Let be an integer, . Consider the problem
where , .
The authors study the solution of with minimal periods and make the following assumptions on the Hamiltonian . (i) convex for any . (ii) There exist constants , and such that . (iii) If is a periodic function with minimal period , rational, and is a periodic function with minimal period , then is necessarily an integer. Let , be prime factors of . For an integer , , define . Define smallest common multiple of , where is relatively prime to and , are uniquely determined by . Then the authors prove the following theorem. Let satisfy (i)– (iii). For fixed , , set and as above. If there exists a , , for some integer such that and , then admits at least distinct solutions with possible minimal periods , and .