Generalized Lagrange’s equations for systems with general constraints and distributed parameters

Abstract

This paper presents a general explicit differential form of Lagrange’s equations for systems with hybrid coordinates and general holonomic and nonholonomic constraints. The appropriate constraint conditions are imposed on d’Alembert–Lagrange principle via Lagrange multipliers to find the correct equations of state of nonholonomic systems. The developed equations take the differential form in terms of the Lagrangian, accounting for the masses and springs that may exist at the boundary. The Lagrangian of hybrid-coordinate systems consists of three parts: the first is due to the rigid-body motion, the second is due to the boundary elements, and the third is the Lagrangian density due to the elastic elements. The developed Lagrange’s equations produce the equations of state and the boundary conditions without the need to carry out the integration by parts generally one needs to do using Hamilton’s principle. Illustrative examples are introduced to show the effectiveness of the developed equations.