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We investigate linear dynamical systems of second order. Uncertaintyquantification is applied, where physical parameters are substituted by randomvariables. A stochastic Galerkin method yields a linear dynamical system ofsecond order with high dimensionality. A structure-preserv\-ing model orderreduction (MOR) produces a small linear dynamical system of second order again.We arrange an associated port-Hamiltonian (pH) formulation of first order forthe second-order systems. Each pH system implies a Hamiltonian functiondescribing an internal energy. We examine the properties of the Hamiltonianfunction for the stochastic Galerkin systems. We show numerical results using atest example, where both the stochastic Galerkin method andstructure-preserving MOR are applied.

Stochastic Galerkin method and port-Hamiltonian form for linear dynamical systems of second order