Physics Notes - Herong's Tutorial Notes

∟Poisson Bracket Expression

∟What Is Poisson Bracket

This section provides a quick introduction to Poisson Bracket, which is an operation of two functions on the phase space (q,p) of a system.

What Is Poisson Bracket? Poisson Bracket is a partial differential expression of two functions on the phase space. Assuming that f(qi, pi, t) and g(qi, pi, t) are 2 functions on the phase space, the Poisson bracket expression is denoted as {f, g} and defined as below: 

Poisson Bracket - Differential Expression on Phase Space

We can also be viewed as the dot product of gradient vector of f and the symplectic gradient vector of g. 

{f, g} = [∂f/∂q, ∂f/∂p] ∙ [∂g/∂p, -∂g/∂q]

Poisson Bracket expression has the following properties.

1. Identity - Poisson Bracket of the same function is an constant zero. 

{ f, f } = 0                                                         (PB.2)

2. Anticommutativity - Also called Antisymmetry. 

{ f, g } = - { g, f }                                               (PB.3)

3. Bilinearity 

{ f, ag+bh } = a{ f, g } + b{ f, h }                        (PB.4)
{ af+bg, h } = a{ f, h } + b{ g, h }                        (PB.5)
where:
  a and b are constants
  h is another function h(qi, pi, t) on the phase space

4. Leibniz's () Rule - Also called the Product Rule. 

{ fg, h } = { f, h }g + f { g, h }                             (PB.6)
{ f, gh } = { f, g }h + g { f, h }                             (PB.7)

5. Jacobi Identity - The sum of the cyclic permutation of double Poisson brackets of three functions is zero. 

{ f, {g, h} } + { g, {h, f} } + { h, {f, g} } = 0            (PB.8)

6. Fundamental Poisson Brackets - The Poisson brackets of the canonical coordinates themselves are called the fundamental Poisson brackets. It is assumed that generalized position coordinates (q1, q2, ..., qn) and generalized momentum coordinates (p1, p2, ..., pn) are independent. That is ∂qi/∂pj = 0 and ∂pi/∂qj = 0. 

Fundamental Poisson Brackets

δij is called Kronecker Delta, which is defined as: 

Kronecker Delta

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

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