時間の再検証

X軸方向の速度:Vx
X軸方向の加速度:Ax
Y軸方向の速度:Vy
Y軸方向の加速度:Ay
物体の場所:(x,y)=(rcosφ,rsinφ)

m[dVx/dt]=fx=f(r)cosφ
m[dVy/dt]=fy=f(r)sinφ

Vx=dx/dt=r'cosφ-rφ'sinφ
Vy=dy/dt=r'sinφ+rφ'cosφ

Ax=dVx/dt
Ax=r''cosφ-r'φ'sinφ-r'(φ'sinφ)-r(φ'sinφ)'
Ax=r''cosφ-r'φ'sinφ-r'(φ'sinφ)-r(φ''sinφ+(φ')2cosφ)
Ax=r''cosφ-r'φ'sinφ-r'φ'sinφ-r'φ'sinφ-r(φ')2cosφ
Ax=r''cosφ-2r'φ'sinφ-r'φ'sinφ-r(φ')2cosφ

Ay=dVy/dt
Ay=r''sinφ+r'φ'cosφ+r'(φ'cosφ)+r(φ'cosφ)
Ay=r''sinφ+r'φ'cosφ+r'φ'cosφ+r(φ''cosφ-(φ')2sinφ)
Ay=r''sinφ+r'φ'cosφ+r'φ'cosφ+r'φ'cosφ-r(φ')2sinφ
Ay=r''sinφ+2r'φ'cosφ+r'φ'cosφ-r(φ')2sinφ

m[dVx/dt]=fx=f(r)cosφ
m[dVy/dt]=fy=f(r)sinφ
cosφ+sinφ=m(cosφdVx/dt+sinφdVy/dt)
Ax=dVx/dt
Ay=dVy/dt
cosφ+sinφ=m(cosφAx+sinφAy)
cosφ+sinφ=(sin2φ+cos2φ)f(r)
cosφ+sinφ=f(r)cosφdVx/dt+sinφdVy/dt=f(r)/m

sinφ-cosφ=m(sinφdVx/dt+cosφdVy/dt)
sinφ-cosφ=cosφsinφf(r)-sinφcosφf(r)=0

cosφ{r''cosφ-2r'φ'sinφ-r'φ'sinφ-r(φ')2cosφ}+
sinφ{r''sinφ+2r'φ'cosφ+
r'φ'cosφ-r(φ')2sinφ}=r''-r(φ')2=f(r)/m
m{r''-r(φ')2}=f(r)

sinφ{r''cosφ-2r'φ'sinφ-r'φ'sinφ-r(φ')2cosφ}
[-cosφ]{r''sinφ+2r'φ'cosφ+
r'φ'cosφ-r(φ')2sinφ}
=2r'φ'+rφ''=0
m(2r'φ'+rφ'')=0

m{r''-r(φ')2}=f(r)
m(2r'φ'+rφ'')=0

r2φ'
S=(1/2)r2φ'
dS/dt={(1/2)r2φ'}'=rr'φ'+(1/2)r2φ''
r'φ'+(1/2)rφ''=0(a)
dφ/dt=hu^2
{(d^2u)/(dφ^2)}=f(1/u)/(h^2u^2)
r=1/u=L/{1+ecos(φ-φ)}
r=L/{1+ecos(φ-φ)}

ここまでがケプラーとニュートンの理論。
ここからが独自の理論。

r'=-[Lesin(φ-φ)/{1+ecos(φ-φ)}^2]φ'
(-[Lesin(φ-φ)/{1+ecos(φ-φ)}^2](φ')^2+(1/2)[L/{1+ecos(φ-φ)}]φ'')=0
(-[esin(φ-φ)/{1+ecos(φ-φ)}](φ')^2+(1/2)φ'')=0
(-[esin(φ-φ)/{1+ecos(φ-φ)}](dφ/dt)^2+(1/2)d(dφ/dt)/dt)=0
[esin(φ-φ)/{1+ecos(φ-φ)}](dφ/dt)dφ+(1/2)d(dφ/dt)=0
elog{1+ecos(φ-φ)}(dφ/dt)-elog{1+ecos(φ-φ)}{d(dφ/dt)/dφ}dφ+(1/2)dφ/dt=0
elog{1+ecos(φ-φ)}(dφ/dt)-elog{1+ecos(φ-φ)}d(dφ/dt)+(1/2)dφ/dt=0
t=log{1+ecos(φ-φ)}
dφ/dt=1/(dt/dφ)={1+ecos(φ-φ)}/{-esin(φ-φ)}
[elog{1+ecos(φ-φ)}+(1/2)](dφ/dt)-e∫log{1+ecos(φ-φ)}d[{1+ecos(φ-φ)}/{-esin(φ-φ)}]=0
∫f(x)d{f’(x)}=f(x) f’(x)
∫f(x)d{f’(x)}=∫f(x)d{df(x)/dx}=(d/dx){∫f(x)df(x)}=(d/dx){f(x)^2/2}=f(x)f’(x)
elog{1+ecos(φ-φ)}[{1+ecos(φ-φ)}/{-esin(φ-φ)}]
[elog{1+ecos(φ-φ)}+(1/2)](dφ/dt)=elog{1+ecos(φ-φ)}[{1+ecos(φ-φ)}/{-esin(φ-φ)}]
[-esin(φ-φ)/{1+ecos(φ-φ)}dφ+(1/2)[-esin(φ-φ)/{1+ecos(φ-φ)}][1/elog{1+ecos(φ-φ)}]dφ=dt]
t=log{1+ecos(φ-φ)}+(1/2)∫[log{1+ecos(φ-φ)}[1/elog{1+ecos(φ-φ)}]-
(1/2)log{1+ecos(φ-φ)}[log^-2{1+ecos(φ-φ)}]{(-e)sin(φ-φ)/1+ecos(φ-φ)}}]dφ
=log{1+ecos(φ-φ)}+(1/2)∫[log^-1{1+ecos(φ-φ)}]{(-e)sin(φ-φ)/{1+ecos(φ-φ)}}]dφ
∫{f'(x)/f(x)}dx=logf(x)
t=(3/2)log{1+ecos(φ-φ)}
t=log{1+ecos(φ-φ)}

t=log(L/r)

ここまで

(xb、yb)=(rbcosφb、rbsinφb)
mbdVxb/dt=fxb=f(rb)cosφb
mbdVyb/dt=fyb=f(rb)sinφb
Vxb=dxb/dt=rb'cosφb-rbφb'sinφb
Vyb=dyb/dt=rb'sinφb+rbφb'cosφb

αxb
αxb=dVxb/dt
αxb=rb''cosφb–rb'φb'sinφb–rb'(φb'sinφb)–rb(φb'sinφa)'
αxb=rb''cosφb–rb'φb'sinφb–rb'(φb'sinφb)–rb(φb''sinφb+(φb')2cosφb)
αxb=rb''cosφb–rb'φb'sinφb–rb'φb'sinφb–rb'φb'sinφb–rb(φb')2cosφb
αxb=r'b'cosφb-2rb'φb'sinφb–rb'φb'sinφb–rb(φb')2cosφb

αyb
αyb=dVyb/dt
αyb=rb''sinφb+rb'φb'cosφb+rb'(φb'cosφb)+rb(φb'cosφb)'
αyb=rb''sinφb+rb'φb'cosφb+rb'φb'cosφb+rb(φb''cosφb-(φb')2sinφb)
αyb=rb''sinφb+rb'φb'cosφb+rb'φb'cosφb+rb'φb'cosφb–rb(φb')2sinφb
αyb=rb''sinφb+2rb'φb'cosφb+rb'φb'cosφb–rb(φb')2sinφb

cosφb+sinφb=mb(cosφb･dVxb/dt+sinφb･dVyb/dt)=(sin2φb+cos2φb)f(rb)=f(rb)
cosφbdVxb/dt+sinφbdVyb/dt=f(rb)/mb

sinφb–cosφb=mb(sinφb･dVxb/dt+cosφb･dVyb/dt)=cosφbsinφbf(rb)-sinφbcosφbf(rb)=0

cosφbrb''cosφb–2rb'φb'sinφb–rb'φb'sinφb-rb(φb')2cosφb}+
sinφ{brb''sinφb+2rb'φb'cosφb+rb'φb'cosφb–rb(φb')2sinφb｝
=rb''–rb(φb')2=f(rb)/m
mb{rb''–rb(φb')2}=f(rb)

sinφb{rb''cosφb - 2rb'φb'sinφb – rb'φb'sinφb – rb(φb')2cosφb｝
(-cosφb){rb''sinφb+2rb'φb'cosφb+rb'φb'cosφb – rb(φb')2sinφb｝
=2rb'φb'+rbφb''=0

mb(2rb'φb'+rbφb'')=0

mb｛rb'' – rb(φb')2｝=f(rb)…⑤
mb(2rb'φb'+rbφb'')=0…⑥

rb2φb'