  In the present paper, we build up trace formulas for both the linear
Hamiltonian systems and Sturm-Liouville systems. The formula connects the
monodromy matrix of a symmetric periodic orbit with the infinite sum of
eigenvalues of the Hessian of the action functional. A natural application is
to study the non-degeneracy of linear Hamiltonian systems. Precisely, by the
trace formula, we can give an estimation for the upper bound such that the
non-degeneracy preserves. Moreover, we could estimate the relative Morse index
by the trace formula. Consequently, a series of new stability criteria for the
symmetric periodic orbits is given. As a concrete application, the trace
formula is used to study the linear stability of elliptic Lagrangian solutions
of the classical planar three-body problem. It is well known that the linear
stability of elliptic Lagrangian solutions depends on the mass parameter
$\bb=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2\in [0,9]$ and the eccentricity
$e\in [0,1)$. Based on the trace formula, we estimate the stable region and
hyperbolic region of the elliptic Lagranian solutions.
