classical notes - Goldstein Classical Mechanics Notes...

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Unformatted text preview: 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 con- tradistinction to ‘quantum’ mechanics. Velocity: v = d r dt . Linear momentum: p = m v . Force: F = d p dt . In most cases, mass is constant and force is simplified: F = d dt ( m v ) = m d v dt = m a . Acceleration: a = d 2 r dt 2 . Newton’s second law of motion holds in a reference frame that is inertial or Galilean. Angular Momentum: L = r × p . Torque: T = r × F . Torque is the time derivative of angular momentum: 1 T = d L dt . Work: W 12 = Z 2 1 F · d r . In most cases, mass is constant and work simplifies to: W 12 = m Z 2 1 d v dt · v dt = m Z 2 1 v · d v dt dt = m Z 2 1 v · d v W 12 = m 2 ( v 2 2- v 2 1 ) = T 2- T 1 Kinetic Energy: T = mv 2 2 The work is the change in kinetic energy. A force is considered conservative if the work is the same for any physically possible path. Independence of W 12 on the particular path implies that the work done around a closed ciruit is zero: I F · d r = 0 If friction is present, a system is non-conservative. Potential Energy: F =-∇ V ( r ) . The capacity to do work that a body or system has by viture of is position is called its potential energy. V above is the potential energy. To express work in a way that is independent of the path taken, a change in a quantity that depends on only the end points is needed. This quantity is potential energy. Work is now V 1- V 2 . 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. 2 1.2 Mechanics of Many Particles Newton’s third law of motion, equal and opposite forces, does not hold for all forces. It is called the weak law of action and reaction....
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classical notes - Goldstein Classical Mechanics Notes...

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