that exhibit chaotic behavior, real variables may be a better choice.

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