User:Z E U S/retartedtwobodyproblem

$$\ddot{\mathbf{r_1}}(t) = -Gm_2\frac{(\mathbf{r_1}(t)-\mathbf{r_2}(t))}{\left| \mathbf{r_2}(t) - \mathbf{r_1}(t) \right|^3}$$

$$\ddot{\mathbf{r_2}}(t) = -Gm_1\frac{(\mathbf{r_1}(t)-\mathbf{r_2}(t))}{\left| \mathbf{r_2}(t) - \mathbf{r_1}(t) \right|^3}$$

$$\ddot{\mathbf{r_1}}(t)=-\frac{Gm_2\left( \mathbf{r_1}(t)-\mathbf{r_2}(t_{r21}) \right)\left( 1-\frac{\left| \mathbf{r_2}(t_{r21}) \right|^2}{c^2}+\left( \mathbf{r_1}(t)-\mathbf{r_2}(t_{r21}) \right)\cdot \frac{\dot{\mathbf{r_2}}(t_{r21}))}{c} \right)-\frac{\mathbf{r_2}(t_{r21})}{c}\left( 1-\frac{\left( \mathbf{r_1}(t)-\mathbf{r_2}(t_{r21}) \right)}{\left| \mathbf{r_1}(t)-\mathbf{r_2}(t_{r21}) \right|}\cdot \frac{\dot{\mathbf{r_2}}(t)}{c} \right)}{\left| \mathbf{r_1}(t)-\mathbf{r_2}(t_{r21}) \right|^3\left( 1-\frac{\left( \mathbf{r_1}(t)-\mathbf{r_2}(t_{r21}) \right)}{\left| \mathbf{r_1}(t)-\mathbf{r_2}(t_{r21}) \right|}\cdot \frac{\dot{\mathbf{r_1}}(t_{r21})}{c} \right)^3}$$

$$t_{r21}=t-\frac{1}{c}\left| \mathbf{r_1}(t)-\mathbf{r_2}(t_{r21}) \right|$$

$$\ddot{\mathbf{r_2}}(t)=-\frac{Gm_1\left( \mathbf{r_2}(t)-\mathbf{r_1}(t_{r12}) \right)\left( 1-\frac{\left| \mathbf{r_2}(t_{r12}) \right|^2}{c^2}+\left( \mathbf{r_2}(t)-\mathbf{r_1}(t_{r12}) \right)\cdot \frac{\dot{\mathbf{r_2}}(t_{r12}))}{c} \right)-\frac{\mathbf{r_2}(t_{r12})}{c}\left( 1-\frac{\left( \mathbf{r_2}(t)-\mathbf{r_1}(t_{r12}) \right)}{\left| \mathbf{r_2}(t)-\mathbf{r_1}(t_{r12}) \right|}\cdot \frac{\dot{\mathbf{r_2}}(t)}{c} \right)}{\left| \mathbf{r_2}(t)-\mathbf{r_1}(t_{r12}) \right|^3\left( 1-\frac{\left( \mathbf{r_2}(t)-\mathbf{r_1}(t_{r12}) \right)}{\left| \mathbf{r_2}(t)-\mathbf{r_1}(t_{r12}) \right|}\cdot \frac{\dot{\mathbf{r_1}}(t_{r12})}{c} \right)^3}$$

$$t_{r12}=t-\frac{1}{c}\left| \mathbf{r_2}(t)-\mathbf{r_1}(t_{r12}) \right|$$