exercise 0.0.1
Consider the space \(T^*S^1\), with coordinates \((q, p)\).- Determine all of the time-1 periodic orbits of the Hamiltonian \(H=|p|^2\).
- Construct a Hamiltonian \(\tilde H\) with the property that
- The non-constant orbits of \(H\) are in bijection with the non-constant orbits of \(\tilde H\) and;
- \(\tilde H\) has only 2 constant orbits.
- Let \(TI(\tilde H)/S^1\) denote the set of time independent orbits of \(\tilde H\) up to reparameterization. It is shown in [BO09] that there exists a time dependent Hamiltonian \(\tilde H_t: T^*S^1\to \RR\) whose
- constant orbits are in bijection with the constant orbits of \(\tilde H\) and;
- whose non-constant orbits are of the form \(\{\gamma_{\min}\}_{\gamma\in TI(\tilde H)}\cup \{\gamma_{\max}\}_{\gamma\in TI(\tilde H)}\).
- Compute \(H_*(LS^1)\), the homology of the loop space of \(S^1\).
- Justify why the generators of \(H_*(LS^1)\) appear in pairs (just like they do \(\SH(T^*S^1)\)).
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exercise 0.0.2
Let \((Q, g)\) be a Riemannian manifold.- Describe a metric on \( g^*:T^*_qQ\to T^*_qQ\) on the cotangent bundle. In local coordinates \((q, p)\) for \(T^*Q\), write down the Hamiltonian vector field \(V_g\) associated to the Hamiltonian \(H(q, p):=g_q^*(p, p): T^*_qQ\to \RR\).
- Recall that a geodesic on \(Q\) is a curve \(\gamma:\RR\to Q\) which is locally length minimizing. In particular, it is a minimizer for the action \[E(\gamma)=\frac{1}{2}\int_a^b g_{\gamma(t)}(\dot \gamma(t), \dot \gamma(t)) dt\] for every \(a,b\in \RR\). Prove that if \(\hat \gamma: \RR\to T^*Q\) is a flow-line of \(V_g\), that \(\gamma:=\pi_Q(\hat \gamma)\) is a geodesic.
- Let \(n>1\). Use the Serre spectral sequence for the fibration \[\Omega S^n\to PS^n\to S^n\] to conclude that \(H_*(\Omega S^n)\simeq \ZZ[x]\). Here, \(\Omega M\) is the based loop space of \(M\), and \(PM\) is the based path space of \(M\). The element \(x\) is of degree \(n-1\) and is determined by \[H_n(S^n)\to \pi_n(S^n)\to \pi_{n-1}(\Omega S^n)\to H_{n-1}(\Omega S^n).\]
- Let \(n>1\). Using the Serre spectral sequence for the fibration \[\Omega S^n\to LS^n\to S^n\] show that as a vector space \(H_*(LS^n)\neq 0, \ZZ/n\ZZ\) or \(\ZZ^2\). Here, \(LM\) is the free loop space of \(L\).
- Show that if there exists a metric \(g\) for \(S^n\) which has no closed geodesics, that there exists a Hamiltonian \(H: S^n\to \RR\) which has no non-constant orbits, and two constant orbits. Conclude that every metric on \(S^n\) has at least one closed geodesic.
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References
[BO09] | Frédéric Bourgeois and Alexandru Oancea. Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces. Duke mathematical journal, 146(1):71--174, 2009. |