Physics Notes - Herong's Tutorial Notes

∟Poisson Bracket Expression

∟Poisson Bracket and Hamilton Equations

This section describes Hamilton Equations in terms of the Poisson brackets.

Hamilton Equations of motion can be expressed in terms of the Poisson brackets.

As we presented earlier in this book, the Hamilton Equations in generalized coordinates are:

Since generalized position coordinates (q₁, q₂, ..., q_n) and generalized momentum coordinates (p₁, p₂, ..., p_n) are independent of time, t, we can rewrite Hamilton Equations as:

Hamilton Equation (PB.14) can be proven as below:

Hamilton Equation (PB.15) can be proven similarly.

As a summary, let's recall how the Hamiltonian, H, is defined:

H = T + V                                    (H.1)
  # H is the Hamiltonian, the total energy
  # T is the kinetic energy
  # V is the potential energy

Then look at the Hamilton Equations (PB.14) and (PB.15) again. We can say that:

• The change of position (speed) is the Poisson bracket of the position and the total energy.
• The change of momentum is the Poisson bracket of the momentum and the total energy.

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 and Hamilton Equations - Updated in 2024, by Herong Yang