[請益] Binet eq, spiral and picard
※ [本文轉錄自 Math 看板 #1JcLAhGG ]
作者: adu (^_^) 看板: Math
標題: [物理] Binet eq, spiral and picard
時間: Thu Jun 12 15:11:35 2014
I have to solve the following physics questions.
Given binet equation:
set m = 1, r = u^-1, L=/=0 is angular momentum, v is angle
F(u^-1) = -(Lu)^2 * [(d^2/dv^2)u + u]
^^^^^^^^^^derive two times
Given archimedean spiral:
r = a + b*v
Given the inverse cube force law:
F(r) = -(r^-3)
Show that:
1.use Binet eq to find a force law produces
orbits in the shape of archimedean spiral.
2.use Binet eq to show the orbits of Cotes spirals when L^2 >1 or ==1 or <1
3.explain why archimedean spiral is not self-semilar,
but logarithmic spiral (r=ae^bv)is.
4.consider the picard iteration, show if a solution X(t) is exact, then
X(t) - Un(t) = o(t^n)
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※ 轉錄者: adu (134.3.109.85), 06/12/2014 15:13:28
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06/12 23:33, , 1F
06/12 23:33, 1F
1.如果直接把a+bv代入Binet會得到一個分母為(a+bv)^4的奇怪式子
2.可以得到二次微分項=-u(1-1/(L^2)) 接下來要如何討論形狀?
3.我用論述的方法說logarithmic spiral可以透過scaling或rotating得到另一個
log. spiral，帶想要用較為數學的方法表示
4.想法是直接算解X(t)，再與Un相減，可自行解決。
※ 編輯: adu (134.3.109.85), 06/13/2014 00:53:04
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