Implicit Hamiltonian systems with symmetry
The classical topic is treated here by exploiting some new and rather promising technical tools. Let \(X\) be a manifold, \(D\subset TX+T^*X\) a vector subbundle, \(D^\perp\subset TX+T^*X\) the subbundle of all \((X,\alpha)\) satisfying \(\langle\alpha\mid\widehat X\rangle+\langle \widehat\alpha\mid X\rangle=0\) for all \((\widehat X,\widehat\alpha)\in D\). Then \(D\) is called a Dirac structure if \(D=D^\perp\) and \(\langle L_{X_1}\alpha_2\mid X_3\rangle+ \langle L_{X_2} \alpha_3\mid X_1\rangle+ \langle L_{X_3}\alpha_1\mid X_2\rangle=0\) for all \((X_i,\alpha_i)\in D\). The author defines an implicit Hamiltonian system by the inclusion \((\dot x,dH)\in D\) where \(H\in C^\infty(X)\) is a given function (the Hamiltonian). The inclusion can be represented by a mixed set of differential and algebraic equations \(F(x,\dot x)=0\). The infinitesimal symmetries \(f\in TX\) are defined by \(L_fD\subset D\). The author shows that the symmetries permit to reduce \(X\) to a quotient manifold \(\widetilde X\) equipped with a reduced Dirac structure \(\widetilde D\).
scientific article; zbMATH DE number 1233896
scientific article