《1 前言》

1 前言

$F=ma。\text{ }\text{ }\text{ }\left(1\right)$

$F=-\frac{GΜm}{r{}^{2}}。\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(2\right)$

《2 导出改进的牛顿第二定律及万有引力定律的变分原理》

2 导出改进的牛顿第二定律及万有引力定律的变分原理

$W\left(0\right)\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}W\left(t\right)。\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\left(3\right)$

$R{}_{\text{W}}=\frac{W\left(t\right)}{W\left(0\right)}-1=0。\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\left(4\right)$

$\Pi ={\int }_{t{}_{1}}^{t{}_{2}}R{}_{\text{W}}^{2}\text{d}t=\mathrm{min}{}_{0}。\text{ }\text{ }\text{ }\left(5\right)$

$\Pi ={\int }_{x{}_{1}}^{x{}_{2}}R{}_{\text{W}}^{2}\text{d}x=\mathrm{min}{}_{0}。\text{ }\text{ }\text{ }\left(6\right)$

$R{}_{\text{W}}=\frac{Q}{{Q}^{\prime }}-1=0。\text{ }\text{ }\text{ }\left(7\right)$

《3 变维分形等形式的改进的牛顿第二定律及万有引力定律》

3 变维分形等形式的改进的牛顿第二定律及万有引力定律

$F=-\frac{GΜm}{r{}^{2}}-\frac{3G{}^{2}Μ{}^{2}mp}{c{}^{2}r{}^{4}}。\text{ }\text{ }\text{ }\left(8\right)$

$F=-\frac{GΜm}{r{}^{2}}\left(1+\frac{a{}_{1}}{r{}^{2}}+\frac{a{}_{2}}{r{}^{4}}+\cdots \right)。\text{ }\text{ }\text{ }\left(9\right)$

$Ν=\frac{C}{r{}^{D}}$

D为常数时, 这种分形可称为常维分形。

$F=-\frac{GΜm}{r{}^{D}}。\text{ }\text{ }\text{ }\left(10\right)$

$D\phantom{\rule{0.25em}{0ex}}=a{}_{1}+a{}_{2}r+a{}_{3}r{}^{2}+\cdots 。\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\left(11\right)$

$F=ma+k{}_{1}a{}^{2}+k{}_{2}a{}^{3}+\cdots 。\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\left(12\right)$

$F=ma{}^{{D}^{\prime }},\text{ }\text{ }\text{ }\left(13\right)$

${D}^{\prime }=k{}_{1}+k{}_{2}a\phantom{\rule{0.25em}{0ex}}+k{}_{3}a{}^{2}+\cdots ‚\text{ }\text{ }\text{ }\left(14\right)$

$F=ma{}^{1+\epsilon }\phantom{\rule{0.25em}{0ex}}‚\text{ }\text{ }\text{ }\left(15\right)$

《4 导出改进的牛顿第二定律及万有引力定律的方法》

4 导出改进的牛顿第二定律及万有引力定律的方法

$\frac{\partial \Pi }{\partial a{}_{i}}=\frac{\partial \Pi }{\partial k{}_{i}}=0。\text{ }\text{ }\text{ }\left(16\right)$

《5 导出改进的牛顿第二定律及万有引力定律的实例》

5 导出改进的牛顿第二定律及万有引力定律的实例

《图1》

Fig.1 A small ball rolls from A to B

$\Pi ={\int }_{-Η}^{0}\left(\frac{v{}_{\text{Ρ}}^{2}}{{v}^{\prime }{}_{\text{Ρ}}^{2}}-1\right){}^{2}\text{d}x=\mathrm{min}{}_{0}。\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(17\right)$

$V=-\frac{GΜm}{\left(D-1\right)r{}_{{\text{Ο}}^{\prime }\text{Ρ}}^{D-1}}\text{ }\text{ }\text{ }\left(18\right)$

$-\frac{GΜm}{\left(D-1\right)r{}_{{\text{Ο}}^{\prime }\text{A}}^{D-1}}=\frac{1}{2}m{v}^{\prime }{}_{\text{Ρ}}^{2}-\frac{GΜm}{\left(D-1\right)r{}_{{\text{Ο}}^{\prime }\text{Ρ}}^{D-1}}‚\text{ }\text{ }\text{ }\left(19\right)$

${v}^{\prime }{}_{\text{Ρ}}^{2}=\frac{2GΜ}{D-1}\left(\frac{1}{r{}_{{\text{Ο}}^{\prime }\text{Ρ}}^{D-1}}-\frac{1}{\left(R+Η\right){}^{D-1}}\right)。\text{ }\text{ }\text{ }\left(20\right)$

$y=y\left(x\right)。\text{ }\text{ }\text{ }\left(21\right)$

$\phantom{\rule{0.25em}{0ex}}\text{d}v/\text{d}t=a‚\text{ }\text{ }\text{ }\left(22\right)$

$\text{d}t=\frac{\text{d}s}{v}=\frac{\left(1+{y}^{\prime }{}^{2}\right){}^{1/2}}{v}\text{d}x‚$

$\text{d}v=a\text{d}t=a\frac{\left(1+{y}^{\prime }{}^{2}\right){}^{1/2}}{v}\text{d}x‚$

$v\text{d}v=a\left(1+{y}^{\prime }{}^{2}\right){}^{1/2}\text{d}x。\text{ }\text{ }\text{ }\left(23\right)$

$F{}_{\text{Ρ}}=\frac{GΜm}{r{}_{{\text{Ο}}^{\prime }\text{Ρ}}^{D}}‚$

$F{}_{\text{a}}\phantom{\rule{0.25em}{0ex}}=\frac{GΜm}{r{}_{{\text{Ο}}^{\prime }\text{Ρ}}^{D}}\phantom{\rule{0.25em}{0ex}}\frac{{y}^{\prime }}{\left(1+{y}^{\prime }{}^{2}\right){}^{1/2}}。\text{ }\text{ }\text{ }\left(24\right)$

$a=\left(\frac{F{}_{\text{a}}}{m}\right){}^{1/\left(1+\epsilon \right)}=\left(\frac{GΜ{y}^{\prime }}{r{}_{{\text{Ο}}^{\prime }\text{Ρ}}^{D}\left(1+{y}^{\prime }{}^{2}\right){}^{1/2}}\right){}^{1/\left(1+\epsilon \right)}‚\text{ }\text{ }\text{ }\left(25\right)\phantom{\rule{0.25em}{0ex}}$

$\begin{array}{l}v\text{d}v=\left(\frac{GΜ{y}^{\prime }}{\left(\left(Η+x\right){}^{2}+\left(R+Η-y\right){}^{2}\right){}^{D/2}\left(1+{y}^{\prime }{}^{2}\right){}^{1/2}}\right){}^{1/\left(1+\epsilon \right)}\left(1+{y}^{\prime }{}^{2}\right){}^{1/2}\text{d}x。\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(26\right)\\ v{}_{\text{Ρ}}^{2}=2{\int }_{-Η}^{x{}_{\text{Ρ}}}\left(\frac{GΜ{y}^{\prime }}{\left(\left(Η+x\right){}^{2}+\left(R+Η-y\right){}^{2}\right){}^{D/2}\left(1+{y}^{\prime }{}^{2}\right){}^{1/2}}\right){}^{1/\left(1+\epsilon \right)}\left(1+{y}^{\prime }{}^{2}\right){}^{1/2}\text{d}x。\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(27\right)\end{array}$

$y=Η+x。\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(28\right)$

$v{}_{\text{Ρ}}^{2}=2{\int }_{-x{}_{\text{Ρ}}}^{Η}\left(\frac{GΜ}{\left(\left(Η-z\right){}^{2}+\left(R+z\right){}^{2}\right){}^{D/2}}\right){}^{1/\left(1+\epsilon \right)}2{}^{\epsilon /2\left(1+\epsilon \right)}\text{d}z。\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(29\right)$

$\Pi {}_{0}=571.421\phantom{\rule{0.25em}{0ex}}5。$

Π0不等于零, 可用最优化方法确定Dε

$\phantom{\rule{0.25em}{0ex}}D=1.999\phantom{\rule{0.25em}{0ex}}89‚\epsilon =\phantom{\rule{0.25em}{0ex}}0.014\phantom{\rule{0.25em}{0ex}}58‚\phantom{\rule{0.25em}{0ex}}\Pi =137.323\phantom{\rule{0.25em}{0ex}}1。$

《6 结论》

6 结论