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Remarks on subharmonics with minimal periods of Hamiltonian systems. (English) Zbl 0789.58030
Let $p>1$ be an integer, $J = \left({0\atop -I\sb N}{I\sb N\atop 0}\right)$. Consider the problem $$J\dot z = H\sb z(z(t),t),\quad z(0) = z(p\ T),\tag {1\sb p}$$ where $z\in \bbfR\sp{2N}$, $H(z,t+T) = H(z,t)$. The authors study the solution of $(1)\sb p$ with minimal periods and make the following assumptions on the Hamiltonian $H$. (i) $H(\cdot,t)$ convex for any $t \in [0,t]$. (ii) There exist constants $a\sb 1 > 0$, $a\sb 2 > 0$ and $1 < \beta < 2$ such that $a\sb 1/\beta \cdot \vert z\vert\sp \beta \leq H(z,t) = a\sb 2/\beta \cdot \vert z\vert\sp \beta$ $\forall z\in \bbfR\sp{2N}$. (iii) If $z = z(t)$ is a periodic function with minimal period $qT$, $q$ rational, and $H\sb z(z(t),t)$ is a periodic function with minimal period $qT$, then $q$ is necessarily an integer. Let $p = p\sb 1\sp{r\sb 1} \dots p\sp{r\sb s}\sb s$, $p\sb 1 < \cdots < p\sb s$ be prime factors of $p$. For an integer $q$, $1 \leq q \leq p$, define $Q\sb q = \min\{L\vert L\vert p, q<L\}$. Define $K\sb q =$ smallest common multiple of $\{j\mid 1 = j=Q\sb q - 1\}: = K'\cdot p\sp{t\sb 1}\sb 1 \cdots p\sp{t\sb s}\sb s$, where $K'$ is relatively prime to $p$ and $t\sb i$, $i = 1,\dots,s$ are uniquely determined by $q$. Then the authors prove the following theorem. Let $H$ satisfy (i)-- (iii). For fixed $q$, $1 \leq q \leq p$, set $Q\sb q$ and $K\sb q = K'\cdot p\sp{t\sb 1}\sb 1\cdots p\sp{t\sb s}\sb s$ as above. If there exists a $t\sb \gamma$, $1 \leq \gamma \leq s$, for some integer $k\geq 1$ such that $a\sb 2/a\sb 1 < (2Q\sb q/k(k+1))\sp{\beta/2}$ and $kt\sb \gamma N < r\sb \gamma$, then $(1)\sb p$ admits at least $2kN$ distinct solutions with possible minimal periods $2\pi p/\ell$, $1 \leq \ell \leq q$ and $\ell\mid p$.

MSC:
 58E30 Variational principles on infinite-dimensional spaces 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 34C25 Periodic solutions of ODE
Keywords:
minimal periods; Hamiltonian
Full Text:
References:
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