Physics Notes - Herong's Tutorial Notes

∟Poisson Bracket Expression

∟Observables in Phase Space

This section describes observables in phase space in terms of the Poisson brackets.

If we represent an observable in a mechanical system as a function in the phase space, f(q_i, p_i, t), then we can express its time derivative in terms of Poisson brackets.

Equation (PB.16) tells us that the total time derivative of any observable, f, can be expressed as its partial time derivative plus the Poisson bracket of f with the Hamiltonian H.

Actually, Hamilton equations (PB.14, PB.15) are special cases of equation (PB.16). In other words, you derive equations (PB.14, PB.15) from (PB.16).

So equation (PB.16) is the only equation you need to remember for a mechanical system. Many text books refer (PB.16) as the Hamilton Equation.

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

 Phase Space and Phase Portrait

►Poisson Bracket Expression

 What Is Poisson Bracket

 Poisson Bracket and Hamilton Equations

►Observables in Phase Space

 Poisson Bracket in Portrait Space Transformation

 References

 Full Version in PDF/ePUB

Observables in Phase Space - Updated in 2024, by Herong Yang