Lagrangian and Hamiltonian Mechanics （拉格朗日和哈密顿力学）.pdfVIP

Lagrangian and Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences and Engineering Prince George’s Community College December 5, 2007 Introduction In this course we have been studying classical mechanics as formulated by Sir Isaac Newton; this is called Newtonian mechanics . Newtonian mechanics is mathematically fairly straightforward, and can be applied to a wide variety of problems. It is not a unique formulation of mechanics, however; other formulations are possible. Here we will look at two common alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics. It is important to understand that all of these formulations of mechanics equivalent. In principle, any of them could be used to solve any problem in classical mechanics. The reason they’re important is that in some problems one of the alternative formulations of mechanics may lead to equations that are much easier to solve than the equations that arise from Newtonian mechanics. Unlike Newtonian mechanics, neither Lagrangian nor Hamiltonian mechanics requires the concept of force ; instead, these systems are expressed in terms of energy. Although we will be looking at the equations of mechanics in one dimension, all these formulations of mechanics may be generalized to two or three dimensions. Newtonian Mechanics We begin by reviewing Newtonian mechanics in one dimension. In this formulation, we begin by writing Newton’s second law, which gives the force F required to give an acceleration a to a mass m: F D ma: (1) Generally the force is a function of x . Since the acceleration a D d 2x=dt 2 , Eq. (