r 1 ¨ ( t ) = − G m 2 ( r 1 ( t ) − r 2 ( t ) ) | r 2 ( t ) − r 1 ( t ) | 3 {\displaystyle {\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}}}}
r 2 ¨ ( t ) = − G m 1 ( r 1 ( t ) − r 2 ( t ) ) | r 2 ( t ) − r 1 ( t ) | 3 {\displaystyle {\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}}}}
r 1 ¨ ( t ) = − G m 2 ( r 1 ( t ) − r 2 ( t r 21 ) ) ( 1 − | r 2 ( t r 21 ) | 2 c 2 + ( r 1 ( t ) − r 2 ( t r 21 ) ) ⋅ r 2 ˙ ( t r 21 ) ) c ) − r 2 ( t r 21 ) c ( 1 − ( r 1 ( t ) − r 2 ( t r 21 ) ) | r 1 ( t ) − r 2 ( t r 21 ) | ⋅ r 2 ˙ ( t ) c ) | r 1 ( t ) − r 2 ( t r 21 ) | 3 ( 1 − ( r 1 ( t ) − r 2 ( t r 21 ) ) | r 1 ( t ) − r 2 ( t r 21 ) | ⋅ r 1 ˙ ( t r 21 ) c ) 3 {\displaystyle {\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 r 21 = t − 1 c | r 1 ( t ) − r 2 ( t r 21 ) | {\displaystyle t_{r21}=t-{\frac {1}{c}}\left|\mathbf {r_{1}} (t)-\mathbf {r_{2}} (t_{r21})\right|}
r 2 ¨ ( t ) = − G m 1 ( r 2 ( t ) − r 1 ( t r 12 ) ) ( 1 − | r 2 ( t r 12 ) | 2 c 2 + ( r 2 ( t ) − r 1 ( t r 12 ) ) ⋅ r 2 ˙ ( t r 12 ) ) c ) − r 2 ( t r 12 ) c ( 1 − ( r 2 ( t ) − r 1 ( t r 12 ) ) | r 2 ( t ) − r 1 ( t r 12 ) | ⋅ r 2 ˙ ( t ) c ) | r 2 ( t ) − r 1 ( t r 12 ) | 3 ( 1 − ( r 2 ( t ) − r 1 ( t r 12 ) ) | r 2 ( t ) − r 1 ( t r 12 ) | ⋅ r 1 ˙ ( t r 12 ) c ) 3 {\displaystyle {\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 r 12 = t − 1 c | r 2 ( t ) − r 1 ( t r 12 ) | {\displaystyle t_{r12}=t-{\frac {1}{c}}\left|\mathbf {r_{2}} (t)-\mathbf {r_{1}} (t_{r12})\right|}
Electromagnetic:
E = − q 4 π ε 0 [ e r ′ r ′ 2 + r ′ c d d t ( e r ′ r ′ 2 ) + 1 c 2 d 2 d t 2 e r ′ ] {\displaystyle \mathbf {E} ={\frac {-q}{4\pi \varepsilon _{0}}}\left[{\frac {\mathbf {e} _{r'}}{r'^{2}}}+{\frac {r'}{c}}{\frac {d}{dt}}\left({\frac {\mathbf {e} _{r'}}{r'^{2}}}\right)+{\frac {1}{c^{2}}}{\frac {d^{2}}{dt^{2}}}\mathbf {e} _{r'}\right]}
B = − e r ′ × E c {\displaystyle \mathbf {B} =-\mathbf {e} _{r'}\times {\frac {\mathbf {E} }{c}}}
m 1 r ¨ 1 ( t ) = μ 0 c 2 q 1 q 2 4 π ( [ r 1 ( t ) − r 2 ( t r ) | r 1 ( t ) − r 2 ( t r ) | − r 1 ( t ) − r 2 ( t r ) | r 1 ( t ) − r 2 ( t r ) | 1 1 − | r ˙ 2 ( t r ) c | 2 ( 1 − r 1 ( t ) − r 2 ( t r ) | r 1 ( t ) − r 2 ( t r ) | ⋅ r ˙ 2 ( t r ) c ) 3 | r 1 ( t ) − r 2 ( t r ) | 2 + r 1 ( t ) − r 2 ( t r ) | r 1 ( t ) − r 2 ( t r ) | × ( ( r 1 ( t ) − r 2 ( t r ) | r 1 ( t ) − r 2 ( t r ) | − r ˙ 2 ( t r ) c ) × r ¨ 2 ( t r ) c ) c ( 1 − r 1 ( t ) − r 2 ( t r ) | r 1 ( t ) − r 2 ( t r ) | ⋅ r ˙ 2 ( t r ) c ) 3 | r 1 ( t ) − r 2 ( t r ) | ] + [ r ˙ 1 ( t ) × [ r 1 ( t ) − r 2 ( t r ) | r 1 ( t ) − r 2 ( t r ) | c × [ r 1 ( t ) − r 2 ( t r ) | r 1 ( t ) − r 2 ( t r ) | − r 1 ( t ) − r 2 ( t r ) | r 1 ( t ) − r 2 ( t r ) | 1 1 − | r ˙ 2 ( t r ) c | 2 ( 1 − r 1 ( t ) − r 2 ( t r ) | r 1 ( t ) − r 2 ( t r ) | ⋅ r ˙ 2 ( t r ) c ) 3 | r 1 ( t ) − r 2 ( t r ) | 2 + r 1 ( t ) − r 2 ( t r ) | r 1 ( t ) − r 2 ( t r ) | × ( ( r 1 ( t ) − r 2 ( t r ) | r 1 ( t ) − r 2 ( t r ) | − r ˙ 2 ( t r ) c ) × r ¨ 2 ( t r ) c ) c ( 1 − r 1 ( t ) − r 2 ( t r ) | r 1 ( t ) − r 2 ( t r ) | ⋅ r ˙ 2 ( t r ) c ) 3 | r 1 ( t ) − r 2 ( t r ) | ] ] ] ) {\displaystyle m_{1}{\mathbf {\ddot {r}} _{1}}(t)={\frac {\mu _{0}c^{2}q_{1}q_{2}}{4\pi }}\left({\left[{\frac {{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}-{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}}{{\frac {1}{1-|{\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}}|^{2}}}(1-{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}})^{3}|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|^{2}}}+{\frac {{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\times {\big (}({\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}-{\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}})\times {\frac {\mathbf {\ddot {r}} _{2}(t_{r})}{c}}{\big )}}{c(1-{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}})^{3}|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\right]}+\left[\mathbf {\dot {r}} _{1}(t)\times \left[{\frac {\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}{c}}\times {\left[{\frac {{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}-{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}}{{\frac {1}{1-|{\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}}|^{2}}}(1-{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}})^{3}|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|^{2}}}+{\frac {{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\times {\big (}({\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}-{\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}})\times {\frac {\mathbf {\ddot {r}} _{2}(t_{r})}{c}}{\big )}}{c(1-{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}})^{3}|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\right]}\right]\right]\right)}
m 2 r ¨ 2 ( t ) = μ 0 c 2 q 1 q 2 4 π ( [ r 2 ( t ) − r 1 ( t r ) | r 2 ( t ) − r 1 ( t r ) | − r ˙ 1 ( t r ) c 1 1 − | r ˙ 1 ( t r ) c | 2 ( 1 − r 2 ( t ) − r 1 ( t r ) | r 2 ( t ) − r 1 ( t r ) | ⋅ r ˙ 1 ( t r ) c ) 3 | r 2 ( t ) − r 1 ( t r ) | 2 + r 2 ( t ) − r 1 ( t r ) | r 2 ( t ) − r 1 ( t r ) | × ( ( r 2 ( t ) − r 1 ( t r ) | r 2 ( t ) − r 1 ( t r ) | − r ˙ 1 ( t r ) c ) × r ¨ 1 ( t r ) c ) c ( 1 − r 2 ( t ) − r 1 ( t r ) | r 2 ( t ) − r 1 ( t r ) | ⋅ r ˙ 1 ( t r ) c ) 3 | r 2 ( t ) − r 1 ( t r ) | ] + [ r ˙ 2 ( t ) × [ r 2 ( t ) − r 1 ( t r ) | r 2 ( t ) − r 1 ( t r ) | c × [ r 2 ( t ) − r 1 ( t r ) | r 2 ( t ) − r 1 ( t r ) | − r ˙ 1 ( t r ) c 1 1 − | r ˙ 1 ( t r ) c | 2 ( 1 − r 2 ( t ) − r 1 ( t r ) | r 2 ( t ) − r 1 ( t r ) | ⋅ r ˙ 1 ( t r ) c ) 3 | r 2 ( t ) − r 1 ( t r ) | 2 + r 2 ( t ) − r 1 ( t r ) | r 2 ( t ) − r 1 ( t r ) | × ( ( r 2 ( t ) − r 1 ( t r ) | r 2 ( t ) − r 1 ( t r ) | − r ˙ 1 ( t r ) c ) × r ¨ 1 ( t r ) c ) c ( 1 − r 2 ( t ) − r 1 ( t r ) | r 2 ( t ) − r 1 ( t r ) | ⋅ r ˙ 1 ( t r ) c ) 3 | r 2 ( t ) − r 1 ( t r ) | ] ] ] ) {\displaystyle m_{2}{\mathbf {\ddot {r}} _{2}}(t)={\frac {\mu _{0}c^{2}q_{1}q_{2}}{4\pi }}\left({\left[{\frac {{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}-{\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}}}{{\frac {1}{1-|{\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}}|^{2}}}(1-{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}})^{3}|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|^{2}}}+{\frac {{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\times {\big (}({\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}-{\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}})\times {\frac {\mathbf {\ddot {r}} _{1}(t_{r})}{c}}{\big )}}{c(1-{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}})^{3}|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\right]}+\left[\mathbf {\dot {r}} _{2}(t)\times \left[{\frac {\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}{c}}\times {\left[{\frac {{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}-{\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}}}{{\frac {1}{1-|{\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}}|^{2}}}(1-{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}})^{3}|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|^{2}}}+{\frac {{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\times {\big (}({\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}-{\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}})\times {\frac {\mathbf {\ddot {r}} _{1}(t_{r})}{c}}{\big )}}{c(1-{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}})^{3}|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\right]}\right]\right]\right)}
t r = t − 1 c | r 1 ( t ) − r 2 ( t r ) | {\displaystyle t_{r}=t-{\frac {1}{c}}|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}
![{\displaystyle \mathbf {E} ={\frac {-q}{4\pi \varepsilon _{0}}}\left[{\frac {\mathbf {e} _{r'}}{r'^{2}}}+{\frac {r'}{c}}{\frac {d}{dt}}\left({\frac {\mathbf {e} _{r'}}{r'^{2}}}\right)+{\frac {1}{c^{2}}}{\frac {d^{2}}{dt^{2}}}\mathbf {e} _{r'}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b56563a9abddcef843da4ef73a9b04bf99e57da6)
![{\displaystyle m_{1}{\mathbf {\ddot {r}} _{1}}(t)={\frac {\mu _{0}c^{2}q_{1}q_{2}}{4\pi }}\left({\left[{\frac {{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}-{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}}{{\frac {1}{1-|{\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}}|^{2}}}(1-{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}})^{3}|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|^{2}}}+{\frac {{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\times {\big (}({\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}-{\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}})\times {\frac {\mathbf {\ddot {r}} _{2}(t_{r})}{c}}{\big )}}{c(1-{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}})^{3}|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\right]}+\left[\mathbf {\dot {r}} _{1}(t)\times \left[{\frac {\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}{c}}\times {\left[{\frac {{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}-{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}}{{\frac {1}{1-|{\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}}|^{2}}}(1-{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}})^{3}|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|^{2}}}+{\frac {{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\times {\big (}({\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}-{\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}})\times {\frac {\mathbf {\ddot {r}} _{2}(t_{r})}{c}}{\big )}}{c(1-{\frac {\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})}{|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{2}(t_{r})}{c}})^{3}|\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t_{r})|}}\right]}\right]\right]\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/672e28d36ab693e52128bd201b2f7ab6b383b79f)
![{\displaystyle m_{2}{\mathbf {\ddot {r}} _{2}}(t)={\frac {\mu _{0}c^{2}q_{1}q_{2}}{4\pi }}\left({\left[{\frac {{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}-{\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}}}{{\frac {1}{1-|{\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}}|^{2}}}(1-{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}})^{3}|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|^{2}}}+{\frac {{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\times {\big (}({\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}-{\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}})\times {\frac {\mathbf {\ddot {r}} _{1}(t_{r})}{c}}{\big )}}{c(1-{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}})^{3}|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\right]}+\left[\mathbf {\dot {r}} _{2}(t)\times \left[{\frac {\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}{c}}\times {\left[{\frac {{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}-{\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}}}{{\frac {1}{1-|{\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}}|^{2}}}(1-{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}})^{3}|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|^{2}}}+{\frac {{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\times {\big (}({\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}-{\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}})\times {\frac {\mathbf {\ddot {r}} _{1}(t_{r})}{c}}{\big )}}{c(1-{\frac {\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})}{|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\cdot {\frac {\mathbf {\dot {r}} _{1}(t_{r})}{c}})^{3}|\mathbf {r} _{2}(t)-\mathbf {r} _{1}(t_{r})|}}\right]}\right]\right]\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71a7849e4782ca00611fdee0365c9f9b611f245d)
