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Remarks on subharmonics with minimal periods of Hamiltonian systems. (English) Zbl 0789.58030

Let p>1 be an integer, J=0 -I N I N 0. Consider the problem

Jz ˙=H z (z(t),t),z(0)=z(pT),(1 p )

where z 2N , H(z,t+T)=H(z,t).

The authors study the solution of (1) p with minimal periods and make the following assumptions on the Hamiltonian H. (i) H(·,t) convex for any t[0,t]. (ii) There exist constants a 1 >0, a 2 >0 and 1<β<2 such that a 1 /β·|z| β H(z,t)=a 2 /β·|z| β z 2N . (iii) If z=z(t) is a periodic function with minimal period qT, q rational, and H z (z(t),t) is a periodic function with minimal period qT, then q is necessarily an integer. Let p=p 1 r 1 p s r s , p 1 <<p s be prime factors of p. For an integer q, 1qp, define Q q =min{L|L|p,q<L}. Define K q = smallest common multiple of {j1=j=Q q -1}:=K ' ·p 1 t 1 p s t s , where K ' is relatively prime to p and t i , i=1,,s are uniquely determined by q. Then the authors prove the following theorem. Let H satisfy (i)– (iii). For fixed q, 1qp, set Q q and K q =K ' ·p 1 t 1 p s t s as above. If there exists a t γ , 1γs, for some integer k1 such that a 2 /a 1 <(2Q q /k(k+1)) β/2 and kt γ N<r γ , then (1) p admits at least 2kN distinct solutions with possible minimal periods 2πp/, 1q and p.


MSC:
58E30Variational principles on infinite-dimensional spaces
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
34C25Periodic solutions of ODE