Abstract
In this paper, we discuss the possibility of using computer algebra tools in the process of modeling and qualitative analysis of mechanical systems and problems from theoretical physics. We describe some constructions—Courant algebroids and Dirac structures—from the so-called generalized geometry. They prove to be a convenient language for studying the internal structure of the differential equations of port-Hamiltonian and implicit Lagrangian systems, which describe dissipative or coupled mechanical systems and systems with constraints, respectively. For both classes of systems, we formulate some open problems that can be solved using computer algebra tools and methods. We also recall the definitions of graded manifolds and Q‑structures from graded geometry. On particular examples, we explain how classical differential geometry is described in the framework of the graded formalism and what related computational questions can arise. This direction of research is apparently an almost unexplored branch of computer algebra.
Highlights
This paper is a part of a large project on “geometrization of mechanics,” which includes a series of works aimed at describing a formalism that is suitable for the qualitative analysis and modeling of a sufficiently wide class of mechanical systems
Recent publications demonstrate advantages of this approach: the consideration of conservation laws, system symmetries, and constraints imposed on the system improves accuracy of the numerical method and, the reliability of modeling results, including the reliable description of qualitative properties of the system
The second problem is more conceptual: it is aimed at facilitating the qualitative analysis of differential equation systems and avoiding “guessing” their correct form
Summary
This paper is a part of a large project on “geometrization of mechanics,” which includes a series of works aimed at describing a formalism (associated with differential or algebraic geometry) that is suitable for the qualitative analysis and modeling of a sufficiently wide class of mechanical systems. In this project, we investigate a broad spectrum of problems: from describing the mathematical model of a system under analysis and defining suitable geometric structures to selecting (or developing) numerical methods that preserve the constructed structures. Contraction of the vector field v with the form α(⋅,⋅, ...):
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