A system of ordinary differential equations that describes the motion of a self-propelling rigid disk is studied. In this system, the disk moves along a straight-line and reflects from a boundary. Interestingly, numerical simulation shows that the angle of reflection is greater than that of incidence. The purpose of this study is to present a mathematical proof for this attractive phenomenon. Moreover, the reflection law is numerically investigated. Finally, existence and asymptotic stability of a square-shaped closed orbit for billiards in square table with inelastic reflection law are discussed.
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Figure 6. The results of linear regression for (1) with $ m_2 = 1 $. Only the results satisfying $ E < 0.001 $ are plotted. (A) The estimated values of $ (c_0, c_1) $ for various $ \delta $ and $ m_0 $. (B) The dependence of $ \sqrt{c_0^2 + c_1^2} $ on $ \delta $. Red, blue, and green circles are results for $ m_0 = 0, 0.5, 1.0 $, respectively
Figure 7. (A) $ \delta $ versus $ \log_{10}E $. The parameters are same as Fig. 6. (B) Numerical data of $ F(\theta_{\mathrm{inc}}{)} $ for $ m_0 = m_2 = 1, \delta = 0.25 $ and fitted curve with $ k = 1 $ and $ k = 2 $
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(A) A laboratory experiment of camphor motions. (B) A trajectory of the center of the disk governed by the moving boundary model (7) below
Reflection from the boundary
A trajectory of the center of the disk governed by the moving boundary model (7). Parameters are
Direction field of
Graphs of
The results of linear regression for (1) with
(A)
Discrete-time model