The Jacobi Identity for the Dirac Bracket

Contents

Notation, and the four identities consumed
Route A: the direct expansion
Route B: the bracket of the induced symplectic form

This appendix proves the one clause of Theorem 30.21 of Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism that the chapter defers: the Dirac bracket Equation (30.25) obeys the Jacobi identity Equation (26.46). Bilinearity, antisymmetry, the Leibniz rule Equation (28.24), the strong vanishing Equation (30.26) and the two statements Equations (30.27) and (30.28) are one-line computations and are done in the chapter; the identity below is what earns the object the name bracket, because without it the correspondence rule Equation (29.31) would send it to something that is not a commutator.

Two independent routes are written out, as the chapter's prose promises. Route A: the direct expansion is the direct expansion: it uses nothing but the Poisson Jacobi identity Equation (26.46), the derivation property Proposition 29.4 and the antisymmetry of \(C^{-1}\), and it is the route that shows where the identity comes from — every one of the four orders in \(C^{-1}\) cancels for its own reason, and three of the four reasons are the Poisson Jacobi identity applied to a different triple. Route B: the bracket of the induced symplectic form is the structural route: the Dirac bracket is the Poisson bracket of the symplectic manifold that Proposition 30.25 constructs on the second-class surface, and the identity is then read off the closure of a two-form. The second is three paragraphs long and is the one to remember; it is written second because it consumes Proposition 30.25, which is proved in the chapter after the Dirac bracket is introduced.

Nothing outside this treatise is used anywhere in this section.

Notation, and the four identities consumed

Throughout, \(\chi_{\alpha}\), \(\alpha=1,\ldots,S\), is a complete set of second-class constraints in the sense of Definition 30.12, \(C_{\alpha\beta}=\pb{\chi_{\alpha}} {\chi_{\beta}}\) is the matrix Equation (30.24), which is invertible on a neighbourhood of the constraint surface by Proposition 30.19 and continuity, and

\begin{equation}\tag{A51.1} D^{\alpha\beta}:=\left(C^{-1}\right)^{\alpha\beta}\ec\qquad D^{\alpha\gamma}C_{\gamma\beta}=\delta^{\alpha}_{\beta}\ec\qquad D^{\alpha\beta}=-D^{\beta\alpha}\ep \end{equation}

The antisymmetry of \(D\) is proved in the chapter, in the antisymmetry step of Theorem 30.21: \(C\transpose=-C\) gives \(\left(C^{-1}\right)\transpose=\left(C\transpose\right)^{-1}=-C^{-1}\). That \(S\) is even, so that such a \(C\) can exist at all, is Proposition 30.19. With this notation Equation (30.25) reads

\begin{equation}\tag{A51.2} \pb{A}{B}_{\text{D}}=\pb{A}{B}+A_{\alpha}D^{\alpha\beta}B_{\beta}\ec \qquad\text{where}\qquad A_{\alpha}:=\pb{A}{\chi_{\alpha}}\ec \end{equation}

the sign having flipped because \(\pb{\chi_{\beta}}{B}=-B_{\beta}\). Two further abbreviations carry the whole computation:

\begin{equation}\tag{A51.3} A_{\alpha\beta}:=\pb{\pb{A}{\chi_{\alpha}}}{\chi_{\beta}} =\pb{A_{\alpha}}{\chi_{\beta}}\ec\qquad \widehat{A}^{\alpha}:=A_{\beta}D^{\beta\alpha}\ep \end{equation}

Note that \(A_{\alpha\beta}\) is not symmetric — its antisymmetric part is Equation (A51.4) below — and that the index of \(\widehat{A}\) is raised from the left, so that \(A_{\alpha}D^{\alpha\beta}B_{\beta}=\widehat{A}^{\beta}B_{\beta}\) while \(D^{\alpha\beta}B_{\beta}=-\widehat{B}^{\alpha}\).

Dimensionally there is nothing to watch: \(D^{\alpha\beta}\) carries the inverse of the dimension of \(C_{\alpha\beta}\), so the correction term of Equation (A51.2) has exactly the dimension of \(\pb{A}{B}\), whatever SI dimensions the individual \(\chi_{\alpha}\) happen to have. Nothing below depends on the constraints being dimensionally homogeneous among themselves.

Lemma A51.1 (The four identities).

For all phase-space functions \(A\), \(B\), \(E\), wherever \(C^{-1}\) is defined,

\begin{align} \pb{A}{C_{\alpha\beta}}&=A_{\alpha\beta}-A_{\beta\alpha}\ec \tag{A51.4}\\ \pb{\pb{B}{E}}{\chi_{\delta}} &=\pb{B}{E_{\delta}}-\pb{E}{B_{\delta}}\ec \tag{A51.5}\\ \pb{A}{D^{\alpha\beta}} &=-D^{\alpha\mu}\pb{A}{C_{\mu\nu}}D^{\nu\beta}\ec \tag{A51.6}\\ T_{\alpha\beta\gamma}+T_{\beta\gamma\alpha}+T_{\gamma\alpha\beta}&=0\ec \qquad T_{\alpha\beta\gamma}:=\pb{C_{\alpha\beta}}{\chi_{\gamma}}\ep \tag{A51.7} \end{align}

Rests on Equation (26.46), Proposition 29.4 and Equation (30.24).

Proof.

Derives Lemma A51.1. Three of the four are the Poisson Jacobi identity Equation (26.46) applied to a different triple, and the fourth is the derivative of an inverse.

Equation (A51.4). Take the triple \(\left(A,\chi_{\alpha},\chi_{\beta}\right)\) in Equation (26.46):

\begin{equation}\tag{A51.8} \pb{A}{\pb{\chi_{\alpha}}{\chi_{\beta}}} +\pb{\chi_{\alpha}}{\pb{\chi_{\beta}}{A}} +\pb{\chi_{\beta}}{\pb{A}{\chi_{\alpha}}}=0\ep \end{equation}

The second term is \(\pb{\chi_{\alpha}}{-A_{\beta}}=+A_{\beta\alpha}\) and the third is \(\pb{\chi_{\beta}}{A_{\alpha}}=-A_{\alpha\beta}\), which gives Equation (A51.4).

Equation (A51.5). Take the triple \(\left(\chi_{\delta},B,E\right)\): \(\pb{\chi_{\delta}}{\pb{B}{E}}+\pb{B}{\pb{E}{\chi_{\delta}}} +\pb{E}{\pb{\chi_{\delta}}{B}}=0\), that is \(-\pb{\pb{B}{E}}{\chi_{\delta}}+\pb{B}{E_{\delta}} -\pb{E}{B_{\delta}}=0\).

Equation (A51.6). The map \(X\longmapsto\pb{A}{X}\) is a derivation (Proposition 29.4), so applying it to \(D^{\alpha\mu}C_{\mu\beta}=\delta^{\alpha}_{\beta}\) gives \(\pb{A}{D^{\alpha\mu}}C_{\mu\beta} +D^{\alpha\mu}\pb{A}{C_{\mu\beta}}=0\); contracting with \(D^{\beta\nu}\) isolates \(\pb{A}{D^{\alpha\nu}}\). This is the only fact about the inverse matrix that the computation needs, and it is why \(D\) may be kept as an unexpanded symbol throughout.

Equation (A51.7). Take the triple \(\left(\chi_{\gamma},\chi_{\alpha},\chi_{\beta}\right)\): each of the three terms of Equation (26.46) is minus one of the \(T\)'s, in the cyclic order stated.

∎

Route A: the direct expansion

Theorem A51.2 (Jacobi identity for the Dirac bracket).

Wherever \(C_{\alpha\beta}\) is invertible, and for all phase-space functions \(A\), \(B\), \(E\),

\begin{equation}\tag{A51.9} \pb{A}{\pb{B}{E}_{\text{D}}}_{\text{D}} +\pb{B}{\pb{E}{A}_{\text{D}}}_{\text{D}} +\pb{E}{\pb{A}{B}_{\text{D}}}_{\text{D}}=0\ec \end{equation}

strongly — as an identity of functions, not merely on the constraint surface. Rests on Definition 30.20, Equation (26.46) and Lemma A51.1.

Proof.

Derives Theorem A51.2. Write \(\mathfrak{S}\) for the sum over the three cyclic images \(\left(A,B,E\right)\to\left(B,E,A\right)\to\left(E,A,B\right)\), so that the left-hand side of Equation (A51.9) is \(\mathfrak{S}\,\pb{A}{\pb{B}{E}_{\text{D}}}_{\text{D}}\).

Step 1: expand once and sort by the number of \(D\) factors. Put \(F:=\pb{B}{E}_{\text{D}}=\pb{B}{E}+B_{\alpha}D^{\alpha\beta}E_{\beta}\). By Equation (A51.2) applied a second time, \(\pb{A}{F}_{\text{D}}=\pb{A}{F}+A_{\gamma}D^{\gamma\delta}F_{\delta}\) with \(F_{\delta}=\pb{F}{\chi_{\delta}}\). Expanding both pieces by the Leibniz rule (Proposition 29.4) and using Equation (A51.5) on \(\pb{\pb{B}{E}}{\chi_{\delta}}\),

\begin{align} \pb{A}{F}&=\pb{A}{\pb{B}{E}} +\pb{A}{B_{\alpha}}D^{\alpha\beta}E_{\beta} +B_{\alpha}D^{\alpha\beta}\pb{A}{E_{\beta}} +B_{\alpha}\pb{A}{D^{\alpha\beta}}E_{\beta}\ec \tag{A51.10}\\ F_{\delta}&=\pb{B}{E_{\delta}}-\pb{E}{B_{\delta}} +B_{\alpha\delta}D^{\alpha\beta}E_{\beta} +B_{\alpha}D^{\alpha\beta}E_{\beta\delta} +B_{\alpha}\pb{D^{\alpha\beta}}{\chi_{\delta}}E_{\beta}\ep \tag{A51.11} \end{align}

Assembling, and using Equation (A51.6) on \(\pb{A}{D^{\alpha\beta}}\) and on \(\pb{D^{\alpha\beta}}{\chi_{\delta}}\), the eight surviving terms sort by the number of factors of \(D\) they carry:

\begin{align} \pb{A}{\pb{B}{E}_{\text{D}}}_{\text{D}} ={}&\underbrace{\pb{A}{\pb{B}{E}}}_{\text{order }0} +\underbrace{\pb{A}{B_{\alpha}}D^{\alpha\beta}E_{\beta} +B_{\alpha}D^{\alpha\beta}\pb{A}{E_{\beta}} +A_{\gamma}D^{\gamma\delta} \left(\pb{B}{E_{\delta}}-\pb{E}{B_{\delta}}\right)} _{\text{order }1}\nn\\ &+\underbrace{\widehat{B}^{\mu}\widehat{E}^{\nu}\pb{A}{C_{\mu\nu}} -\widehat{A}^{\delta}\widehat{E}^{\alpha}B_{\alpha\delta} +\widehat{A}^{\delta}\widehat{B}^{\beta}E_{\beta\delta}} _{\text{order }2} +\underbrace{\widehat{A}^{\delta}\widehat{B}^{\mu}\widehat{E}^{\nu} T_{\mu\nu\delta}}_{\text{order }3}\ep \tag{A51.12} \end{align}

The three order-two terms and the order-three term are obtained by substituting Equation (A51.6) and then absorbing each \(A_{\gamma}D^{\gamma\delta}\) and \(B_{\alpha}D^{\alpha\mu}\) into the hatted abbreviation of Equation (A51.3), remembering from it that \(D^{\nu\beta}E_{\beta}=-\widehat{E}^{\nu}\), which is where the sign of the first order-two term and of the order-three term comes from.

Step 2: order zero. \(\mathfrak{S}\,\pb{A}{\pb{B}{E}}=0\) is the Poisson Jacobi identity Equation (26.46) itself.

Step 3: order one. Introduce the trilinear object

\begin{equation}\tag{A51.13} G(U,V,W):=\pb{U}{V_{\alpha}}D^{\alpha\beta}W_{\beta}\ep \end{equation}

The first order-one term is \(G(A,B,E)\). For the second, relabel \(\alpha\leftrightarrow\beta\) and use \(D^{\beta\alpha}=-D^{\alpha\beta}\): \(B_{\alpha}D^{\alpha\beta}\pb{A}{E_{\beta}}=-G(A,E,B)\). For the third, the same manoeuvre on each of the two pieces gives \(A_{\gamma}D^{\gamma\delta}\pb{B}{E_{\delta}}=-G(B,E,A)\) and \(-A_{\gamma}D^{\gamma\delta}\pb{E}{B_{\delta}}=+G(E,B,A)\). The order-one part of Equation (A51.12) is therefore

\begin{equation}\tag{A51.14} G(A,B,E)-G(A,E,B)-G(B,E,A)+G(E,B,A)\ep \end{equation}

Now apply \(\mathfrak{S}\). A cyclic sum of \(G\) is a sum of three terms that depends only on which of the two orientations of the triple its arguments carry, so \(\mathfrak{S}G(B,E,A)=\mathfrak{S}G(A,B,E)\) and \(\mathfrak{S}G(E,B,A)=\mathfrak{S}G(A,E,B)\): in each case the three summands are the same three terms in a different order. Hence the first term of Equation (A51.14) cancels the third and the second cancels the fourth, and the cyclic sum vanishes. Concretely, the summand \(G(A,B,E)\) produced by the first term of the \(\left(A,B,E\right)\) copy is cancelled by the summand \(-G(A,B,E)\) produced by the third term of the \(\left(E,A,B\right)\) copy; the pairing of the second and fourth terms is the same one cycle over. This is the step that a reader cannot reconstruct without being told which term meets which, and it is the reason Equation (A51.5) had to be used in Step 1: without it the third order-one term is a bracket of a bracket and does not have the shape Equation (A51.13) at all.

Step 4: order two. Replace \(\pb{A}{C_{\mu\nu}}\) by \(A_{\mu\nu}-A_{\nu\mu}\), which is Equation (A51.4), and relabel so that every term carries the free index pair in the order \(\left(p,q\right)\):

\begin{equation}\tag{A51.15} X(A,B,E)=\widehat{B}^{p}\widehat{E}^{q}A_{pq} -\widehat{B}^{q}\widehat{E}^{p}A_{pq} -\widehat{A}^{q}\widehat{E}^{p}B_{pq} +\widehat{A}^{q}\widehat{B}^{p}E_{pq}\ec \end{equation}

the second term coming from \(-\widehat{B}^{p}\widehat{E}^{q}A_{qp}\) by the interchange \(p\leftrightarrow q\). Now collect the cyclic sum by which of \(A_{pq}\), \(B_{pq}\), \(E_{pq}\) each term carries. The coefficient of \(A_{pq}\) receives \(\widehat{B}^{p}\widehat{E}^{q}-\widehat{B}^{q}\widehat{E}^{p}\) from \(X(A,B,E)\), \(+\widehat{B}^{q}\widehat{E}^{p}\) from the fourth term of \(X(B,E,A)\) and \(-\widehat{E}^{q}\widehat{B}^{p}\) from the third term of \(X(E,A,B)\); the four contributions cancel in pairs. The coefficients of \(B_{pq}\) and of \(E_{pq}\) are the same computation read one and two cycles on, so \(\mathfrak{S}X=0\).

Step 5: order three. By Equation (A51.7) the totally contracted coefficient is a cyclic sum of \(T\). Writing the order-three term of Equation (A51.12) as \(\widehat{A}^{p}\widehat{B}^{q}\widehat{E}^{r}T_{qrp}\) and applying \(\mathfrak{S}\), which permutes \(\widehat{A},\widehat{B},\widehat{E}\) cyclically, the three summands are \(\widehat{A}^{p}\widehat{B}^{q}\widehat{E}^{r}\) times, in turn, \(T_{qrp}\), \(T_{rpq}\) and \(T_{pqr}\). Their sum vanishes by Equation (A51.7).

All four orders vanish separately, so Equation (A51.9) holds. No constraint was set to zero anywhere in the argument, so the identity is strong, as claimed.

∎

Remark A51.3 (What each order used).

It is worth recording the audit, because the four cancellations look alike and are not. Order zero is the Poisson Jacobi identity on the triple \(\left(A,B,E\right)\). Order one is the Poisson Jacobi identity on \(\left(\chi_{\delta},B,E\right)\), which turns a bracket of a bracket into two terms of the shape Equation (A51.13), plus the antisymmetry of \(D\). Order two is the Poisson Jacobi identity on \(\left(A,\chi_{\alpha},\chi_{\beta}\right)\), which is what makes the derivative of the inverse matrix collapse onto the second brackets \(A_{\alpha\beta}\) already present in the sum, plus the antisymmetry of \(D\). Order three is the Poisson Jacobi identity on three constraints \(\left(\chi_{\gamma},\chi_{\alpha},\chi_{\beta}\right)\). The antisymmetry of \(D\) enters at orders one, two and three, and never on its own: at every order it does no more than allow the terms to be written with their indices in a common order, and the vanishing is always supplied by Equation (26.46). So the Dirac bracket inherits its Jacobi identity from the Poisson bracket four times over, once for each way of feeding a constraint into the triple.

Route B: the bracket of the induced symplectic form

The second proof is shorter, gives the reason rather than the verification, and is the one worth remembering. Its input is Proposition 30.25, proved in the chapter: with \(C_{\alpha\beta}\) invertible at \(x\), the tangent space splits as \(T_{x}M=W\oplus T_{x}\Sigma_{\chi}\) along the projector Equation (30.42), the two summands are \(\omega\)-orthogonal, and the restriction \(\omega_{\Sigma}\) of the symplectic form to \(T_{x}\Sigma_{\chi}\) is non-degenerate, so that \(\left(\Sigma_{\chi},\omega_{\Sigma}\right)\) is a symplectic manifold.

Proposition A51.4 (The Dirac bracket is a projected Hamiltonian flow).

Let \(X_{A}\) be the Hamiltonian vector field of \(A\) in the ambient space, Equation (28.5), and let \(P\) be the projector Equation (30.42). Then, for all \(A\) and \(B\),

\begin{equation}\tag{A51.16} \pb{B}{A}_{\text{D}}=\dd B\left(P\left(X_{A}\right)\right)\ep \end{equation}

On \(\Sigma_{\chi}\) the vector field \(P(X_{A})\) is the Hamiltonian vector field of the restriction \(A|_{\Sigma_{\chi}}\) with respect to \(\omega_{\Sigma}\), and consequently

\begin{equation}\tag{A51.17} \pb{A}{B}_{\text{D}}\Big|_{\Sigma_{\chi}} =\pb{A|_{\Sigma_{\chi}}}{B|_{\Sigma_{\chi}}}_{\omega_{\Sigma}}\ep \end{equation}

Rests on Proposition 30.25, Equation (28.7) and Equation (30.42).

Proof.

Derives Proposition A51.4. By Equation (28.7), \(\pb{u}{v}=\dd u\left(X_{v}\right)\) for every pair of functions. Hence \(\pb{B}{A}=\dd B\left(X_{A}\right)\), \(\pb{B}{\chi_{\alpha}}=\dd B\left(X_{\chi_{\alpha}}\right)\) and \(\pb{\chi_{\beta}}{A}=\dd\chi_{\beta}\left(X_{A}\right)\), so that Equation (30.25) reads

\begin{equation}\tag{A51.18} \pb{B}{A}_{\text{D}} =\dd B\left(X_{A}\right) -\dd B\left(X_{\chi_{\alpha}}\right) \left(C^{-1}\right)^{\alpha\beta} \dd\chi_{\beta}\left(X_{A}\right) =\dd B\!\left(X_{A}-X_{\chi_{\alpha}} \left(C^{-1}\right)^{\alpha\beta} \dd\chi_{\beta}\left(X_{A}\right)\right)\ec \end{equation}

and the argument of \(\dd B\) is exactly \(P\left(X_{A}\right)\) of Equation (30.42). That is the whole of Equation (A51.16), and it is the whole point: the Dirac bracket differs from the Poisson bracket only in that the Hamiltonian vector field is projected onto the constraint surface before it is used.

For the second statement, let \(v\in T_{x}\Sigma_{\chi}\). The \(\omega\)-orthogonality of the splitting gives \(\omega\!\left(X_{\chi_{\alpha}},v\right)=\dd\chi_{\alpha}(v)=0\), so

\begin{equation}\tag{A51.19} \omega_{\Sigma}\!\left(P\left(X_{A}\right),v\right) =\omega\!\left(X_{A},v\right) -\left(C^{-1}\right)^{\alpha\beta} \dd\chi_{\beta}\left(X_{A}\right)\, \omega\!\left(X_{\chi_{\alpha}},v\right) =\dd A(v)\ep \end{equation}

Since \(P\left(X_{A}\right)\) is tangent to \(\Sigma_{\chi}\) and \(\dd A(v)\) for tangent \(v\) depends on \(A\) only through \(A|_{\Sigma_{\chi}}\), this says precisely that \(P\left(X_{A}\right)\) is the \(\omega_{\Sigma}\)-Hamiltonian vector field of \(A|_{\Sigma_{\chi}}\). Feeding that back into Equation (A51.16) with the roles of \(A\) and \(B\) exchanged, and using antisymmetry of both brackets, gives Equation (A51.17).

∎

Second derivation of the Jacobi identity. Derives Theorem A51.2. This is a second, independent proof of Theorem A51.2. Fix a point \(x\) at which \(C_{\alpha\beta}\) is invertible, and let \(c_{\alpha}:=\chi_{\alpha}(x)\). The shifted functions \(\chi_{\alpha}-c_{\alpha}\) have the same brackets as the \(\chi_{\alpha}\), hence the same invertible matrix \(C_{\alpha\beta}\), and they define the level surface \(\Sigma_{c}\) through \(x\). Everything in Proposition 30.25 and in Proposition A51.4 therefore applies verbatim to \(\Sigma_{c}\), and the Dirac bracket built from \(\chi_{\alpha}-c_{\alpha}\) is, term by term in Equation (30.25), the Dirac bracket built from the \(\chi_{\alpha}\): constants drop out of every bracket. So the level surfaces of the constraints foliate a neighbourhood of \(\Sigma_{\chi}\) by symplectic manifolds \(\left(\Sigma_{c},\omega_{c}\right)\), and by Equation (A51.17) the Dirac bracket of two functions restricted to \(\Sigma_{c}\) is the Poisson bracket of \(\left(\Sigma_{c},\omega_{c}\right)\) applied to their restrictions.

Now \(\omega_{c}\) is the pullback \(i_{c}^{*}\omega\) of the ambient symplectic form along the inclusion, so \(\dd\omega_{c}=i_{c}^{*}\dd\omega=0\): it is closed because \(\omega\) is, and non-degenerate by Proposition 30.25. A symplectic manifold's Poisson bracket satisfies the Jacobi identity — that is Equation (26.46) on \(\left(\Sigma_{c},\omega_{c}\right)\), which by Theorem 28.12 may be computed in canonical coordinates, where it is the same computation as in the ambient space. Hence the cyclic sum Equation (A51.9), restricted to \(\Sigma_{c}\), vanishes.

Finally, the value of the cyclic sum at \(x\) is determined by its restriction to the leaf through \(x\): by Equation (A51.17) each of its three terms is, at \(x\), the corresponding double bracket of \(\left(\Sigma_{c},\omega_{c}\right)\) applied to restrictions. Since \(x\) was an arbitrary point of the neighbourhood on which \(C^{-1}\) exists, Equation (A51.9) holds there identically.

∎

Remark A51.5 (Why the leaves, and not the surface alone).

The last paragraph is the step at which a shorter-looking argument fails, and the failure is worth naming. It is tempting to say that Equation (30.26) lets one add any multiple of a constraint to either argument without changing the Dirac bracket, so that an identity holding on \(\Sigma_{\chi}\) holds everywhere. It does not: by the Leibniz rule, \(\pb{A}{c^{\alpha}\chi_{\alpha}}_{\text{D}} =\chi_{\alpha}\pb{A}{c^{\alpha}}_{\text{D}}\), which vanishes on \(\Sigma_{\chi}\) and not off it. What Equation (30.26) does say — and this is the right reading — is that every \(\chi_{\alpha}\) is a Casimir of the Dirac bracket, so its level sets are unions of symplectic leaves, and the foliation used above is not an artifice but the canonical structure the bracket itself defines. The Dirac bracket makes a neighbourhood of \(\Sigma_{\chi}\) a Poisson manifold in the sense of Definition 28.36, whose symplectic leaves are the \(\Sigma_{c}\); Theorem A51.2 is the assertion that it is a Poisson bracket at all.

Remark A51.6 (What is quoted here).

Nothing. Both routes are carried out from material proved in this treatise: the Poisson Jacobi identity Equation (26.46), the derivation property Proposition 29.4, the relation between the bracket and the symplectic form Equation (28.7), Darboux's theorem Theorem 28.12 in the form proved in Darboux's Theorem: Local Canonical Coordinates, and the splitting of Proposition 30.25. The one hypothesis that is not proved but assumed is the regularity of the constraint set (Remark 30.7), which is what makes \(\Sigma_{\chi}\) a submanifold in the first place; it is an assumption about the system under study and is stated as such in the chapter.

Remark A51.7.

The Jacobi Identity for the Dirac Bracket discharges the derivation owed at Theorem 30.21 of Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism, whose proof establishes every other property of the Dirac bracket and defers this one. With it the object of Definition 30.20 is a bracket in the full sense, so that Postulate 30.47 may send it to a commutator, and the three facts the chapter draws together are seen to be one fact in three costumes: the Jacobi identity proved here, the degree-of-freedom count of Theorem 30.17, and the reduced Liouville measure of Proposition 30.25 are the tangential, the dimensional and the volumetric readings of the single statement that \(\left(\Sigma_{\chi},\omega_{\Sigma}\right)\) is a symplectic manifold. Remark 30.26 makes the same point from the measure side; Example 30.22 is the worked instance, where the leaves are the tangent bundles of a surface in \(\R^{3}\) and the reduced bracket is the one Lagrangian Mechanics obtains by eliminating a coordinate.