Final Equations

Lagrangian and Euler-Lagrange Equation

$$L=T-U$$ $$\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}_i}-\frac{\partial L}{\partial q_i}=0$$

Reduced Mass Equation

$$\frac{m_1{m}_2}{m_1+m_2}=\mu$$

Centrifugal Potential

$$U_{cf}=\frac{l^2}{2\mu r^2}$$

Effective Potential

$$U_{eff}=U+U_{cf}$$

Orbit

$$\frac{{\partial }^{2}u}{\partial {\theta }^2}+u=-\frac{\mu }{l^2}\frac{1}{u^2}F\left(\frac{1}{u}\right)$$

Continuous Distribution Inertia Tensor

$$I_{ij}=\int \rho \left({\delta }_{ij}\sum _k{x}_k^2-x_i{x}_j\right)dV$$

$A_{jk}$ matrices

$$A_{jk}=\frac{{\partial }^{2}u}{\partial q_j\partial q_k}$$

Eigenfrequency and Eigenvector relations

$$\det [\bar{A}-{\omega }^2\bar{m}]=0$$ $$[\bar{A}-{\omega }^2\bar{m}]\cdot \vec{v}=0$$

${\mu }_r$ formula

$${\mu }_r=\frac{2}{L}\underset{0}{\overset{L}{\int }}q(x,0)\sin \left(\frac{r\pi x}{L}\right)dx$$

${\nu }_r$ formula

$${\nu }_r=-\frac{2}{{\omega }_{r}L}\underset{0}{\overset{L}{\int }}\dot{q}(x,0)\sin \left(\frac{r\pi x}{L}\right)dx$$

position of strings formula

$$q(x,t)=\sum _{r=1}^{\infty }\sin \left(\frac{r\pi x}{L}\right)({\mu }_r\cos ({\omega }_{r}t)-{\nu }_r\sin ({\omega }_{r}t))$$