Recent zbMATH articles in MSC 37J46 https://www.zbmath.org/atom/cc/37J46 2022-04-28T18:15:44.554837Z Werkzeug Minimal periodic problem for brake orbits of first-order Hamiltonian systems https://www.zbmath.org/1482.37064 2022-04-28T18:15:44.554837Z "Zhang, Xiaofei" https://www.zbmath.org/authors/?q=ai:zhang.xiaofei "Liu, Chungen" https://www.zbmath.org/authors/?q=ai:liu.chungen "Lu, Xinnian" https://www.zbmath.org/authors/?q=ai:lu.xinnian A brake orbit for a Hamiltonian system is a periodic solution of Hamilton's equations for which the generalized momenta are zero at two different points. This paper considers the autonomous system defined by $$\dot{z} = J \nabla H(z(t))$$, $$z(-t) = Nz(t)$$ and $$z(t+\tau) = z(t)$$ where $$H \in C^2(\mathbb{R}^{2n}, \mathbb{R})$$, $$H(z) = H(N(z)$$ for all $$z \in \mathbb{R}^{2n}$$, $$t \in \mathbb{R}$$ and $$\tau > 0$$. Matrices $$N$$ and $$J$$ are defined by $N =\left( \begin{matrix} -I_n & 0\\ 0 & I_n \end{matrix}\right),$ and $J =\left( \begin{matrix} 0 & -I_n\\ I_n & 0 \end{matrix}\right),$ where $$I_n$$ is the $$n \times n$$ identity matrix. To find a brake orbit solution of this system the authors look at the $$L_0$$-boundary problem defined by $$\dot{z} = J \nabla H(z(t))$$ with $$z(0) \in L_0$$, $$z(\tau/2) \in L_0$$ $$t \in [0, \tau/2]$$, and where $$L_0 = \{ 0 \} \times \mathbb{R}^n$$. The main result of the paper is about the existence of a non-trivial brake orbit under a number of technical conditions on the Hamiltonian function $$H$$. In particular there are conditions on the Hessian of $$H(z)$$ with respect to $$q$$ where $$z = (p,q)$$ and $$p, q \in \mathbb{R}^n$$. With even stricter conditions, the brake orbit has a minimal period of $$\tau$$ or $$\tau/2$$. The authors argue that brake orbit problems can be transformed into $$L_0$$-boundary value problems which are amenable to treatment as Lagrangian boundary value problems. The authors rely on several related papers to provide the necessary background. Reviewer: William J. Satzer Jr. (St. Paul)