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 = dr dt. Linear momentum: p = mv. Force: F = dp dt. In most cases, mass is constant and force is simplified: F = d dt (mv) = m dv dt = ma. Acceleration: a = d2r dt2. 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. Work: W12 = ∫ 2 1 F dr. In most cases, mass is constant and work simplifies to: W12 = m ∫ 2 1 dv dt vdt = m ∫ 2 1 v dv dt dt = m ∫ 2 1 v dv W12 = m 2 (v22 − v21) = T2 − T1 Kinetic Energy: T = mv2 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 W12 on the particular path implies that the work done around a closed ciruit is zero:∮ F dr = 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 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.

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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.

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 = ∑ miri∑ mi = ∑ miri M. 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 Newton’s third law, have no effect on the motion of the center of mass. F(e) ≡M d 2R dt2 = ∑ i F(e)i. Motion of center of mass is unaffected. This is how rockets work in space.

Total linear momentum: P = ∑ i mi dri dt =M dR dt. Conservation Theorem for the Linear Momentum of a System of Particles: If the total external force is zero, the total linear momentum is conserved. 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 dt = N(e). Torque is also called the moment of the external force about the given point. 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. Total Angular Momentum: L = ∑ i ri × pi = R×Mv+ ∑ i r′i × p′i.

Total angular momentum about a point O is the angular momentum of mo- tion concentrated at the center of mass, plus the angular momentum of motion about the center of mass. 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 = 12 ∑ i miv 2 i. Total kinetic energy: T = 1 2 ∑ i miv 2 i = 1 2 Mv2 + 1 2 ∑ i miv ′2 i.

Kinetic energy, like angular momentum, has two parts: the K.E. Zoids Game For Android there. Obtained if all the mass were concentrated at the center of mass, plus the K.E.

Of motion about the center of mass. Program De Jocuri 3d there. Total potential energy: V = ∑ i Vi + 1 2 ∑ i,j i 6=j Vij.