# Poisson brackets and canonical coordinates for constrained system

Let's have some Hamiltonian which involves the set of first class constraints $\varphi$ and set of constraints $\kappa$, which play role of canonical conjugated momentums for $\varphi$,. They're obeyed the conditions ($[A, B]_{P}$ is the Poisson bracket) $$\tag 1 det [\varphi_{i}, \kappa_{j}]_{P}\neq 0, \quad [\kappa_{i},\kappa_{j}]_{P} = 0, \quad [\varphi_{i}, \varphi_{j}]_{P} = \sum_{k}d^{k}_{ij}\varphi_{k}$$ (in a strong sense). Also there is the statement that for some observed function $f(q, p)$ $$\tag 2 [f_{a}, \varphi_{b} ] = \sum c^{ab}_{i}\varphi_{i}.$$ Without constraints we have $n$ pairs of independent canonical coordinates $(q, p)$, but constraints reduce it to $n - k$. Let's assume canonical Poisson bracket, $$\tag 3 [A, B]_{P} = \sum \left(\frac{\partial A}{\partial q_{i}}\frac{\partial B}{\partial p_{i}} - \frac{\partial B}{\partial q_{i}}\frac{\partial A}{\partial p_{i}}\right).$$ There is a statement that if we introduce the new set of coordinates, $(q, p) \to \eta = (\varphi , Q, p_{i}, P)$, where $\kappa_{j} = P_{j}$ (here $Q, P$ is the set of canonical coordinates which leave independent after solving the equations of constraints), then there is the statement that $(3)$ may be written as $$\tag 4 [A, B]_{P} = \sum_{\eta}[\eta_{i}, \eta_{j}]_{P}\frac{\partial A}{\partial \eta_{i}}\frac{\partial B}{\partial \eta_{j}} = \sum_{Q, P}\left( \frac{\partial A}{\partial Q_{i}}\frac{\partial B}{\partial P_{i}} - \frac{\partial A}{\partial P_{i}}\frac{\partial B}{\partial Q_{i}}\right).$$ How to prove the last identity in $(4)$ by using $(1), (2)$?

We interpret OP's question as essentially asking the following. (If this is not what OP is asking we expect at least the proof method to be very similar.)

Given $$\tag{1} \det \{\varphi_{i}, \kappa_{j}\}_{PB}\neq 0, \quad \{\kappa_{i},\kappa_{j}\}_{PB} ~=~ 0, \quad \{\varphi_{i}, \varphi_{j}\}_{PB} ~=~ \sum_{k}d^{k}_{ij}\varphi_{k},$$ in strong sense, where $d^{k}_{ij}$ are smooth coefficient functions, does there exists a set of constraints $\tilde{\varphi}_i$ equivalent to the set of constraints $\varphi_i$, such that the set of coordinates $(\tilde{\varphi}_i,\kappa_{i})$ can be extended into a full set of local Darboux coordinates in phase space?

Hints:

1. The statement is only true in a sufficiently small local neighborhood. Moreover, the statement is only true if we impose adequate regularity conditions on the constraints, such as e.g. linearly independence and constant rank requirements, cf. e.g. Ref. 1.

2. This exercise is a special case of Exercise 1.22 in Ref. 1. The statement can be viewed as a generalization of Caratheodory–Jacobi–Lie theorem, which in turn, is a generalization of Darboux' theorem.

3. The proof technique is very similar to the proof of Darboux' theorem. The $\kappa_i$ coordinates are already on Darboux form, so we only need to find the other Darboux coordinates. The idea is to use Weinstein's splitting theorem to establish the local existence of Darboux coordinates step by step, see e.g. Refs. 3-4.

References:

1. M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, 1994.

2. D.M. Gitman and I.V. Tyutin, Quantization of fields with constraints, 1990.

3. A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom. 18 (1983) 523; Chapter 2.

4. Akhil Mathew, Poisson manifolds and the splitting theorem, blogpost december 2009.