|
СЕМИНАРЫ |
Научный семинар «Актуальные проблемы геометрии и механики» имени проф. В. В. Трофимова
|
|||
|
Интегрируемость в элементарных функциях систем с переменной диссипацией М. В. Шамолин Московский государственный университет имени М. В. Ломоносова |
|||
Аннотация: In this work, the questions of integrability through a finite set of elementary functions of nonconservative dynamical systems of the following form $$ \dot{\alpha}=f_{\alpha}(\omega,\sin\alpha,\cos\alpha), \quad \dot{\omega}_{k}=f_{k}(\omega,\sin\alpha,\cos\alpha), \quad k=1,\ldots,n $$ are studied. The system is given on $S^{1}\{\alpha\bmod2\pi\}\times \mathbb{R}^{n}\{\omega\}$, \begin{gather*} f_{\alpha}(-t_{1},-t_{2},t_{3})=-f_{\alpha}(t_{1},t_{2},t_{3}), \quad f_{\alpha}(t_{1},t_{2},-t_{3})=f_{\alpha}(t_{1},t_{2},t_{3}), \\ f_{k}(-t_{1},-t_{2},t_{3})=-f_{k}(t_{1},t_{2},t_{3}), \quad f_{k}(t_{1},t_{2},-t_{3})=-f_{k}(t_{1},t_{2},t_{3}). \end{gather*} Such a system corresponds to the system $$ \frac{d\omega_{k}}{d\alpha}= \frac{f_{k}(\omega,\sin\alpha,\cos\alpha)} {f_{\alpha}(\omega,\sin\alpha,\cos\alpha)}, $$ which, by using the substitution $$ \frac{d\omega_{k}}{d\tau}= \frac{f_{k}(\omega,\tau,\varphi_{k}(\tau))} {f_{\alpha}(\omega,\tau,\varphi_{\alpha}(\tau))}, $$ where $\varphi_{\lambda}(-\tau)=\varphi_{\lambda}(\tau)$, In the work, the case where the functions |