"Hamiltonian mechanics is geometry in phase spase."
The Poisson bracket arises naturally in Hamiltonian mechanics, and since this theory has an elegant geometric interpretation, I'm interested in knowing the geometrical interpretation of the Poisson bracket.
I've read somewhere that the Poisson bracket of two functions $f$ and $g$ of the dynamical variables $(q_i,p_i,t)$ given by:
$$\{f,g\}=\sum_i \left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)$$
is the dot product in phase space of the 'ordinary' gradient of $f$ and the symplectic gradient of $g$. Let me illustrate this interpretation..
As I know, the gradient in phase space is $(\partial_q,\partial_p)$, and the symplectic gradient is $(\partial_p,-\partial_q)$ (i.e is just the gradient rotated by $90^\circ$ clockwise). I think the dot product is apparent now.
Questions.
- What is the meaning/significance of the dot product interpretation?
- Is there any other geometrical interpretation of the Poisson bracket?
Best Answer
The importance comes from the equations of motions: $$ \dot{q}=\frac{\partial H}{\partial p}\, ,\qquad \dot{p}=-\frac{\partial H}{\partial q} $$ which can be rewritten as $$ \dot{q}=\{q,H\}\, ,\qquad \dot{p}=\{p,H\} $$ In particular, for an arbitrary function of $f(p,q)$, we have $$ \frac{d}{dt}f(p,q,t)=\{f,H\}+\frac{\partial f}{\partial t}\, . $$ Geometrically, changes of coordinates in phase space (i.e. canonical transformation) preserve the Poisson bracket, i.e. the transformation $(q,p)\to (Q(q,p),P(q,p)$ is such that $$ \{q,p\}=\{Q,P\}=0\, , $$ and so in this sense, and thinking of $q$ and $p$ as "basis vectors", canonical transformation preserve the Poisson bracket much like rotation preserve the dot product between two vectors. In this interpretation a quantity is conserved (i.e. $\dot f(q,p,t)=0$) when it is Poisson-orthogonal to the Hamiltonian, for instance.