- A straighfoward computation shows that \(\varphi_t(q,p)=(q-2pt,p)\). Therefore the flow is 1-periodic if and only if \(2p=2k\pi, k\in \ZZ\), i.e. \(p=k\pi\).
- At the zero-section \(S^n\subset T^*S^n\) we have \(X_H\restr{S^n}=0\). Consider therefore a {\it perturbed} Hamiltonian \(\tilde H=H+\epsilon h\), where \(h(q,p):=\eta(p)\sin q\) with \(\eta:\RR\to \RR\) a cut-off function centered at \(p=0\). Note that \(\tilde H=H\) whenever \(\eta=0\), so outside a neighborhood of the zero-section we have a bijection between orbits of \(\tilde H\) and orbits of \(H\), all of which are non-constant. Furthermore, we have \(X_{\tilde H}=(-2p+\epsilon\eta(p)\cos q)\partial_q +\epsilon\eta'(p)\sin q\partial_p\) from which we deduce that, for \(\epsilon\) small enough, the only constant orbits occur when \(p=0\) and \(\cos q=0\), a total of two points. It remains to argue that there cannot be other non-constant orbits in this region. This follows from the fact that we can make \(|X_{\tilde H}|\) as small as we want --- again, by choosing suitably small \(\epsilon\) and cut-off width --- so that any periodic orbit must necessarily have period \(T>>1\).
- We will use the time-dependent Hamiltonian \(\tilde H_t\) to compute the symplectic cohomology \(SH^\bullet(T^*S^1)\). Note \(\tilde H_t\) has:
- Two constant orbits ocurrying at \((q,p)=\left(\pm \frac{\pi}{2},0\right)\), call them \(\gamma^0_{max,min}\);
- Two non-constant orbits \(\gamma_{max,min}^k\) for each \(k\in \ZZ\setminus\set{0}\) ocurrying near \(p_k=k\pi\).

- We first note that \(\pi_0(LS^1)=\ZZ\), where \(k\in \ZZ\) distinguishes homotopy classes. Let \(LS^1(k)\subset LS^1\) be the connected component consisting of loops of homotopy class equal to \(k\in\pi_1(S^1)\cong\ZZ\). Then we have a splitting \[ H_*(LS^1)\cong\oplus_{k\in \ZZ}H_*(LS^1(k))\] There is an obvious inclusion \(S^1\into LS^1(k), p\mapsto (t\mapsto pe^{i2\pi k t})\), and one can see the map \(LS^1(k)\to S^1\) that records the base-point of the loop is a homotopy inverse. Therefore \(S^1\simeq LS^1(k)\) have the same homology, and we conclude \[ H_*(LS^1)=\oplus_{k\in \ZZ} \RR^2\cong \RR[x]\oplus\RR[x]\]
- Each component \(LS^1(k)\) contributes two generators to \(H_*(LS^1)\). Consider a generator of \(H_*(LS^1)\) coming from \(H_0(LS^1(k))\), which is just a point \(\gamma\in LS^1(k)\). Then there is an obvious 1-cycle \(\Gamma\in C_1(LS^1(k)), \partial\Gamma=0\) that we can build, namely \(\Gamma(s)=\gamma(\cdot+s)\), the 'rotation' of the loop \(\gamma\). Note that recording the basepoint \(\Gamma(s)(0)=\gamma(s)\) of each rotated loop we obtain the fundamental cycle of \(S^1\), so according to our identification \(S^1\simeq LS^1(k)\) the 1-cycle \(\Gamma\) is precisely the degree one generator of \(H_*(LS^1(k))\).