![]() In the variational formulation the equations of motion are formulated in terms of the difference of the kinetic energy and the potential energy. The Newtonian formulation of the equations of motion is intrinsically a particle-by-particle description. The motion of the system is determined by considering how the individual component particles respond to these forces. In the Newtonian formulation the forces can often be written as derivatives of the potential energy of the system. In contrast to the Newtonian formulation of mechanics, the variational formulation of mechanics describes the motion of a system in terms of aggregate quantities that are associated with the motion of the system as a whole. Mechanics, as invented by Newton and others of his era, describes the motion of a system in terms of the positions, velocities, and accelerations of each of the particles in the system. ![]() 1 For a great variety of systems realizable motions of the system can be formulated in terms of a variational principle. This is the variational strategy: for each physical system we invent a path-distinguishing function that distinguishes realizable motions of the system by having a stationary point for each realizable path. ![]() However, there is an alternate strategy that provides more insight and power: we could look for a path-distinguishing function that has a minimum on the realizable paths-on nearby unrealizable paths the value of the function is higher than it is on the realizable path. Newton's equations of motion are of this form at each moment Newton's differential equations must be satisfied. For example, the output could be a number, and we could try to arrange that this number be zero only on realizable paths. We want this function to have some characteristic behavior when its input is a realizable path. The path-distinguishing function that we seek takes a configuration path as an input and produces some output. The motion of the juggling pin is specified by giving the position and orientation of the pin as a function of time. The juggling pin rotates as it flies through the air the configuration of the juggling pin is specified by giving the position and orientation of the pin. Such a description of the motion of the system is called a configuration path the configuration path specifies the configuration as a function of time. The motion of a system can be described by giving the position of every piece of the system at each moment. How can we distinguish motions of a system that can actually occur from other conceivable motions? Perhaps we can invent some mathematical function that allows us to distinguish realizable motions from among all conceivable motions. We can imagine that the juggling pin might pause in midair or go fourteen times around the head of the juggler before being caught, but these motions do not happen. There are many conceivable ways a system could move that never occur. Complex physical objects, such as juggling pins, can be modeled as myriad particles with fixed spatial relationships maintained by stiff forces of interaction. It is also a remarkable discovery that the same mathematical tools used to describe the motions of the planets can be used to describe the motion of the juggling pin.Ĭlassical mechanics describes the motion of a system of particles, subject to forces describing their interactions. In fact, the skill of juggling depends crucially on this predictability. That mathematics can be used to describe natural phenomena is a remarkable fact.Ī pin thrown by a juggler takes a rather predictable path and rotates in a rather predictable way. The effort to formulate these regularities and ultimately to understand them led to the development of mathematics and to the discovery that mathematics could be effectively used to describe aspects of the physical world. The subject of this book is motion and the mathematical tools used to describe it.Ĭenturies of careful observations of the motions of the planets revealed regularities in those motions, allowing accurate predictions of phenomena such as eclipses and conjunctions. To formulate the aims of mechanics in this way, without serious ![]() Position in space with “time.” I should load myĬonscience with grave sins against the sacred spirit of lucidity were I The purpose of mechanics is to describe how bodies change their Structure and Interpretation of Classical Mechanics: Lagrangian Mechanics ⇡
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |