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In this paper, the Conley conjecture, which were recently proved by Franksand Handel \cite{FrHa} (for surfaces of positive genus), Hingston \cite{Hi}(for tori) and Ginzburg \cite{Gi} (for closed symplectically asphericalmanifolds), is proved for $C^1$-Hamiltonian systems on the cotangent bundle ofa $C^3$-smooth compact manifold $M$ without boundary, of a time 1-periodic$C^2$-smooth Hamiltonian $H:\R\times T^\ast M\to\R$ which is strongly convexand has quadratic growth on the fibers. Namely, we show that such a Hamiltoniansystem has an infinite sequence of contractible integral periodic solutionssuch that any one of them cannot be obtained from others by iterations. If $H$also satisfies $H(-t,q, -p)=H(t,q, p)$ for any $(t,q, p)\in\R\times T^\ast M$,it is shown that the time-one map of the Hamiltonian system (if exists) hasinfinitely many periodic points siting in the zero section of $T^\ast M$. If$M$ is $C^5$-smooth and $\dim M>1$, $H$ is of $C^4$ class and independent oftime $t$, then for any $\tau>0$ the corresponding system has an infinitesequence of contractible periodic solutions of periods of integral multiple of$\tau$ such that any one of them cannot be obtained from others by iterationsor rotations. These results are obtained by proving similar results for theLagrangian system of the Fenchel transform of $H$, $L:\R\times TM\to\R$, whichis proved to be strongly convex and to have quadratic growth in the velocitiesyet.

The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems