论文标题
Brachistochrone的有趣曲目
An interesting track for the Brachistochrone
论文作者
论文摘要
如果粒子必须首先从A垂直降低1 m，然后水平移动1 m到B，则需要一个时间$ t（=τ_1+τ_2=τ_3= 3/\ sqrt {2G}）= 0.67 $ s。在重力和没有摩擦的情况下，如果它在两个点A和B高的两个点A和B之间的线性轨道上向下向下，则需要时间$ t（=τ_4= 2/\ sqrt {g}）= 0.63 $ s。从历史上看，在这两个极端之间，伯努利（1718）证明，这些点A和B之间最快的轨道是循环的，而下降时间最少，$ t =τ_b=τ_b= 0.58 $ s。除其他有趣的情况外，我们在这里研究了粒子/珠在$ y（x）=（1-x^ν）=（1-x^ν）^{1/ν}的A和B之间的无摩擦运动。我们发现，当$ν\ in（0.09653，0.31749），τ_4<t <τ_3$以及$ν\ in（0.31749，1），τ_b<t <t <τ_4$时。但最值得注意的是，如果$ν\ in（0，ν_c= 0.09653）$，凹面曲线变得非常陡峭/深度（$ = 1/\ sqrt {2G} =τ_2）<τ_b$。功能$ t（ν$）在$ν=ν_c$下遭受跳跃不连续性，我们提供一些分辨率。
If a particle has to fall first vertically 1 m from A and then move horizontally 1 m to B, it takes a time $t(=τ_1+τ_2=τ_3=3/\sqrt{2g})=0.67$ s. Under gravity and without friction, if it sides down on a linear track inclined at $45^0$ between two points A and B of 1 m height, it takes time $t(=τ_4=2/\sqrt{g})=0.63$ s. Between these two extremes, historically, Bernoulli (1718) proved that the fastest track between these points A and B is cycloid with the least time of descent $t=τ_B=0.58$ s. Apart from other interesting cases, here we study the frictionless motion of a particle/bead on an interesting track/wire between A and B given by $y(x)=(1-x^ν)^{1/ν}.$ For $ν> 1$ the track becomes convex and $t>>τ_4$, and when $ν>1.22$, the motion with zero initial speed is not possible. We find that when $ν\in (0.09653, 0.31749), τ_4<t <τ_3$ and when $ν\in ( 0.31749, 1),τ_B < t < τ_4$. But most remarkably, the concave curve becomes very steep/deep if $ν\in (0, ν_c=0.09653)$, then $t=0.2258$ s $< τ_B$, this is as though a particle would travel 1 meter horizontally with a speed equal $\sqrt{2g}$ m/sec to take the time ($=1/\sqrt{2g}=τ_2) < τ_B$. The function $t(ν$) suffers a jump discontinuity at $ν=ν_c$, we offer some resolution.
