Equations of motion are not all independent, because coordinates are no longer all independent 2.
Classical Mechanics Goldstein Solution Download View SolutionDownload View solution Manual Classical Mechanics, Goldstein.pdf as PDF for free.Goldstein Classical Mechanics Notes Michael Good May 30, 2004 1 Chapter 1: Elementary Principles 1.1 Mechanics of a Single Particle Classical mechanics incorporates special relativity. Classical refers to the contradistinction to quantum mechanics. Velocity: v dr. dt Linear momentum: p mv. Force: dp. dt In most cases, mass is constant and force is simplified: F F d dv (mv) m ma. Torque: T r F. Torque is the time derivative of angular momentum: 1 dL. T A force is considered conservative if the work is the same for any physically possible path. The capacity to do work that a body or system has by viture of is position is called its potential energy. Work is now V1 V2. The change is -V. Energy Conservation Theorem for a Particle: If forces acting on a particle are conservative, then the total energy of the particle, T V, is conserved. The Conservation Theorem for the Linear Momentum of a Particle states that linear momentum, p, is conserved if the total force F, is zero. The Conservation Theorem for the Angular Momentum of a Particle states that angular momentum, L, is conserved if the total torque T, is zero. Mechanics of Many Particles Newtons third law of motion, equal and opposite forces, does not hold for all forces. R M mi Center of mass moves as if the total external force were acting on the entire mass of the system concentrated at the center of mass. Internal forces that obey Newtons third law, have no effect on the motion of the center of mass. The strong law of action and reaction is the condition that the internal forces between two particles, in addition to being equal and opposite, also lie along the line joining the particles. Then the time derivative of angular momentum is the total external torque: dL N(e). Conservation Theorem for Total Angular Momentum: L is constant in time if the applied torque is zero. Linear Momentum Conservation requires weak law of action and reaction. Angular Momentum Conservation requires strong law of action and reaction. If the center of mass is at rest wrt the origin then the angular momentum is independent of the point of reference. Total Work: W12 T2 T1 where T is the total kinetic energy of the system: T Total kinetic energy: T 1 2 P i mi vi2. X 1 1X mi vi2 M v 2 mi vi02. Kinetic energy, like angular momentum, has two parts: the K.E. K.E. of motion about the center of mass. The term on the right is called the internal potential energy. For rigid bodies the internal potential energy will be constant. For a rigid body the internal forces do no work and the internal potential energy remains constant. ![]()
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