15.Бине

$\tiny |\vec{mom_{O}}\vec{V_{O}}|=2\frac{d\sigma}{dt}=c,\\ c=|\vec{mom_{O}}\vec{V_{O}}|=r_{0}V_{0}sin(\vec{r_{0}}, \vec{V_{0}}) <=> \\ |\vec{mom_{O}}\vec{V_{O}}|=V_{\varphi }r=r^{2}\frac{d\varphi}{dt} =>\\ r^{2}\frac{d\varphi}{dt}=c\\ V^{2}=V_{r}^{2}+V_{\varphi}^{2}=(\frac{dr}{dt})^{2}+r^{2}(\frac{d\varphi}{dt})^{2}\\ V_{r}=\frac{dr}{dt}=\frac{dr}{d\varphi}\frac{d\varphi}{dt}=\frac{c}{r^{2}}\frac{dr}{d\varphi}, V_{\varphi}=r\frac{d\varphi}{dt}=\frac{c}{r} u=\frac{1}{r}\\ \frac{du}{d\varphi}=-\frac{1}{r^{2}}\frac{dr}{d\varphi}\\ V_{r}=-c\frac{du}{d\varphi}, V_{\varphi}=cu;\\ V^{2}=c^{2}((\frac{du}{d\varphi})^{2})+u^{2}\\ d(\frac{mV^{2}}{2})=F_{r}dr\\ \frac{c^2m}{2}\frac{d}{d\varphi}((\frac{du}{d\varphi})^{2}+u^{2})=-\frac{F_{r}}{u^{2}}\frac{du}{d\varphi}\\ \mathbf{F_{r}=-mc^2u^2(\frac{d^2u}{d\varphi^2}+u)}$