This thesis is centred around higher-order invariant variational problems defined on Lie groups. We are mainly motivated by applications in computational anatomy and quantum control, but the general framework is relevant in many other contexts as well. We first develop a higher-order analog of Euler–Poincaré reduction theory for variational problems with symmetry and discuss the important examp...

We study Euler–Poincaré systems (i.e., the Lagrangian analogue of LiePoisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d’Alembert type. Then we derive an abstract Kelvin-Noether theorem for these equations. We also explore their rela...

Abstract Isospectral flows appear in a variety of applications, e.g. the Toda lattice solid state physics or discrete models for two-dimensional hydrodynamics. The isospectral property thereby often corresponds to mathematically physically important conservation laws. most prominent feature these systems, i.e. eigenvalues matrix variable, should therefore be retained when discretizing systems. ...

We introduce the concept of m-shifted symplectic Lie n-groupoids and Morita equivalences between them. then build various models for 2-shifted structure on classifying stack in this setting construct explicit