Xijun Hu — Page 3 | Searcharxiv

arXiv2013

Trace formula for linear Hamiltonian systems with its applications to elliptic Lagrangian solutions

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.

math-ph↗

arXiv2007

Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces

There is a long standing conjecture in Hamiltonian analysis which claims that there exist at least $n$ geometrically distinct closed characteristics on every compact convex hypersurface in $\R^{2n}$ with $n\ge 2$. Besides many partial results, this conjecture has been only completely solved for $n=2$. In this paper, we give a confirmed answer to this conjecture for $n=3$. In order to prove this result, we establish first a new resonance identity for closed characteristics on every compact convex hypersurface $\Sg$ in $\R^{2n}$ when the number of geometrically distinct closed characteristics on $\Sg$ is finite. Then using this identity and earlier techniques of the index iteration theory, we prove the mentioned multiplicity result for $\R^6$. If there are exactly two geometrically distinct closed characteristics on a compact convex hypersuface in $\R^4$, we prove that both of them must be irrationally elliptic.

math.SG↗