Application of V. V. Kozlov's theorem for proving of nonexistence of analytical integrals for some problems of mechanics (Q2895492)

Consider a Hamiltonian system with two degrees of freedom which Hamiltonian \(H\) depends analytically both on the canonical variables and on a real parameter \(\alpha\). Let the Fourier's expansion of \(H\) around certain equilibrium point reads NEWLINE\[NEWLINEH =\lambda_1 I_1 + \lambda_2 I_2 + \sum_{|k|+|n|=4} B_{k,n}(I_1,I_2,\alpha) e^{ik\varphi_1+in\varphi_2} + (\text{terms with } |k|+|n|>4).NEWLINE\]NEWLINE Here \(I_j,\varphi_j\) are action-angle variables. Let the frequencies \(\lambda_j(\alpha)\) are purely imaginary and non-resonant, i.e. \(k\lambda_1+n\lambda_2\) does not vanish identically in \(\alpha\) for any pair of integers \((k,n) \not= (0,0)\). Suppose however that certain linear combination, say \(\lambda_1-3\lambda_2\), vanishes if the parameter \(\alpha=\alpha_0\), and \(B_{1,-3} \neq 0\). Then a theorem of V. V. Kozlov claims: there does not exist any independent on \(H\) first integral, analytical on \(I_j,\varphi_j\) and \(\alpha\) .NEWLINENEWLINEThe authors apply the above formulated result for a heavy axially-symmetric top whose fixed point lies on its equatorial plane. In the Hamiltonian NEWLINE\[NEWLINEH=\frac{p_\theta^2}{2A} + \frac{p_\varphi^2}{2C} + \frac{(p_\psi-p_\varphi \cos\theta )^2}{2A \sin^2\theta} + \sin\theta \sin\varphiNEWLINE\]NEWLINE \(A\) and \(C\) are positive constants, while the integral \(p_\psi\) has been considered as the parameter \(\alpha\). It is proved that an additional analytic integral exists if and only if \(A=C\) (purely symmetric top) or \(A=2C\) (Kowalevski's top).NEWLINENEWLINENext the authors study the double flat pendulum \(OAB\); \(O\) is the fixed point, (the mass of \(A)=m_1 \geq 0\), (the mass of \(B)=m_2>0\), the lengths \(|OA|=l_1>0\), \(|AB|=l_2>0\), NEWLINE\[NEWLINE\begin{multlined} H=\frac{1}{2}(m_1+m_2) l_1^2 \dot{\varphi}_1^2 + m_2 l_1 l_2 \cos(\varphi_1-\varphi_2) \dot{\varphi}_1\dot{\varphi}_2 + \frac{1}{2}m_2 l_2^2 \dot{\varphi}_2^2 \\ - (m_1+m_2) l_1 g \cos\varphi_1 - m_2 l_2 g \cos\varphi_2\end{multlined}NEWLINE\]NEWLINE with \(\varphi_j\) and \(p_j:=m_j \dot{\varphi}_j\) being angle-action variables and \(g=\text{const}.>0\). It is proved: an additional analytic integral could exist only if \(m_1=0\).NEWLINENEWLINEFor the entire collection see [Zbl 1236.70002].