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In classical Hamiltonian mechanics evolution of any observable (scalar function on a manifold in hand) is given as $$\frac{dA}{dt} = \{A,H\}+\frac{\partial A}{\partial t}$$

So Poisson bracket is a binary, skew-symmetric operation $$\{f,g\} = - \{f,g\}$$ which is bilinear $$\{\alpha f+ \beta g,h\} = \alpha \{f, g\}+ \beta \{g,h\}$$ satisfies Leibniz rule: $$\{fg,h\} = f\{g,h\} + g\{f,h\}$$ and Jacobi identity: $$\{f,\{g,h\}\} + \{g,\{h,f\}\} + \{h,\{f,g\}\} = 0$$

How to physically interpret these properties in classical mechanics? What physical characteristic each of them is connected to? For example I suppose the anticommutativity has to do something with energy conservation since because of it $\{H,H\} = 0$.

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Suggestion to the question formulation (v2): Use curly brackets $\{,\}$ for the Poisson bracket instead of square brackets $[,]$. –  Qmechanic Jun 21 '13 at 18:42

3 Answers 3

Let's assume no explicit time dependence and that our Poisson bracket $\{,\}$ - I prefer curly brackets so square ones $[,]$ can be used to denote the commutator of vector fields - is non-singular, ie there's a corresponding symplectic product $\omega$.

The time derivative $$ \frac{\mathrm d}{\mathrm dt}=\{\,\cdot\,,H\} $$ is actually the Lie derivative with respect to the Hamiltonian vector field $X_H$ given by $$ X_H\rfloor\omega \equiv \mathrm dH $$ in disguise as can be seen by $$ \{f,H\} \equiv \omega(X_f,X_H)=(X_f\rfloor\omega)(X_H)=\mathrm df(X_H)=\mathcal{L}_{X_H}f $$ As $\mathcal{L}_{X_H}$ is a linear differential operator respecting the Leibniz rule, so is $\{\,\cdot\,,H\}$.

Antisymmetry translates to $$ \mathcal{L}_{X_f}g = -\mathcal{L}_{X_g}f $$ ie the change of $g$ with respect to the Hamiltonian flow induced by $f$ is the negative of the change in $f$ with respect to the Hamiltonian flow induced by $g$.

Rewriting the Jacobi identity as $$ \{f ,\{g,h\}\} = \{\{f,g\},h\} - \{\{f,h\},g\} $$ we see that $$ \mathcal{L}_{X_{\{g,h\}}}f=\left(\mathcal{L}_{X_h}\mathcal{L}_{X_g} - \mathcal{L}_{X_g}\mathcal{L}_{X_h}\right)f = \mathcal{L}_{[X_h,X_g]}f $$ ie $f\mapsto X_f$ is a Lie-algebra homomorphism.

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The choice of square versus curly brackets has significance, and can not be made freely. $\left\{ ,\right\}$ indicates an "anti-commutator". Sticking with this non-standard notation will cause you trouble eventually. –  dmckee May 13 '13 at 16:29
@dmckee: notation is domain-specific, and it's quite common to use curlies for Poisson brackets, both in introductory and advanced literature –  Christoph May 13 '13 at 16:58
So basically you state that Poisson structure arises from the fact that we describe system evolution as a vector field generated by a Hamiltonian and Poisson structure is just desired properties of vector fields lifted (not a term) to scalar functions, am I right? –  Yrogirg May 14 '13 at 13:57
What is the physical meaning of your treatment of antisymmetry? Why should it be so? –  Yrogirg May 15 '13 at 12:33

The physical interpretation is integrability conditions being satisfied on the manifold. From the first equation, if you would take A not depending on 't' explicitly then dA/dt = [A,H]. The Poisson bracket contains in it the dynamics involved in canonically conjugate variables and in classical mechanics, we can measure them simultaneously. Apart from this, laws of conservation can be explicitly seen in this representation.

One important factor to note is that, Poisson brackets are valid only for exact differentials and they follow the canonical transformations. In fact, canonical transformations are nothing but invariance of Poisson brackets.

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I edited the formatting slightly - for some reason it was being displayed with a scroll bar! –  twistor59 May 13 '13 at 12:05

If we consider for simplicity a 2d phase space (q,p), then we can interpretate the poisson bracket between two functions f(q,p) and g(q,p) as the vector product of their gradients, which are vector fields in this plane:

$[f,g]=(\nabla f\times \nabla g)\cdot \mathbf{e}_z$

where $e_z$ is a unit vector perpendicular to the plane.

From that definition all the properties are obvious.

We can imagine the following physical analogy for the equation of motion, the gradient of the hamiltonian act like magnetic field $B$ and the gradient of the function is the velocity $v$, in formulas:

$\partial_tf\, \mathbf{e}_z= \nabla f\times \nabla H = \mathbf{v} \times \mathbf{B}$

which is the expression of the Lorentz force.

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