Non-holonomic Systems in View of Hamiltonian Principle

The aim of the paper is to outline some important attributes of non-holonomic systems, which appear in dynamics of deformable systems interacting with neighborhood. The paper is oriented to theoretical way of investigation. Its core consists in characterization of basic and generalized non-holonomic systems inspired by civil and mechanical engineering, but coming also frequently from other disciplines. Definition of a dynamic system consists of specification of the system itself and relevant constraints representing links with surrounding environment. The governing differential system itself is deduced from a definition based on the Hamiltonian principle. A new form of the generalized Lagrange equation system is derived assuming higher time derivatives of displacement components in the kinetic energy definition, as they emerge due to interaction of mechanical and other physical fields. Linear and nonlinear definitions of non-holonomic constraints including arbitrary time derivative order, which originate from interaction of mechanical and other physical fields are discussed. Consequently, the constraints can be of a very general character; they include many variants from a simple geometric coupling with fixed points and interaction with the movement trajectory to a soft relation to surrounding area via complicated time-dependent constraints of deterministic or random types. Lagrangian multiplier techniques are employed incorporating the non-holonomic constraints of simple or higher order into the complete mathematical model. Comparison with corresponding equation systems obtained by means of the virtual works principle is done. Several particular mathematical models deduced by this conventional way including classical Lagrangian equation system are cited and interpreted in view of the new model following from the Hamiltonian principle. Strengths and shortcomings of both procedures are evaluated and domains of the new approach preference are outlined. Four illustrating examples are included to demonstrate the large variety of dynamic systems. Relation to some branches beyond classical definition of dynamics are mentioned in order to demonstrate the general character of the theoretical background discussed and its applicability in domains apparently far from mechanical or civil engineering.