In his classic book, *Lectures on Quantum Mechanics* [1], Dirac introduces a generic theory of many particles through the theory's Lagrangian $L$, which, assuming (as usual) that $L$ depends only on the particle positions $q_n$ and velocities $\dot{q}_n$, leads directly to the Euler-Lagrange equations of motion, obtained by minimizing the action $S=\int dt L$:

(1)\begin{align}

\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}_n}\right)=\frac{\partial L}{\partial q_n}.\label{eq:EL}

\end{align}

Dirac then switches to the Hamiltonian formalism by introducing the canonical momenta

(2)\begin{align}

p^n=\frac{\partial L}{\partial\dot{q}_n}.\label{eq:p}

\end{align}

(My preference is to be fussy and always distinguish contravariant and covariant indices. Dirac used lower indices in all expressions.)

Dirac then moves on to **introduce**, what he calls, the **primary constraints** of the theory, given in the form

(3)\begin{align}

\phi_m(p^n,q_n)=0.\label{eq:constraint}

\end{align}

Dirac then defines the Hamiltonian the usual way, given by

\begin{align}

H=p^n\dot{q}_n-L,

\end{align}

where the Einstein summation convention ($x^iy_i\equiv\sum_ix^iy_i$) applies.

The variation of the Hamiltonian is given by

\begin{align}

\delta H=\delta p^n\dot{q}_n-\frac{\partial L}{\partial q_n}\delta q_n.

\end{align}

Dirac also observes that, as a consequence of the constraints $\phi_m$, the Hamiltonian is not uniquely determined; any linear combination of the $\phi_m$ can be added to $H$ to go over to another Hamiltonian,

\begin{align}

H^*=H+c^m\phi_m,

\end{align}

where the quantities $c_m$ can be functions of the $p$-s and $q$-s.

Inexplicably, however, Dirac next "deduces" (his word) the Hamiltonian equations of motion in the form,

(4)\begin{align}

\dot{q}_n&=\frac{\partial H}{\partial p^n}+u^m\frac{\partial\phi_m}{\partial p^n},\\

-\dot{p}^n=-\frac{\partial L}{\partial q_n}&=\frac{\partial H}{\partial q_n}+u^m\frac{\partial\phi_m}{\partial q_n},\label{eq:H2}

\end{align}

where the unknown coefficients $u_m$ are now *not* functions of the $p$-s and $q$-s. In the paragraph preceding these equations, Dirac actually indicates that the $p$-s and the $q$-s cannot be varied independently, because of the constraints (3).

Notice, however, that on the left-hand side of Eq. (4), Dirac uses the Euler-Lagrange equation (1), which was obtained from the *unconstrained* Lagrangian $L$. This is clearly self-contradictory: the system is either constrained by the $\phi_m$ or it isn't, but in either case, the constraints should be consistently applied (or not applied).

The Lagrangian of the *constrained* system is obtained by introducing the constraints in the form of Lagrange-multipliers. Specifically, we replace the Lagrangian $L(q_n,\dot{q}_n)$ with

\begin{align}

L^*(q_n,\dot{q}_n,\lambda^m)=L(q_n,\dot{q}_n)+\lambda^m\psi_m(q_n,\dot{q}_n),\label{eq:L*}

\end{align}

where the $\lambda_m$ are *not* functions of the $q$-s and $\dot{q}$-s, nor are they constants: they are new (nondynamical) degrees of freedom. Varying $L^*$ with respect to the $\lambda^m$ yields the constraint equations

\begin{align}

\psi_m(q_n,\dot{q}_n)=0,

\end{align}

while variation with respect to the $q$-s yields

\begin{align}

\frac{d}{dt}\left(\frac{\partial L^*}{\partial\dot{q}_n}\right)=\frac{\partial L^*}{\partial q_n}.\label{eq:EL*}

\end{align}

Although we have a new Lagrangian, it does not follow automatically that we can perform a Legendre transformation unambiguously and obtain a corresponding new Hamiltonian. However, we can express the modified action $S^*=\int dt L^*$ using the *original* Hamiltonian and canonical momenta, which are defined through a Legendre transformation of $L$ and their existence is not dependent on the Euler-Lagrange equations (1):

\begin{align}

S^*=\int~dt\left[p^n\dot{q}_n-H+\lambda^m\psi_m\right].

\end{align}

We note that Eq. (2) can always be solved for the $\dot{q}_n$:

\begin{align}

\dot{q}_n=v_n(p^n,q_n),

\end{align}

provided the Hessian metric,

\begin{align}

H^{ij}=\frac{\partial^2L}{\partial\dot{q}_i\partial\dot{q}_j},

\end{align}

is non-singular [2]. This allows us to write

\begin{align}

\psi_m(q_n,\dot{q}_n)=\psi_m(q_n,v_n(p^n,q_n))=\phi_m(p^n,q_n),

\end{align}

and write the action in the form,

\begin{align}

S^*=\int~dt\left[p^n\dot{q}_n-H+\lambda^m\phi_m\right].

\end{align}

The variation of this action is given by

\begin{align}

\delta S^*=\int&~dt\left[\delta p^n\dot{q}_n+p^n\delta\dot{q}_n-\frac{\partial H}{\partial p^n}\delta p^n-\frac{\partial H}{\partial q_n}\delta q_n+\delta\lambda^m\psi_m+\lambda^m\frac{\partial \phi_m}{\partial q_n}\delta q_n+\lambda^m\frac{\partial \phi_m}{\partial p^n}\delta p^n\right].\nonumber

\end{align}

Proceeding the usual way, we observe that as the operators $d/dt$ and $\delta$ commute, we can eliminate $\delta\dot{q}$ at the cost of introducing a surface term, which does not contribute to the variation of $S^*$:

\begin{align}

p^n\delta\dot{q}_n=p^n\frac{d}{dt}\delta q_n=\frac{d}{dt}(p^n\delta q_n)-\dot{p}^n\delta q_n,

\end{align}

leading us to

\begin{align}

\delta S^*=\int&~dt\left[\delta p^n\dot{q}_n-\dot{p}^n\delta q_n-\frac{\partial H}{\partial p^n}\delta p^n-\frac{\partial H}{\partial q_n}\delta q_n+\delta\lambda^m\psi_m+\lambda^m\frac{\partial \phi_m}{\partial q_n}\delta q_n+\lambda^m\frac{\partial \phi_m}{\partial p^n}\delta p^n\right],\nonumber

\end{align}

or, after collecting like terms,

\begin{align}

\delta S^*=\int~dt&\left\{\delta p^n\left[\dot{q}_n-\frac{\partial H}{\partial p^n}+\lambda^m\frac{\partial \phi_m}{\partial p^n}\right]-\left[\dot{p}^n+\frac{\partial H}{\partial q_n}-\lambda^m\frac{\partial \phi_m}{\partial q_n}\right]\delta q_n+\delta\lambda^m\psi_m\right\},\nonumber

\end{align}

leading to the Hamiltonian equations of motion,

\begin{align}

\dot{q}_n&=\frac{\partial H}{\partial p^n}-\lambda^m\frac{\partial \phi_m}{\partial p^n},\\

-\dot{p}^n&=\frac{\partial H}{\partial q_n}-\lambda^m\frac{\partial \phi_m}{\partial q_n},\\

\phi_m(p^n,q_n)&=0.

\end{align}

These are the same equations (after equating $\lambda^m=-u^m$) that Dirac "deduced". Clearly, Dirac's instincts were in the right place (unsurprisingly, given his reputation.) However, we obtained this result without resorting to ill-defined concepts such as "strong" and "weak" equations, and without any heuristic appeals to deduction, using rigorous mathematics instead. Furthermore, the role of the $\lambda_m$ as Lagrange-multipliers is now clear, and no use is made of the unconstrained Euler-Lagrange equation (1), which is clearly not satisfied in the constrained system.

[1] P. A. M. Dirac, *Lectures on Quantum Mechanics* (Dover Publications, 2001).

[2] H. Kleinert, *Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets* (World Scientific, 2006), 4th ed.