周期関数時間a

Vx
Ax
Vy
Ay
(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=rcosφ-r'φ'sinφ-r'(φ'sinφ)-r(φsinφ+(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=rsinφ+r'φ'cosφ+r'φ'cosφ+r(φcosφ-(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)}