Geodesics of Hofer's metric on the group of Hamiltonian diffeomorphisms (Q1345231)

Let \(\mathcal D\) be the Lie group of symplectomorphisms of the standard linear symplectic space \((\mathbb{R}^{2n}, \omega)\) which can be joined to the identity by a smooth path, i.e. an isotopy generated by a smooth compactly supported Hamiltonian. The length of a smooth path \(\ell\), defined as \(\int^ b_ a \|\ell(t)\| dt\) where \(\|\;\|\) is the norm on the Lie algebra of \(\mathcal D\), gives rise to a metric on \(\mathcal D\). It is shown that each point of \(\mathcal D\) has a flat \(C^ 1\)- neighbourhood, which allows a description of the geodesics on \(\mathcal D\). Applications to classical mechanics are given.