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The period function of Hamiltonian systems with separable variables. (English) Zbl 1451.37085

The authors consider planar Hamiltonian systems \[ \dot x=-H_y(x,y), \qquad \dot y=H_x(x,y), \] with Hamiltonian function of the form \(H(x,y)=F_1(x)+F_2(y)\), where \(F_1,F_2\) are analytic functions with a local minimum at the origin, so that the dynamical system driven by the system \(\dot x=-F'_2(y), \dot y=F'_1(x)\) has a critical center at the origin. Denoting by \(\gamma_h\) the periodic orbit inside the energy level \(H=h\), the period function is given by \[ T(h)=\int_{\gamma_h} \frac{dx}{F'_2(y)}. \]
The authors look for conditions implying the monotonicity of the period function as well as the existence of at most one critical periodic orbit. Several sufficient conditions are presented. They include those already known as well as some new ones. Finally the authors investigate the asymptotic properties of the period function in the special case \(F_1(x)=F_2(-x)\).

MSC:

37J46 Periodic, homoclinic and heteroclinic orbits of finite-dimensional Hamiltonian systems
34C25 Periodic solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
70H05 Hamilton’s equations
70K42 Equilibria and periodic trajectories for nonlinear problems in mechanics
70K70 Systems with slow and fast motions for nonlinear problems in mechanics
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