Connection between Poisson Brackets and Symplectic Form

Question

Jose and Saletan say the matrix elements of the Poisson Brackets (PB) in the $ {q,p} $ basis are the same as those of the inverse of the symplectic matrix $ \Omega^{-1} $, whereas the matrix elements of the symplectic form are equal to those of the symplectic matrix $ \Omega $ .

I have no problem with the latter statement, but I do with the first. That's because the PBs are introducing by writing Hamilton's equation as

$$ \dot{\xi^j} = \omega^{jk}{\frac{\partial H }{\partial \xi^k}} ,$$

where $\omega^{jk}$ are the elements of the $\Omega$, and then taking the Lie derivative of a dynamical variable $f$ along the dynamical vector field, which gives

$$L_{\Delta}f = (\partial_j f) \omega^{jk}(\partial_k H)+ \partial_{t}f.$$

It is later said that the term containing the $\omega^{jk}$ is the PB $\left \{ f,H \right \}$, which I have no problem at all, as it gives the right expression for the PBs of the canonical coordinates when the $\omega^{jk}$ are the elements of the symplectic matrix $\Omega$, i.e., in the way it was first introduced through Hamilton's equations. However, as I have mentioned, in a later consideration they say that that the $\omega^{jk}$ of the PBs are the elements of $\Omega^{-1}$, which made me confused, especially because at several times in the book they use $\omega^{jk}$ as components of $\Omega$, and $\Omega_{jk}$ as components of $\Omega^{-1}$, at various derivations. However, I do not believe the statement on the book about the elements of the PBs being those of $\Omega^{-1}$ is wrong, because this is used in the derivation of the preservation of the symplectic form under canonical transformations. Therefore, I think there is a misconception on my behalf somewhere, which I do not know where it is, and I would be thankful if anyone could shed some light on this.

Answer 1

It seems that OP just wants to trace a sign convention in a specific book (Ref. [JS]). To answer this most convincingly, we should document where we read what.

• In the case of canonical coordinates (also known as Darboux coordinates), [JS] uses a convention where the positions $q^i$ are ordered before the momenta $p_i$, $$\xi^i~=~(q^1, \ldots, q^n,p_1, \ldots p_n),$$ cf. e.g. p. 215 in [JS].

• On top of p. 230 in [JS] is written:

  The elements $\omega_{jk}$ of the symplectic form and the $\omega^{jk}$ of the Poisson bracket [...] are the matrix elements of $\Omega$ and $\Omega^{-1}$, respectively.

Since we are only after a sign convention, let us for simplicity restrict to canonical/Darboux coordinates. Then $\Omega^2=-1_{2n}$, and hence the difference between $\Omega$ and $\Omega^{-1}$ boils down to a sign.

• On p. 216 eqs. (5.42b) and (5.43) read $$ \tag{5.42b} \omega_{\ell j} \dot{\xi}^j~=~ \partial_{\ell} H, $$ $$ \tag{5.43} \Omega ~=~\begin{bmatrix} 0_n & -I_n \cr I_n & 0_n \end{bmatrix}. $$

• On p. 218 eq. (5.47) defines the Poisson bracket $$ \tag{5.47} \{f,g\} ~\equiv~ (\partial_jf) \omega^{jk} (\partial_kg) ~\equiv~ \frac{\partial f}{\partial q^{\alpha}}\frac{\partial g}{\partial p_{\alpha}}-\frac{\partial f}{\partial p_{\alpha}}\frac{\partial g}{\partial q^{\alpha}}, $$ leading to Hamilton's equation of motion (5.50), $$ \tag{5.50} \dot{\xi}^j~=~ \{\xi^j , H\}. $$

We conclude that [JS] has the convention that

$$ \Omega^{-1} ~=~\begin{bmatrix} 0_n & I_n \cr -I_n & 0_n \end{bmatrix}. $$

So far so good.

• On p.228 is written

$$ \tag{5.75} \omega ~=~ dq^{\alpha} \wedge dp_{\alpha}. $$

Comparing with $\Omega$ and $\omega_{ij}$, we conclude that [JS] has the convention that

$$ \omega ~=~ \frac{1}{2} \omega_{ij} d\xi^j \wedge d\xi^i ~=~ -\frac{1}{2} \omega_{ij} d\xi^i \wedge d\xi^j . $$

Note the opposite ordering of $i$ and $j$! This is probably the point where OP and many others instead would have liked to defined it oppositely as

$$\tag{Opposite JS} \omega ~=~ \frac{1}{2} \omega_{ij} d\xi^i \wedge d\xi^j,$$

and

$$\tag{Opposite JS} \omega ~=~ dp_{\alpha} \wedge dq^{\alpha}.$$

References:

[JS] Jose and Saletan, Classical Dynamics: A contemporary Approach, 1998.

Answer 2

Let's see: A symplectic form on a manifold $P$ (the phasespace) is a nondegenerate closed two-form $\omega$.

This gives you for every function $f \colon P \to \mathbf{R}$ a vectorfield $\xi_f$ defined by

$$i_{\xi_f}\omega = -df$$,

where $i$ denotes the interior product.

Then for two functions $f,g \colon P \to \mathbf{R}$ the Poissonbracket is

$$ \{f,g\} = \xi_f g = \omega(\xi_f,\xi_g) = - \{g,f\}$$

Now in local coordinates $\xi_f = \xi^i_f \partial_i$, so you get

$$ i_{\xi_f}\omega(\partial_j) = \omega(\xi^i_f \partial_i,\partial_j) = \xi^i_f \omega(\partial_i,\partial_j) = \xi^i_f \omega_{ij} = -df(\partial_j) = -\partial_j f$$

Since $\omega$ is invertible this means $\xi^i_f = -\omega^{ij}\partial_j f$, hence $$\{f,g\} = \omega^{ij}\partial_i f \partial_j g$$

Answer 3

I don't have Jose and Saletan, but see Theorem 18.1.3, in particular (18.6), in my book Classical and Quantum Mechanics via Lie algebras

Answer 4

The issue here is the raising and lowering of indices. The form $\omega_{ij}$ with lower indices is not the same as the two-tensor $\omega^{ij}$, although with a flat metric (as your examples have), the two are the same (since raising and lowering an index makes no difference). But there is a sign issue--- the two tensors are antisymmetric, and you could define raising the index to swap the index position too. In this case, you get an extra minus sign.

The issue is important, because the symplectic matrix has one down and one up index, and squares to -1:

$$ \omega^i_j \omega ^j_k = -\delta^i_k $$

(assuming no swap definition). Then lowering and raising with the metric,

$$ \omega^{ij} = g^{jk}\omega^i_k $$

$$ \omega_{ij} = g_{ik}\omega^k_i $$

so that if you multiply the entries as matrices,

$$ \omega^{ij}\omega_{jk} = \omega^i_k g^{kj}g_{jl}\omega^l_k = -\delta^i_k $$

since the g's are inverse to each other, so you get cancellation in the middle. The result is that the entries of the upper index $\omega$ are the inverse matrix (up to a sign) of the lower index $\omega$, and this is what the authors are trying to say.

They are either sloppy about the sign, or have an index flipping convention that fixes it up, I don't know. But the sign on the inverse is the reason for your confusion. I wouldn't use their terminology--- I would say that the upper index $\omega$ has entries which are the negative inverse of the lower index $\omega$'s, but they probably fix up any signs using their experience and intuition, so that the formulas end up correct in the end.

EDIT: Qmechanic found the reference

It's a swap convention. They flip the indices for the tensor vs. the form, absorbing the minus sign. This is not a great convention, but it's what they do. Thanks Qmechanic for figuring it out.