Vector fields whose flow preserves a symplectic form up to a constant, such as simple mechanical systems with friction, are called “conformal”. We develop a reduction theory for symmetric conformal Hamiltonian systems, analogous to symplectic reduction theory. This entire theory extends naturally to Poisson systems: given a symmetric conformal Poisson vector field, we show that it induces two r...

We define quantum bi-Hamiltonian systems, by analogy with the classical case, as derivations in operator algebras which are inner derivations with respect to two compatible associative structures. We find such structures by means of the associative version of Nijenhuis tensors. Explicit examples, e.g. for the harmonic oscillator, are given.

We study linear Hamiltonian systems using bilinear and quadratic differential forms. Such a representation-free approach allows to use the same concepts and techniques to deal with systems isolated from their environment and with systems subject to external influences, and allows to study systems described by higher-order differential equations, thus dispensing with the usual point of view in c...

Either from a control theoretic viewpoint or from an analysis viewpoint it is necessary to convert smooth systems to discrete systems, which can then be implemented on computers for numerical simulations. Discrete models can be obtained either by discretizing a smooth model, or by directly modeling at the discrete level itself. One of the goals of this paper is to model port-Hamiltonian systems...

where / is a smooth function. The basic theory of numerical methods for (1) has been known for more than thirty years, see e.g. [8]. This theory, in tandem with practical experimentation, has led to the development of general software packages for the efficient solution of (1). It is perhaps remarkable that both the theory and the packages do not take into account any structure the problem may ...

A big-isotropic structure E is an isotropic subbundle of T M ⊕ T * M , endowed with the metric defined by pairing. The structure E is said to be the explicit expression of X H and of the integrability conditions of E under the regularity condition dim(pr T * M E) = const. We show that the port-controlled, Hamiltonian systems (in particular, constrained mechanics) [1, 4] may be interpreted as we...

where H is the Hamiltonian (”energy”) and {. , .} is a Poisson bracket on an infinite dimensional phase space, called Poisson manifold. Unlike finite dimensional Hamiltonian systems, which are ordinary differential evolution equations on finite dimensional phase spaces, for which general existence and uniqueness theorems for solutions exist, this is not the case for PDEs. There are no general e...

The theory of differential forms began with a discovery of Poincaré who found conservation laws of a new type for Hamiltonian systems—The Integral Invariants. Even in the absence of non-trivial integrals of motion, there exist invariant differential forms: a symplectic two-form, or a contact one-form for geodesic flows. Some invariant forms can be naturally considered as “forms on the quotient....