Spectrum of the second variation

Second variation of a smooth optimal control problem at a regular extremal is a symmetric Fredholm operator. We study asymptotics of the spectrum of this operator and give an explicit expression for its determinant in terms of solutions of the Jacobi equation. In the case of the least action principle for the harmonic oscillator we obtain a classical Euler identity $\prod\limits_{n=1}^\infty\left(1-\frac{x^2}{(\pi n)^2}\right)=\frac{\sin x}{x}$. General case may serve as a rich source of new nice identities.