The constant c is useless in the limit process. Let's take it as 0. If the limit space average of time average is directly applied to this formula, we need to find an orbit. An obvious orbit is that our integrand function is formal at this time, and then it becomes an irrelevant number after integrating n times. Obviously, this transformation is not helpful to the problem, so we need to find another set of reasonable unrelated frequencies. Due to many reasons such as solvability, we think that such frequencies should be regarded as a linear combination, which is independent of each other's rationality, so we can infer that they are independent of each other's rationality. Remember, we can notice the identity: so if we choose these n- 1 variables (0 when i= 1) as our frequency (obviously irrelevant to rationality), at this time, for other I and J, it is not equal to 1, so we have the integrand function in the above integral, which is as follows: I haven't learned much about special functions. I wonder if there is a simplified formula for this integral? In the case of triangles: Even for double integrals, the amount of calculation is particularly touching. The numerical calculation is probably correct (right triangle and equilateral triangle are verified). I don't know which summer afternoon, in the classroom, I wrote a four-bar Lagrange function, which physics has long forgotten. When I first derived this formula, I was so excited ... the conservation of energy, momentum and angular momentum. These three conservation laws can all be derived from Lagrange equation! ! I was so excited at that time ... Oh, by the way, I must add Hamilton's minimum dose principle. I thought I had discovered the ultimate truth of the world! And Hamilton-jacoby equation, from which Schrodinger equation is derived!