Physics Notes - Herong's Tutorial Notes

∟Introduction of Generalized Coordinates

∟Hamilton's Principle in Generalized Coordinates

This section provides a quick introduction to the Hamilton's Principle in Generalized Coordinates.

The nice thing about the Hamilton's Principle is that it it independent of coordinate systems. So the Hamilton's Principle stay the same: the true position function q(t) of a system is a stationary point of the Action S[q(t)] functional.

Mathematically, a stationary point of Action S[q(t)] functional is a function q(t), where the functional differential of S[q(t)] is zero. This can be expressed as: 

d(S[q(t)]) / d(q(t)) = 0
  # functional differential of S[q(t)]

or:
  d(∫ L(q,q',t)dt) / dq = 0
    # Replaced S with its definition

Hamilton's Principle is also called Stationary Action.

Table of Contents

 About This Book

 Introduction of Space

 Introduction of Frame of Reference

 Introduction of Time

 Introduction of Speed

 Newton's Laws of Motion

 Introduction of Special Relativity

 Time Dilation in Special Relativity

 Length Contraction in Special Relativity

 The Relativity of Simultaneity

 Introduction of Spacetime

 Minkowski Spacetime and Diagrams

 Introduction of Hamiltonian

 Introduction of Lagrangian

►Introduction of Generalized Coordinates

 Generalized Coordinates and Generalized Velocity

 Simple Pendulum Motion in Generalized Coordinates

►Hamilton's Principle in Generalized Coordinates

 Lagrange Equations in Generalized Coordinates

 Lagrange Equations on Simple Pendulum

 What Is Generalized Momentum

 What Is Legendre Transformation

 Hamilton Equations in Generalized Coordinates

 Phase Space and Phase Portrait

 References

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