Constant-speed ramps for a central force field

Figure 1.  A mass $ M $ sliding along $ \alpha $ under the effect of a central force $ {\mathbf F}({\mathbf r}) $ and the friction force

Figure 2.  Case $ v = 1 $. The phase portrait of the system (17), with $ \mu = 0.5 $. The origin $ (0,0) $ is the only equilibrium point and it is a stable focus

Figure 3.  Case $ v = 1 $. The purple part of the logarithmic spiral (left) is the TreadmillSled of the non-circular ramp (right)

Figure 4.  The phase portrait of the system (21). Left: $ v = 2 $ and $ \mu = 0.1 $. Right: $ v = 0.5 $ and $ \mu = 0.3 $

Figure 5.  TreadmillSleds that are half-lines. Left: $ v = 2 $, $ \mu = 0.1 $. Right: $ v = 0.5 $, $ \mu = 0.3 $

Figure 6.  Constant-speed ramps whose TreadmillSleds are half-lines of Figure 5, left. Here $ v = 2 $, $ \mu = 0.1 $. Left: parametrization (23) for $ \mathbf{a} $. Right: parametrization (23) for $ -\mathbf{a} $

Figure 7.  Constant-speed ramps $ \alpha_{ls} $ whose TreadmillSleds $ \gamma_{ls} $ are the half-lines of Figure 5, right. Here $ v = 0.5 $ and $ \mu = 0.3 $. Left: parametrization (23) for $ \mathbf{a} $. Right: parametrization (23) for $ -\mathbf{a} $

Figure 8.  Constant-speed ramps whose TreadmillSleds are not half-lines. Here $ v = 2 $ and $ \mu = 0.1 $. Left: the TreadmillSled which is asymptotic to the line of vector $ \mathbf{a} $. Right: the constant-speed ramp

Figure 9.  Constant-speed ramps whose TreadmillSleds are not half-lines. Here $ v = 0.8 $ and $ \mu = 0.3 $. Left: the TreadmillSled. Right: the constant-speed ramp

Figure 10.  Constant-speed ramps that are not spirals. Left: $ v = 2 $ and $ \mu = 0.1 $. Right: $ v = 0.8 $ and $ \mu = 0.3 $