Symmetries and constraints in mechanics of continua

TL;DR

This paper explores longstanding and complex issues related to constraints in continuum mechanics, highlighting conceptual confusions and the need for clearer understanding of various constrained systems and their differences.

Contribution

The paper provides a critical analysis of the conceptual and theoretical challenges in the mechanics of continua, emphasizing the distinctions between different types of constraints and their implications.

Findings

Identifies confusions between field theory and continuum mechanics.

Highlights differences between holonomic and nonholonomic constraints.

Calls for conceptual clarification in the theory of constrained continua.

Abstract

In spite of its long history and classical character which goes back even to d'Alembert and Lagrange, the problems of constraints in mechanics of continua is still mysterious and full of misunderstandings. Let us mention the problem of difference between special solutions of unconstrained systems and constraints dynamics, holonomic and nonholonomic constrained, various versions of so-called vakonomy, sub-Riemannian structures, servoconstraints, and programme motion. There are some strange confusions between field theory and mechanics of continua; many problems call for conceptual cleaning, although both the mechanics of continua and field theory are formally based on partial differential equations. The author is deeply indebted to professor Czes\law Wo\'{z}niak for his help during many years, and deep understanding and support.