# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Remarks on subharmonics with minimal periods of Hamiltonian systems. (English) Zbl 0789.58030

Let $p>1$ be an integer, $J=\left(\genfrac{}{}{0pt}{}{0}{-{I}_{N}}\genfrac{}{}{0pt}{}{{I}_{N}}{0}\right)$. Consider the problem

$J\stackrel{˙}{z}={H}_{z}\left(z\left(t\right),t\right),\phantom{\rule{1.em}{0ex}}z\left(0\right)=z\left(p\phantom{\rule{4pt}{0ex}}T\right),\phantom{\rule{2.em}{0ex}}\left({1}_{\mathrm{p}}\right)$

where $z\in {ℝ}^{2N}$, $H\left(z,t+T\right)=H\left(z,t\right)$.

The authors study the solution of ${\left(1\right)}_{p}$ with minimal periods and make the following assumptions on the Hamiltonian $H$. (i) $H\left(·,t\right)$ convex for any $t\in \left[0,t\right]$. (ii) There exist constants ${a}_{1}>0$, ${a}_{2}>0$ and $1<\beta <2$ such that ${a}_{1}/\beta ·{|z|}^{\beta }\le H\left(z,t\right)={a}_{2}/\beta ·{|z|}^{\beta }$ $\forall z\in {ℝ}^{2N}$. (iii) If $z=z\left(t\right)$ is a periodic function with minimal period $qT$, $q$ rational, and ${H}_{z}\left(z\left(t\right),t\right)$ is a periodic function with minimal period $qT$, then $q$ is necessarily an integer. Let $p={p}_{1}^{{r}_{1}}\cdots {p}_{s}^{{r}_{s}}$, ${p}_{1}<\cdots <{p}_{s}$ be prime factors of $p$. For an integer $q$, $1\le q\le p$, define ${Q}_{q}=min\left\{L|L|p,q. Define ${K}_{q}=$ smallest common multiple of $\left\{j\mid 1=j={Q}_{q}-1\right\}:={K}^{\text{'}}·{p}_{1}^{{t}_{1}}\cdots {p}_{s}^{{t}_{s}}$, where ${K}^{\text{'}}$ is relatively prime to $p$ and ${t}_{i}$, $i=1,\cdots ,s$ are uniquely determined by $q$. Then the authors prove the following theorem. Let $H$ satisfy (i)– (iii). For fixed $q$, $1\le q\le p$, set ${Q}_{q}$ and ${K}_{q}={K}^{\text{'}}·{p}_{1}^{{t}_{1}}\cdots {p}_{s}^{{t}_{s}}$ as above. If there exists a ${t}_{\gamma }$, $1\le \gamma \le s$, for some integer $k\ge 1$ such that ${a}_{2}/{a}_{1}<{\left(2{Q}_{q}/k\left(k+1\right)\right)}^{\beta /2}$ and $k{t}_{\gamma }N<{r}_{\gamma }$, then ${\left(1\right)}_{p}$ admits at least $2kN$ distinct solutions with possible minimal periods $2\pi p/\ell$, $1\le \ell \le q$ 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