Physics Notes - Herong's Tutorial Notes

∟Poisson Bracket Expression

∟Poisson Bracket in Portrait Space Transformation

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

The Poisson brackets are invariant under a portrait space transformation from (q_i, p_i) to (Q_j, P_j):

{f, g}Q,P = {f, g}q,p               (PB.17)

where: 
  Qj = Qj(qi, pi)                     (PB.18)
  Pj = Pj(qi, pi)                      (PB.19)

Let's try to prove (PB.17) by starting with the Poisson bracket definition and the multivariable chain rule:

Rearranging terms on the right side, we have:

Now apply (PB.20) to {Q_j, f}, and simplify it with fundamental Poisson brackets (PB.9 - PB.12). Then apply anticommutativity (PB.3) to get a simplified expression for {f, Q_j}:

Similarly, we can get a simplified expression for {f, P_j}:

Similarly, we can replace {f, Q_j} and {f, P_j} in (PB.20) with (PB.21) and (PB.22) to prove (PB.17):

Note that a portrait space transformation is also called a canonical transformation.

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

Poisson Bracket in Portrait Space Transformation - Updated in 2024, by Herong Yang