Recent zbMATH articles in MSC 37J46https://www.zbmath.org/atom/cc/37J462022-04-28T18:15:44.554837ZWerkzeugMinimal periodic problem for brake orbits of first-order Hamiltonian systemshttps://www.zbmath.org/1482.370642022-04-28T18:15:44.554837Z"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.xinnianA 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)