Differential Mechanics

Author: Zhiming Ou

Publisher:    3265 Public Way

ISBN: 978-1-0677109-6-5

Summary

A mechanical system is a collection of particles that (1) have momentum, (2) occupy a region in space with or without a boundary, and (3) have a life span for existence. The particles may have mass or may be massless. Massless particles include oscillators, wave packets, phonons, magnetons, and some virtual particles like gravitons, strings.

Classically, there are 3 ways to describe a mechanical system: (1) by field theory, a field is a region in space that is distributed with energies and forces, (2) by action, which the time-integral of the Lagrangian function, an expression for the energy of the system, (3) by wavefunction, whose magnitude is the probability for the system to fall into a certain region. The results from these 3 descriptions are the same.

In this book, I describe the classical quantities in a mechanical system by their infinitely small parts, or the differential elements of the quantity. Only in this way, we can reveal the secrets of the universe, without relying on anybody’s postulates or axioms.

 

Content

Chapter 1 Stable States of an Object       3

  • 1 Rigid Body 3
  • 2 Gases 5
  • 3 Liquids 7
  • 4 Plasma and Condensate State 10

 

Chapter 2 Force Fields       12

  • 1 Fundamental Theorem of Vector Calculus 12
  • 2 Interaction between two masses 14
  • 3 Cavendish’s Experiment 20
  • 4 Potential Energy in a Gravitational Field 23
  • 5 Other types of forces: Elasticity, Tension, Friction, resistance 23

Chapter 3 The Action         29

  • 1 The Lagrangian 29
  • 2 Various Mechanical Systems 32

Systems with Uniform Time

Systems in Uniform Space

Inertial Systems

 

Chapter 4 Centered/Bounded Motions and Vibrations        37

  • 1 Centered Motion 37

Central force, area velocity, Orbit of the Particle

  • 2 Forces due to Rotation 43

Euler Force, Coriolis Force, Centrifugal Force

  • 3 Orbital angular Momentum and torque 47

 

Chapter 5 Fluids       53

  • 1 The continuity equation 53
  • 2 Stress Vector and stress tensor 56
  • 3 Cauchy Momentum Equation 62
  • 4 Navier-Stokes Equation 64

 

Chapter 6 Relativistic Motion       68

  • 1 Concept of Space and Time 68
  • 2 The Lorentz Transformation 73
  • 3 Relativistic Momentum and Energy 81