Fundamental Principles of Lagrangian Dynamics: Mechanical Systems with Non-ideal, Holonomic, and Nonholonomic Constraints

This paper deals with the foundations of analytical dynamics. It obtains the explicit equations of motion for mechanical systems that are subjected to non-ideal holonomic and nonholonomic equality constraints. It provides an easy incorporation of such non-ideal constraints into the framework of Lagrangian dynamics. It bases its approach on a fundamental principle that includes non-ideal constraints and that reduces to D’Alembert’s Principle in the special case when all the constraints become ideal. Based on this, the problem of determining the equations of motion for the constrained system is reformulated as a constrained minimization problem. This yields a new fundamental minimum principle of analytical dynamics that reduces to Gauss’s Principle when the constraints become ideal. The solution of this minimization problem then yields the explicit equations of motion for systems with non-ideal constraints. An illustrative example showing the use of this general equation for a system with sliding Coulomb friction is given