Colloquium

Colloquium
岩井敏洋 The geometry and the mechanics of generalized pseudo-rigid bodies 7月17日(金) 13時00分 The configuration space of a generalized pseudo-rigid bodies is the linear group, $\mathcal{P}:={\rm GL}^+(n,{\bf R})$, of non-singular matrices with positive determinant. It admits the two-sided action of $\SO{n}\times \SO{n}$, which is not free on $\mathcal{P}$, and hence $\mathcal{P}$ is not made into a principal fiber bundle with respect to this action. In spite of this, if $\mathcal{P}$ is restricted to an open dense subspace $\dot{\mathcal{P}}$, the isotropy subgroup at each point of $\dot{\mathcal{P}}$ is a finite discrete group, so that the quotient space $(\SO{n}\times \SO{n})\backslash \dot{\mathcal{P}}$ becomes a manifold, and further one can define a connection on $\dot{\mathcal{P}}$, which will be called a bi-connection. The bi-connection is used to reduce the pseudo-rigid body system on $T^*\dot{\mathcal{P}}$ with the $\SO{n}\times \SO{n}$ symmetry. As an application of the reduction procedure, relative equilibria are discussed in relation with the reduced Hamilton equations of motion. A necessary and sufficient condition is given for a relative equilibrium in terms of an amended potential on the reduced phase space.

岩井敏洋

The geometry and the mechanics of generalized pseudo-rigid bodies

7月17日(金) 13時00分

The configuration space of a generalized pseudo-rigid bodies is the linear group, $\mathcal{P}:={\rm GL}^+(n,{\bf R})$, of non-singular matrices with positive determinant. It admits the two-sided action of $\SO{n}\times \SO{n}$, which is not free on $\mathcal{P}$, and hence $\mathcal{P}$ is not made into a principal fiber bundle with respect to this action. In spite of this, if $\mathcal{P}$ is restricted to an open dense subspace $\dot{\mathcal{P}}$, the isotropy subgroup at each point of $\dot{\mathcal{P}}$ is a finite discrete group, so that the quotient space $(\SO{n}\times \SO{n})\backslash \dot{\mathcal{P}}$ becomes a manifold, and further one can define a connection on $\dot{\mathcal{P}}$, which will be called a bi-connection. The bi-connection is used to reduce the pseudo-rigid body system on $T^*\dot{\mathcal{P}}$ with the $\SO{n}\times \SO{n}$ symmetry. As an application of the reduction procedure, relative equilibria are discussed in relation with the reduced Hamilton equations of motion. A necessary and sufficient condition is given for a relative equilibrium in terms of an amended potential on the reduced phase space.