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Section A.1 Linear Algebra Foundations
Subsection A.1.1 Poisson Bracket
Definition A.1.1. Poisson Bracket.
The
Poisson bracket between two scalar functions
\(A=A(q,p)\) and
\(B=B(q,p)\) is given by:
\begin{equation*}
\{ A, B \} = \frac{\partial A}{\partial q} \frac{\partial B}{\partial p} - \frac{\partial A}{\partial p} \frac{\partial B}{\partial q} \, .
\end{equation*}
Properties:
-
Antisymmetry: \(\{A, b\} = - \{B, A\} \)
-
Jacobi identity: \(\{A, \{B, C\}\} + \{B, \{C, A\}\} + \{C, \{A, B\}\} = 0 \)
-
Leibniz rule: \(\{A, BC\} = \{A, B\}C + B\{A, C\} \)
