Lagrangian formulation pdf


Lagrangian Formulation •That’s the energy formulation – now onto the Lagrangian formulation. •This is a formulation. It gives no new information – there’s no advantage to it. •But, easier than dealing with forces: • “generalized coordinates” – works with any convenient coordinates, don’t have to set up a coordinate system 1. The Lagrangian formulation 2. Lagrangian systems 3. Hamilton’s principle (also called the least action principle) 4. The Hamiltonian formalism 5. The Hamilton-Jacobi formalism 6. Integrable systems 7. Quasi-integrable systems 8. From order to chaos In each chapter, the reader will find: • A clear, succinct and rather deep summary of all ... The focus of Chapter 8 is primarily on the Newton-Euler formulation, because it uses some of the geometric tools we have already developed, and it results in an efficient recursive algorithm for calculating the inverse dynamics. In this video, though, we start with the Lagrangian formulation, due to its conceptual simplicity. Lagrange’s equations rather than Newton’s. The first is that Lagrange’s equations hold in any coordinate system, while Newton’s are restricted to an inertial frame. The second is the ease with which we can deal with constraints in the Lagrangian system. We’ll look at these two aspects in the next two subsections. –12– coordinate formulation, the Eulerian displacements can be used to directly compute strain and avoid drift away from the object’s rest configuration. We pause here to offer a second derivation which demonstrates the general nature of the method. 3.2 Linear Modal Eulerian-on-Lagrangian Simulation Lagrange’s equations rather than Newton’s. The first is that Lagrange’s equations hold in any coordinate system, while Newton’s are restricted to an inertial frame. The second is the ease with which we can deal with constraints in the Lagrangian system. We’ll look at these two aspects in the next two subsections. –12– Chapter 4. Lagrangian Dynamics (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7) 4.1 Important Notes on Notation In this chapter, unless otherwise stated, the following notation conventions will be used: 1. Einstein’s summation convention. Whenever an index appears twice (an only