Symmetry breaking of infinite-dimensional dynamic system

The relationship between the symmetry breaking and the energy dissipation of dynamic systems is the foundation of the geometric mechanics, the investigation of which will establish a bridge between the structure-preserving theory and the assessment approach of the structure-preserving property for the employed numerical scheme. In this letter, two typical factors inducing the symmetry breaking for the infinite-dimensional dynamic system, including the symmetry breaking of the coefficient matrices and the space–time dependence of the Hamiltonian function, are investigated in detail. Based on the multi-symplectic theory, the local energy variations for dynamic systems with the mentioned symmetry breaking factors are deduced and the specific forms of which for a flexible cantilever with the variable bending rigidity under an external excitation are presented, which shows the local energy dissipation explicitly and provides the possibility of reproduction the local energy dissipation for the infinite-dimensional dynamic system in the numerical simulation.