Invariants at fixed and arbitrary energy. A unified geometric approach

Invariants at arbitrary and fixed energy (strongly and weakly conserved quantities) for 2-dimensional Hamiltonian systems are treated in a unified way. This is achieved by utilizing the Jacobi metric geometrization of the dynamics. Using Killing tensors we obtain an integrability condition for quadratic invariants which involves an arbitrary analytic function $S(z)$. For invariants at arbitrary energy the function $S(z)$ is a second degree polynomial with real second derivative. The integrability condition then reduces to Darboux's condition for quadratic invariants at arbitrary energy. The four types of classical quadratic invariants for positive definite 2-dimensional Hamiltonians are shown to correspond to certain conformal transformations. We derive the explicit relation between invariants in the physical and Jacobi time gauges. In this way knowledge about the invariant in the physical time gauge enables one to directly write down the components of the corresponding Killing tensor for the Jacobi metric. We also discuss the possibility of searching for linear and quadratic invariants at fixed energy and its connection to the problem of the third integral in galactic dynamics. In our approach linear and quadratic invariants at fixed energy can be found by solving a linear ordinary differential equation of the first or second degree respectively.Comment: Some misprints corrected with respect to the printed versio

Hierarchy of general invariants for bivariate LPDOs

We study invariants under gauge transformations of linear partial differential operators on two variables. Using results of BK-factorization, we construct hierarchy of general invariants for operators of an arbitrary order. Properties of general invariants are studied and some examples are presented. We also show that classical Laplace invariants correspond to some particular cases of general invariants.Comment: to appear in J. "Theor.Math.Phys." in May 200

On the decidability of the existence of polyhedral invariants in transition systems

Automated program verification often proceeds by exhibiting inductive invariants entailing the desired properties.For numerical properties, a classical class of invariants is convex polyhedra: solution sets of system of linear (in)equalities.Forty years of research on convex polyhedral invariants have focused, on the one hand, on identifying "easier" subclasses, on the other hand on heuristics for finding general convex polyhedra.These heuristics are however not guaranteed to find polyhedral inductive invariants when they exist.To our best knowledge, the existence of polyhedral inductive invariants has never been proved to be undecidable.In this article, we show that the existence of convex polyhedral invariants is undecidable, even if there is only one control state in addition to the "bad" one.The question is still open if one is not allowed any nonlinear constraint

Counterexample-Guided Polynomial Loop Invariant Generation by Lagrange Interpolation

We apply multivariate Lagrange interpolation to synthesize polynomial quantitative loop invariants for probabilistic programs. We reduce the computation of an quantitative loop invariant to solving constraints over program variables and unknown coefficients. Lagrange interpolation allows us to find constraints with less unknown coefficients. Counterexample-guided refinement furthermore generates linear constraints that pinpoint the desired quantitative invariants. We evaluate our technique by several case studies with polynomial quantitative loop invariants in the experiments