You should have a look in Landau - Lifchitz book "Mechanics" chapter 3 where a similar computation is done.

The answer to this question is attached.

Sadrilla Abdullaev

Thank you very much for your interest in my question and for your kind responses, Dr. Cessac and Dr. Abdullaev!

However, the approximate formula I asked about is not the exact one kindly presented in an attachment by Dr. Abdullaev, and appearing also in Chapter 3 of the book of Landau and Lifchitz, in Goldstein book, and in many other sources.

Please note that the square root come first, applied to a quotient. In the denominator of the quotient appears the definite integral between 0 and A of x^3 f(x).

I couldn´t derive it using methods like averaging or the asymptotic Krilov-Bogoliubov-Mitropolsky method. But I applied it in several cases and works fairly well when compared with well known ans more precise results.

Dear Professor Roberto Suárez Antola:

You are right, there are several non-conventional approximations for second order oscillators. Perhaps if you could provide me an insight where you collect that formula, I can help because I also derived my own method to determine limit cycles in Lienard systems.

Best wishes,

Andrés García, UIniversidad Nacional del Sur, Argentina

PD: MY e-mail is [email protected]

Dear Professor García:

Thank you very much for your kind answer.

I sent to your e-mail a file with further details about the posed problem.

I could not attach the above mentioned file to this message, although I believe it could be of some interest to Dr. Cessac and to Dr. Abdullaev also.

Please find attached some comments about the approximate analytical formula for the period of a conservative oscillator that motivated my question to people in RG.

I don't have an answer to my question yet, but perhaps these comments could be of certain interest to persons that would like to think about this problem.