In a Hamiltonian system with impacts (or "billiard with potential"), a point particle moves about the interior of a bounded domain according to a background potential and undergoes elastic collisions at the boundaries. When the background potential is identically zero, this is the hard-wall billiard model. Previous results on smooth billiard models (where the hard-wall boundary is replaced by a steep smooth billiard-like potential) have clarified how a smooth billiard may be rigorously approximated using a hard-wall billiard. These results are extended here to models with smooth background potential satisfying some natural conditions. This generalization is then applied to geometric models of collinear triatomic chemical reactions. (The models are far from integrable n-degree-of-freedom systems with n ≥ 2.) The application demonstrates that the simpler analytical calculations for the hard-wall system may be used to obtain qualitative information with regard to the solution structure of the smooth system and to quantitatively assist in finding solutions of the soft impact system by continuation methods. In particular, stable periodic triatomic configurations are easily located for the smooth highly nonlinear two-and three-degree-of-freedom geometric models.
BACKGROUND: Deep brain stimulation is an effective and safe technique. Displacement of the electrode relative to the optimal stimulation site can lead to insuffi cient effect and sometimes to the need of operative electrode re-position. OBJECTIVE: This study was aimed to analyse targeting accuracy of deep brain stimulation electrode implantation to subthalamic nucleus (STN) and globus pallidus internus (Gpi). It detected possible causes of inaccuracy and prevalent shift to certain direction. METHODS: Targeting accuracy was analysed in 47 patients with Parkinson´s disease (PD) and 11 patients with dystonia with bilateral implantation of deep brain stimulation electrodes between years 2009 and 2016. RESULTS: A shift of electrode to prevalent direction was observed on the left side to medial and posterior and on the right side to lateral direction. Greater shift was observed on the left side and in a higher angulation of trajectory laterally. Movement of the electrode, because of its traction in anchoring device, was identifi ed as a possible factor for prevalent electrode shift. Calibration of stereotactic coordinates to correct prevalent shift was used. CONCLUSION: Targeting inaccuracy is the result of accumulation of errors in individual steps of electrode implantation. Direction of the shift can be random or it can be toward a prevalent direction. A correction of prevalent error can prevent a suboptimal electrode placement (Tab. 3, Fig. 11, Ref. 29). Text in PDF www.elis.sk.
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