Periodic orbits of Lagrangian systems with prescribed action or period (Q2802118)

Let \(M\) be a closed connected smooth Riemannian manifold. A smooth convex Lagrangian is a smooth function \(L:TM \to \mathbb R\) such that \(L\) restricted to each \(T_xM\) has a positive definite Hessian. \(L\) is said to be quadratic at infinity if there exists \(R>0\) such that for each \(x\in M\) and \(|v|_x>R\), \(L(x,v)\) is of the form NEWLINE\[NEWLINE L(x,v)=\frac{1}{2}|v|_x^2+\theta_x(v)-V(x),NEWLINE\]NEWLINE where \(\theta\) is a smooth 1-form on \(M\), and \(V:M\to \mathbb R\) is smooth.NEWLINENEWLINELet \(\Lambda\) denote the set of absolutely continuous curves \(x:[0,1]\to M, x(0)=x(1)\) with finite \(L^2\)-norm which is a Hilbert manifold. Then the free-time action functional \(\mathcal A_L:\mathbb R^+\times \Lambda \to \mathbb R\) given by NEWLINE\[NEWLINE \mathcal A_L(b,x)= \int_0^1bL(x(t), \dot{x}(t)/b)dt, NEWLINE\]NEWLINE is a \(C^{1,1}\)-function, whose critical points are periodic orbits of the Lagrangian.NEWLINENEWLINEThe following two theorems are proved.NEWLINENEWLINETheorem 1: For every convex Lagrangian quadratic at infinity there exist positive numbers \(a_0, \alpha\) such that for every \(a>a_0\) there is a periodic orbit such that its period \(T\) and average energy \(e\) satisfy NEWLINE\[NEWLINE \frac{1}{\alpha a} \leq T \leq \frac{\alpha}{a}\quad \text{ and }\quad \frac{a^2}{\alpha}\leq e \leq \alpha a^2. NEWLINE\]NEWLINENEWLINENEWLINETheorem 2. Under the same assumptions there are positive numbers \(T_0, \sigma\) such that for every \(0<T<T_0\) there is a periodic orbit of period \(T\) such that its action \(a\) and energy \(e\) satisfy NEWLINE\[NEWLINE \frac{1}{\sigma T\leq a} \leq \frac{\sigma}{T}\quad \text{ and }\quad \frac{1}{\sigma T^2}\leq e \leq \frac{\sigma}{T^2}. NEWLINE\]NEWLINE Note that existence of periodic orbits of every period follows from Theorem 2 by taking iterations.NEWLINENEWLINEA minimax type argument is used in the proof.