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I am learning there is an important connection between Hamiltonian formalisms and Symplectic Geometry. It seems like the Newtonian Mechanics are described on what is called the standard symplectic space, which is given by:

$$\Omega=\bigg(\matrix{0&I_n\\-I_n&0}\bigg)$$

This is a $2n\times2n$ matrix, where the $I_n$'s are identity matrices of $n\times n$. So the dimension of this space is proportional to the number of degrees of freedom of the system (in phase space there are $2n$ d.o.f.).

My question is: how do they construct this matrix? Can there exist new machanical systems (relativistic, quantum) so that this matrix is modified? In such case, how are these constructed?


I would appreciate good sources to study this kind of things, because I only found wikipedia pages, and also the book Mathematical Methods of Classical Mechanics by V. I. Arnold, but I find it too formal, and hard to follow.

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  1. The geometric setting for a Hamiltonian theory is often taken to be a $2n$-dimensional real symplectic manifold $(M,\omega)$, where $\omega$ is a closed non-degenerate real 2-form.

  2. In a coordinate neighborhood $U$, the 2-form $\omega$ is given as $$\left.\omega\right|_{U}~=~\frac{1}{2}\sum_{I,J=1}^{2n}\omega_{IJ}(z)~ \mathrm{d}z^I \wedge\mathrm{d}z^J,\tag{1}$$ where $\omega_{IJ}=-\omega_{JI}$ is a non-degenerate antisymmetric real matrix.

  3. The inverse matrix $$ (\omega^{-1})^{IJ}~=~\{z^I,z^J\}_{PB}\tag{2} $$ gives rise to a Poisson bracket.

  4. The Darboux theorem states that there locally (in a sufficiently small open neighborhood $U$) exist Darboux/canonical coordinates $$z^I~=~\left(q^1,\ldots, q^n,p_1, \ldots, p_n\right),\tag{3}$$ where $\omega_{IJ}$ is on the form $$ \omega ~=~\begin{bmatrix} {\bf 0}_{n \times n} & -{\bf 1}_{n \times n} \cr {\bf 1}_{n \times n} & {\bf 0}_{n \times n} \end{bmatrix}, \tag{4}$$ or equivalently, $$\left.\omega\right|_{U}~=~\sum_{i=1}^n\mathrm{d}p_i \wedge\mathrm{d}q^i.\tag{5}$$

  5. It may be helpful to note that one cannot diagonalize a non-degenerate antisymmetric real matrix in a real vector space (because the eigenvalues are imaginary), so the canonical form (4) is in some sense the best one could hope for, up to sign conventions and coordinate permutations.

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  • $\begingroup$ thank you for your answer, so $\omega$ and its inverse only depends on the coordinates? that is why the standard basis has such form, because one always considers Darboux coordinates? $\endgroup$
    – rsaavedra
    Oct 27, 2016 at 15:34
  • $\begingroup$ $\uparrow$ Yes & Yes. $\endgroup$
    – Qmechanic
    Oct 27, 2016 at 18:00

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