Lagrangian Dynamics

Abstract

The variational form of the equations of dynamics is derived from its vector counterpart (Newton’s law) through the principle of virtual work, extended to dynamics thanks to d’Alembert principle. This leads to Hamilton’s principle and the Lagrange equations for discrete systems; a set of examples illustrate special features such as gyroscopic effects. The Lagrange equations are extended to systems with kinematic constraints and conservation laws are established in special conditions (Jacobi integral, conservation of generalized momentum). This chapter includes the treatment of the prestresses using the Green strain tensor to measure the deformation in prestressed flexible bodies. The geometric stiffness is discussed and its relation to buckling is established and illustrated by mechanisms aimed at introducing a negative stiffness (such mechanisms are used in vibration isolation of precision structures). The chapter ends with a set of problems.

Le bon sens est la chose la mieux

répartie au monde, parce que nul

ne se plaint de ne pas en avoir

assez.

Descartes, Discours de la

Méthode (1637)