The Routh theorem for mechanical systems with unknown first integrals

The First Mean Value Theorem for Integrals

For simplicity, we use the following convention: X is a non empty set, S is a σ-field of subsets of X, M is a σ-measure on S, f , g are partial functions from X to R, and E is an element of S. One can prove the following three propositions: (1) If for every element x of X such that x ∈ dom f holds f(x) ≤ g(x), then g − f is non-negative. (2) For every set Y and for every partial function f from...

The Fundamental Theorem on Quadratic First Integrals.

Stabilization of positive systems with first integrals

Positive systems possessing %rst integrals are considered. These systems frequently occur in applications. This paper is devoted to two stabilization problems. The %rst is concerned with the design of feedbacks to stabilize a given level set. Secondly, it is shown that the same feedback allows to globally stabilize an equilibrium point if it is asymptotically stable with respect to initial cond...

A mean value theorem for systems of integrals

Abstract. More than a century ago, G. Kowalewski stated that for each n continuous functions on a compact interval [a, b], there exists an n-point quadrature rule (with respect to Lebesgue measure on [a, b]), which is exact for given functions. Here we generalize this result to continuous functions with an arbitrary positive and finite measure on an arbitrary interval. The proof relies on a ver...

Determining Liouvillian first integrals for dynamical systems in the plane

Here we present/implement an algorithm to find Liouvillian first integrals of dynamical systems in the plane. In [1], we have introduced the basis for the present implementation. The particular form of such systems allows reducing it to a single rational first order ordinary differential equation (rational first order ODE). We present a set of software routines in Maple 10 for solving rational ...