classical notes - Goldstein Classical Mechanics Notes...

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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
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T = d L dt . Work: W 12 = 2 1 F · d r . In most cases, mass is constant and work simplifies to: W 12 = m 2 1 d v dt · v dt = m 2 1 v · d v dt dt = m 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: 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
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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. Center of mass: R = m i r i m i = m i r i M .
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