PoissonBracket

The Poisson bracket is an important mathematical concept in classical mechanics and mathematical physics, particularly in the context of Hamiltonian mechanics. It is a binary operation that acts on pairs of functions defined on a symplectic manifold or a Poisson manifold, capturing the fundamental structure of the underlying dynamics.

Given a phase space with canonical coordinates $(q, p)$, where $q$ represents the generalized coordinates and $p$ represents the generalized momenta of a mechanical system, the Poisson bracket of two scalar functions $f(q, p)$ and $g(q, p)$ is defined as:

$$\lbrace f, g \rbrace = \sum_i \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right)$$

Here, the sum runs over all the generalized coordinates and momenta, and the partial derivatives are computed with respect to the corresponding variables.

The Poisson bracket has several important properties:

1. Skew-symmetry: $\lbrace f, g \rbrace = -\lbrace g, f \rbrace$
2. Linearity: $\lbrace af + bg, h \rbrace = a\lbrace f, h \rbrace + b\lbrace g, h \rbrace$ for any scalar constants $a$ and $b$
3. Leibniz rule: $\lbrace fg, h \rbrace = f\lbrace g, h \rbrace + g\lbrace f, h \rbrace$
4. Jacobi identity: $\lbrace f, \lbrace g, h \rbrace \rbrace + \lbrace g, \lbrace h, f \rbrace \rbrace + \lbrace h, \lbrace f, g \rbrace \rbrace = 0$

In Hamiltonian mechanics, the Poisson bracket plays a crucial role in describing the time evolution of a dynamical system. The time derivative of a function $f$ with respect to the Hamiltonian $H$ is given by:

$$\frac{d f}{d t} = \lbrace f, H \rbrace + \frac{\partial f}{\partial t}$$

This equation, called the Poisson's equation, is the foundation of the Hamilton's equations of motion.

In the context of symplectic manifolds, the Poisson bracket is closely related to the symplectic form. In the more general setting of Poisson manifolds, the Poisson bracket is derived from a Poisson tensor, which is a bivector field that generalizes the symplectic form to cases where the non-degenerate condition is not satisfied.

Complex variables

Let the position and momentum variables be denoted by $q$ and $p$ respectively. We can define a complex canonical variable $z$ and its conjugate $z^*$ as:

$$ z = q + i \cdot p \\ z^* = q - i \cdot p $$

Here, $i$ is the imaginary unit, and $q$ and $p$ are real numbers. Now, we can express the position and momentum in terms of the complex canonical variables:

$$ q = \frac{z + z^}{2} \ p = \frac{z - z^}{2i} $$

In terms of these complex variables, the Poisson bracket between two functions $A(z, z^)$ and $B(z, z^)$ is defined as:

$$ {A, B} = \frac{\frac{\partial A}{\partial z} \cdot \frac{\partial B}{\partial z^} - \frac{\partial A}{\partial z^} \cdot \frac{\partial B}{\partial z}}{i} $$