Matematica Financiera

L=T-V=12μr2+r2o2+Gm1 m2r
μr=μro2-Gm1 m2r2
r=1u
drdu=1u2
u=u'σ
L=μr2σ=1u2μσ
σ=Lμu2
u=Lμu'u2r=drduu=-1u2Lμu'u2=-Lμu'
r=drdσσ=-Lμu''Lμu2=L2μ2u''u2
μr=μrσ2-Gm1 m2r2
μ-L2μ2u''u2=μLμu22u- Gm1 m2u2
-L2μ2u''u2=L2μu3-Gm1 m2u2-L2μ2u''u2=L2μu3-Gm1 m2u2
u''+u=Gm1 m2μL2
λ=L2μGm1 m2
u''+u=1λ
u=αCosσ+βSenσ
u'=-α Senσ+β Cosσ
u'=-αCosσ-βSenσ
u=αCosσ+1λu''=-αCosσ
u''+u=αCosσ+1λ-αCosσ=1λ
u=αCosσ+1λ=1+εCosσλ
L=1-rsrc2t2-r21-rsr-r2σ2
rs=2GMc2
1-rsrc2t2-r21-rsr-r2σ2=c2c2γ=12∂L∂t=1-rsrc2t
γ=t1-rsr
t=γ1-rsr
1-rsrc2t2-r21-rsr-r2σ2=c2
1-rsuc2γ21-rsu2--Lμu'21-rsu-1u2Lμu22=c2c2γ21-rsu-L2μ2u'21-rsu-L2μ2u2=c2
c2γ2-L2μ2u'2-L2μ2u2+rsL2μ2u3=c2-rsc2u
L2μ2u'2=c2γ2-1+rsc2u-L2μ2u2+rsL2μ2u32L2μ2u'u''=rsc2u'-2L2μ2uu'+3rsL2μ2u2u'
u''=rsμ2c22L2-u+3rs2u2
u''+u=1λ+3rs2u2
u=1λ1+εCosσ
u''+u=1λ+3rs2u2=1λ+3rs21λ1+εCosσ2u''+u=1λ+3rs2λ21+2εCosσ+ε2Cos2σ
u''+u=1λ+3rs2λ2+3rs2λ22εCosσ+3rs2λ2ε2Cos2σ
u1''+u1=1λ
u2''+u2=3rs2λ2
u3''+u3=3rs2λ22εCosσu4''+u4=3rs2λ2ε2Cos2σ
u1=1λ1+εCosσ
u2=3rs2λ2
u=n=0∞anσn
u'=n=1∞ann σn-1=n=n+1n=0∞an+1n+1σn
u''=n=1∞an+1n+1nσn-1=n=n+1n=0∞an+2n+2n+1σn
3rs2λ22εCosσ=3rs2λ22εn=0∞-1nσ2n2n!=3rsλ2εn=0∞-1nσ2n2n!
n=0∞an+2n+2n+1σn+n=0∞anσn=3rsλ2εn=0∞-1nσ2n2n!