Equations of motion for general nonholonomic systems from the d'Alembert principle via an algebraic method

TL;DR

This paper introduces an algebraic method to derive equations of motion for general nonlinear nonholonomic systems from the d'Alembert principle, unifying classical approaches without integrations.

Contribution

It presents an algebraic procedure that treats Cetaev and vakonomic methods equally, avoiding integrations and clarifying their relationship.

Findings

Algebraic derivation of nonholonomic equations of motion

Comparison of Cetaev and vakonomic methods

Discussion on transpositional relation and method compatibility

Abstract

The aim of this study is to present an alternative way to deduce the equations of motion of general (i.e., also nonlinear) nonholonomic constrained systems starting from the d'Alembert principle and proceeding by an algebraic procedure. The two classical approaches in nonholonomic mechanics -- Cetaev method and vakonomic method -- are treated on equal terms, avoiding integrations or other steps outside algebraic operations. In the second part of the work we compare our results with the standard forms of the equations of motion associated to the two method and we discuss the role of the transpositional relation and of the commutation rule within the question of equivalence and compatibility of the Cetaev and vakonomic methods for general nonholonomic systems.