Poisson brackets YouTube

Conservation in Hamilton's form

Hamilton's form means we can derive conserved quantities, and hence cconservation laws, without the need to consider symmetries of any physical system.

For example, suppose we had some function $$A = A(q_i, p_i)$$ then the condition necessary for this to be a conserved quantity is $$\begin{align*}0 &= \dot{A}\\&= \sum_{i} \left\{ \frac{\partial A}{\partial q_i} \dot{q_i} + \frac{\partial A}{\partial p_i} \dot{p_i} \right\}\\&= \sum_{i} \left\{ \frac{\partial A}{\partial q_i} \left( \frac{\partial H}{\partial p_i} \right) + \frac{\partial A}{\partial p_i} \left( -\frac{\partial H}{\partial q_i} \right) \right\}\\&= \sum_{i} \left\{ \frac{\partial A}{\partial q_i} \frac{\partial H}{\partial p_i} - \frac{\partial A}{\partial p_i} \frac{\partial H}{\partial q_i} \right\}\end{align*}$$

Special case of a Poisson bracket

Given any two functions of position and momenta, $$\begin{align*}f &= f(q_i, p_i)\\g &= g(q_i, p_i)\end{align*}$$ we define the Poisson bracket of $f$ and $g$ to be $$\{f, g\} = \sum_{i} \left\{ \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right\}$$

Thus, we see that the expression for the time derivative of a function of the position and momenta of a given system, $$A = A(q_i, p_i)$$ can be written as a Poisson bracket of $A$ with the Hamiltonian, $H$. That is, $$\dot{A} = \{A, H\}$$

Thus, given a Hamiltonian, $H$, the condition that the quantity, $A$, is conserved, is then $$\{A, H\} = 0$$

We can work out the Poisson brackets of the time derivatives of the positions and momenta, $$\begin{align*}\dot{q_i} &= \{q, H\}\\&= \frac{\partial q}{\partial q} \frac{\partial H}{\partial p} - \frac{\partial q}{\partial p}\frac{\partial H}{\partial q}\\&= 1 \cdot \frac{\partial H}{\partial p} - 0 \cdot \frac{\partial H}{\partial q}\\&= \frac{\partial H}{\partial p}\end{align*}$$ and $$\begin{align*}\dot{p_i} &= \{p, H\}\\&= \frac{\partial p}{\partial q} \frac{\partial H}{\partial p} - \frac{\partial p}{\partial p}\frac{\partial H}{\partial q}\\&= 0 \cdot \frac{\partial H}{\partial p} - 1 \cdot \frac{\partial H}{\partial q}\\&= -\frac{\partial H}{\partial q}\end{align*}$$ which are just Hamilton's equations.

We shall return to Poisson brackets later in the course.