周期関数時間d

A
Xa
B
Vxab
αxab
Ya
B
Vyab
αyab
Xa
C
Vxac
αxac
Ya
C
Vyac
αyac

B
(xab,yab)=(rabcosφb,rabsinφb)
C
(xac,yac)=(raccosφc,racsinφc)

mb*dVxab/dt=fxab=f(rab)cosφb
mb*dVyab/dt=fyab=f(rab)sinφb
mc*dVxac/dt=fxac=f(rac)cosφc
mc*dVyac/dt=fyac=f(rac)sinφc

Vxab=dxab/dt=rab'cosφb–rabφb'sinφb
Vyab=dyab/dt=rab'sinφb+rabφb'cosφb
Vxac=dxac/dt=rac'cosφc–racφc'sinφc
Vyac=dyac/dt=rac'sinφc+racφc'cosφc

αxab
αxab=dVxab/dt
αxab=rab''cosφb–rab'φb'sinφb–rab'(φb'sinφb)–rab(φb'sinφa)'
αxab=rabcosφb–rab'φb'sinφb–rab'(φb'sinφb)–rab(φbsinφb+(φb')2cosφb)
αxab=rab''cosφb–rab'φb'sinφb–rab'φb'sinφb–rab'φb'sinφb–rab(φb')2cosφb
αxab=ra'b'cosφb-2rab'φb'sinφb–rab'φb'sinφb–rab(φb')2cosφb

αxac
αxac=dVxac/dt
αxac=rac''cosφc–rac'φc'sinφc–rac'(φc'sinφc)–rac(φc'sinφa)'
αxac=raccosφc–rac'φc'sinφc–rac'(φc'sinφc)–rac(φcsinφc+(φc')2cosφc)
αxac=rac''cosφc–rac'φc'sinφc–rac'φc'sinφc–rac'φc'sinφc–rac(φc')2cosφc
αxac=ra'c'cosφc-2rac'φc'sinφc–rac'φc'sinφc–rac(φc')2cosφc

αyab
αyab=dVyab/dt
αyab=rab''sinφb+rab'φb'cosφb+rab'(φb'cosφb)+rab(φb'cosφb)'
αyab=rabsinφb+rab'φb'cosφb+rab'φb'cosφb+rab(φbcosφb-(φb')2sinφb)
αyab=rab''sinφb+rab'φb'cosφb+rab'φb'cosφb+rab'φb'cosφb–rab(φb')2sinφb
αyab=rab''sinφb+2rab'φb'cosφb+rab'φb'cosφb–rab(φb')2sinφb

αyac
αyac=dVyac/dt
αyac=rac''sinφc+rac'φc'cosφc+rac'(φc'cosφc)+rac(φc'cosφc)'
αyac=racsinφc+rac'φc'cosφc+rac'φc'cosφc+rac(φccosφc-(φc')2sinφc)
αyac=rac''sinφc+rac'φc'cosφc+rac'φc'cosφc+rac'φc'cosφc–rac(φc')2sinφc
αyac=rac''sinφc+2rac'φc'cosφc+rac'φc'cosφc–rac(φc')2sinφc

cosφb+sinφb=mb(cosφb･dVxab/dt+sinφb･dVyab/dt)=(sin2φb+cos2φb)f(rab)=f(rab)
cosφb･dVxab/dt+sinφb･dVyab/dt=f(rab)/mb
sinφb–cosφb=mb(sinφb･dVxab/dt+cosφb･dVyab/dt)=cosφbsinφbf(rab)-sinφbcosφbf(rab)=0

cosφb+sinφb=mb(cosφb･dVxab/dt+sinφb･dVyab/dt)=(sin2φb+cos2φb)f(rab)=f(rab)
cosφb･dVxab/dt+sinφb･dVyab/dt=f(rab)/mb
sinφb–cosφb=mb(sinφb･dVxab/dt+cosφb･dVyab/dt)=cosφb sinφb f(rab)-sinφb cosφbf(rab)=0

cosφc+sinφc=mc(cosφc･dVxac/dt+sinφc･dVyac/dt)=(sin2φc+cos2φc)f(rac)=f(rac)
cosφc･dVxac/dt+sinφc･dVyac/dt=f(rac)/mc
sinφc–cosφc=mc(sinφc･dVxac/dt+cosφc･dVyac/dt)=cosφcsinφcf(rac)-sinφccosφcf(rac)=0

cosφc+sinφc=mc(cosφc･dVxac/dt+sinφc･dVyac/dt)=(sin2φc+cos2φc)f(rac)=f(rac)
cosφc･dVxac/dt+sinφc･dVyac/dt=f(rac)/mc
sinφc–cosφc=mc(sinφc･dVxac/dt+cosφc･dVyac/dt)=cosφc sinφc f(rac)-sinφc cosφcf(rac)=0

dVxab/dt
dVyab/dt
cosφb{rab''cosφb–2rab'φb'sinφb–rab'φb'sinφb-rab(φb')2cosφb}+
sinφb{rab''sinφb+2rab'φb'cosφb+rab'φb'cosφb–rab(φb')2sinφb}
=rab''–rab(φb')2=f(rab)/m
mb{rab''–rab(φb')2}=f(rab)

sinφb{rab''cosφb-2rab'φb'sinφb–rab'φb'sinφb–rab(φb')2cosφb}-
cosφb{rab''sinφb+2rab'φb'cosφb+rab'φb'cosφb–rab(φb')2sinφb}
=2rab'φb'+rabφb''=0

mb(2rab'φb'+rabφb'')=0

mb{rab''–rab(φb')2}=f(rab)
mb(2rab'φb'+rabφb'')=0

rab2φb'

Sb=(1/2)rab2φb'

dSb/dt={(1/2)rab2φb'}'=rabrab'φb'+(1/2)rab2φb''　
rab'φb'+(1/2)rabφb''=0

dVxac/dt
dVyac/dt
cosφc{rac''cosφc–2rac'φc'sinφc–rac'φc'sinφc-rac(φc')2cosφc}+
sinφc{rac''sinφc+2rac'φc'cosφc+rac'φc'cosφc–rac(φc')2sinφc}
=rac''–rac(φc')2=f(rac)/m
mc{rac''–rac(φc')2}=f(rac)

sinφc{rac''cosφc-2rac'φc'sinφc–rac'φc'sinφc–rac(φc')2cosφc}-
cosφc{rac''sinφc+2rac'φc'cosφc+rac'φc'cosφc–rac(φc')2sinφc}
=2rac'φc'+racφc''=0

mc(2rac'φc'+racφc'')=0

mc{rac''–rac(φc')2}=f(rac)
mc(2rac'φc'+racφc'')=0