周期関数時間c

(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=rbcosφb–rb'φb'sinφb–rb'(φb'sinφb)–rb(φbsinφb+(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=rbsinφb+rb'φb'cosφb+rb'φb'cosφb+rb(φbcosφb-(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{brbsinφb+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){rbsinφb+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)
mb(2rb'φb'+rbφb'')=0

rb2φb'
Sb=(1/2)rb2φb'
dSb/dt={(1/2)rb2φb'}'=rbrb'φb'+(1/2)rb2φb''
rb'φb'+(1/2)rbφb''=0

B
Xb
Vxa
αxa
Yb
Vya
αya
A
(xa,ya)=(racosφa,rasinφa)

ma*dVxa/dt=fxa=f(ra)cosφa
ma*dVya/dt=fya=f(ra)sinφa

Vxa=dxa/dt=ra'cosφa–raφa'sinφa
Vya=dya/dt=ra'sinφa+raφa'cosφa

αxa
αxa=dVxa/dt
αxa=ra''cosφa–ra'φa'sinφa–ra'(φa'sinφa)–ra(φa'sinφa)'
αxa=racosφa–ra'φa'sinφa–ra'(φa'sinφa)–ra(φasinφa+(φa')2cosφa)
αxa=ra''cosφa–ra'φa'sinφa–ra'φa'sinφa–ra'φa'sinφa–ra(φa')2cosφa
αxa=r'a'cosφa-2ra'φa'sinφa–ra'φa'sinφa–ra(φa')2cosφa

αya
αya=dVya/dt
αya=ra''sinφa+ra'φa'cosφa+ra'(φa'cosφa)+ra(φa'cosφa)'
αya=rasinφa+ra'φa'cosφa+ra'φa'cosφa+ra(φacosφa-(φa')2sinφa)
αya=ra''sinφa+ra'φa'cosφa+ra'φa'cosφa+ra'φa'cosφa–ra(φa')2sinφa
αya=ra''sinφa+2ra'φa'cosφa+ra'φa'cosφa–ra(φa')2sinφa

cosφa+sinφa=ma(cosφa*dVxa/dt+sinφa*dVya/dt)=(sin2φa+cos2φa)f(ra)=f(ra)
cosφa*dVxa/dt+sinφa*dVya/dt=f(ra)/ma
sinφa–cosφa=ma(sinφa*dVxa/dt+cosφa*dVya/dt)=cosφasinφaf(ra)-sinφacosφaf(ra)=0
cosφa{ra''cosφa–2ra'φa'sinφa–ra'φa'sinφa-ra(φa')2cosφa}+
sinφa{rasinφa+2ra'φa'cosφa+ra'φa'cosφa–ra(φa')2sinφa}=ra–ra(φa')2=f(ra)/m
ma{ra''–ra(φa')2}=f(ra)

sinφa{ra''cosφa-2ra'φa'sinφa–ra'φa'sinφa–ra(φa')2cosφa}-
cosφa{rasinφa+2ra'φa'cosφa+ra'φa'cosφa–ra(φa')2sinφa}=2ra'φa'+raφa=0
ma(2ra'φa'+raφa'')=0

ma{ra''–ra(φa')2}=f(ra)
ma(2ra'φa'+raφa'')=0

ra2φa'

Sa=(1/2)ra2φa'
dSa/dt={(1/2)ra2φa'}'=rara'φa'+(1/2)ra2φa''
ra'φa'+(1/2)raφa''=0

dφb/dt=hbub^2
dφa/dt=haua^2

{(d^2ub)/(dφb^2)}=f(1/ub)/(hb^2*ub^2)
{(d^2ua)/(dφa^2)}=f(1/ua)/(ha^2*ua^2)

rb=1/ub=Lb/{1+ebcos(φb-φb)}
ra=1/ua=La/{1+eacos(φa-φa)}