Chapter 13 Lagrange Equations
Lagrange formalism constitutes the foundation of analytical mechanics, a term coined by Lagrange himself 1 . Lagrange method provides another tool to derive the same equations found by applying Newton-Euler formalism, and hence brings about nothing fundamentally new. Lagrange himself stated in the preface of his first edition
“There already exist several treatises on mechanics, but the purpose of this one is entirely new. I propose to condense the theory of this science and the method of solving the related problems to general formulas whose simple application produces all the necessary equations for the solution of each problem. I hope that my presentation achieves this purpose and leaves nothing lacking.”“In addition, this work will have another use. The various principles presently available will be assembled and presented from a single point of view in order to facilitate the solution of the problems of mechanics. Moreover, it will also show their interdependence and mutual dependence and will permit the evaluation of their validity and scope.”
However, contrary to the Newton-Euler formalism, Lagrange method takes a unified, global view of the system. The advantages of Lagrange equations are well known: their wide range and ease of applicability, the freedom of choice of coordinates to describe the system, the elimination of forces and moments of constraint. Hence, when the Newton-Euler formalism requires a careful and often difficult strategy to find the equations of motion so as to avoid unwanted equations, Lagrange method will yield these equations in an automatic and systematic way. The two formalisms are also different in an important way. Newton-Euler formalism is vectorial, while Lagrange formalism is scalar: the contribution of the inertial effects is found from the system’s kinetic energy while the contribution of the mechanical actions is expressed through their virtual power. It is also important to point out that Lagrange equations do not yield a solution for the complete analysis of a dynamic system. Their use is also problematic for kinematically constrained systems or for the determination of the unknown contact actions which take place through the system’s geometric joints.
To derive Lagrange equations, we take the same approach adopted to derive the Kinetic Energy Theorem: we choose a specific virtual velocity field and apply D’Alembert Principle of Virtual Power. In order to obtain as many equations as the body’s degrees of freedom, the kinematic screw of a rigid body is decomposed into components with respect to the body’s coordinates, known as the body’s partial kinematic screws. Then, two types of terms are obtained: the virtual power of volume and surface forces leads to power coefficients, whereas the virtual power of inertial forces is related to the body’s kinetic energy by Lagrange kinematic formula. The derivation of Lagrange equations is first done for a single body, then for a system of rigid bodies. A special case is considered to take into account the holonomic and non-holonomic constraint equations to arrive at Lagrange equations with multipliers. Finally, we derive Painlevé equation which can lead to a first integral under certain conditions.
