Hamiltonian evolution of curves in classical affine geometries
The author of this very interesting paper studies geometric Poisson brackets for the manifold \(M=(G\times \mathbb R^n)/G\cong \mathbb R^n\) endowed with an affine geometry (in Klein sense) determined by the affine action of \(G\times \mathbb R^n\) on \(\mathbb R^n\) given by the equality \((g,v)\cdot u=gu+v\), \(G\subset\text{GL}(n, \mathbb R)\) is a semisimple group. There is a natural Hamiltonian structure \(\{\;,\;\}({\mathbf k})\) associated with \(M\) and defined on the space of differential invariants of parametrized curves with a monodromy in the group. There exist curves \(u(x)\in \mathbb R^n\) such that \(u(x+T)=m u(x)\) for all \(x\), where \(m\in G\) and \(T\in \mathbb R\) is the period. These curves have periodic differential invariants, such that other conditions can be imposed to ensure that the coefficients vanish at infinity. The first result is that the evolution of curves on the manifold induces a Hamiltonian evolution on the invariants of the flow. Consider \(G\subset\text{GL}(n)\), that is a classical Lie group, and \(u_t=F(u_x,u_{xx},\dots)\) the evolution of \(u(t,x)\in \mathbb R^n\) which is invariant under the affine action \(G\times \mathbb R^n\). This evolution induces another evolution on its differential invariants. If \(G\neq\text{GL}(n,\mathbb R)\), and if the evolution induced on the invariants is Hamiltonian with respect to the reduced geometric bracket, then \(u_t=F(u_x,u_{xx},\dots)\) is arc-length preserving. The second result is that for \(G\neq\text{GL}(n,\mathbb R)\) geometric Poisson brackets are trivial extensions of Poisson brackets for the unparametrized case. That is, it vanishes on functionals that depend only on the parameter of arc-length type and can be restricted to the submanifold where the invariant of arc-length type is constant.
scientific article; zbMATH DE number 5506318
scientific article