Hamiltonian format of Pontryagin's maximum principle

1. Hamiltonian format is native for the maximum principle regardless of any special restrictions imposed on the control equations of the optimal problem under consideration. It assigns canonically to any optimal problem a Hamiltonian system with parameters, complemented with the maximum condition, which “dynamically” eliminates the parameters in the process of solving the initial value problem for the Hamiltonian system as we proceed along the trajectory. Thus the extremals of the problem are generated as simultaneous solutions of a regular Hamiltonian system with parameters and the maximum condition (to which all singularities of the problem are “relegated”), and not as a Hamiltonian flow, i.e. as a family of solutions of an initial value problem of a Hamiltonian system (without parameters), to which the Euler–Lagrange equation could be reduced in the regular case of the classical calculus of variations.
2. I shall give here an invariant formulation of the Hamiltonian format of the maximum principle for the time-optimal problem.
Let the control system be given by the equation
$$ \frac{dx}{dt}=X(x,u)=X, \qquad u\in U, $$
where $X$ is a vector field on the configuration space $M$, $x\in M$, $u$ is the control parameter, $U$ is the set of admissible values of $u$.
With the vector field $X$ we canonically associate a scalar-valued fiberwise linear function $H_X$ on the cotangent bundle $T^*M$,
$$ H_X(\xi,u)\overset{\text{def}}=\langle\xi,X(\pi\xi,u)\rangle, \qquad \xi\in T^*M, \quad u\in U, $$
where $\pi\colon T^*M\to M$ is the canonical projection. Hence a family of Hamiltonian vector fields $\overrightarrow H_X$ defined by the family of Hamiltonians $H_X$ canonically corresponds to the time-optimal problem. Every initial value $\xi\in T^*M$, $\xi\notin M$, defines an extremal of the time optimal problem as a trajectory $\xi(t)$, $\xi(0)=\xi$, of the Hamiltonian vector field $\overrightarrow H_X$, from which the parameter $u\in U$ is “dynamically” eliminated by the maximum condition
$$ H_X(\xi(t),u(t))=\max_{v\in U}H_X(\xi(t),v). $$
Every solution of the optimal problem could be obtained in the described way.
3. Since the time-optimal problem is completely defined by the vector field $X(x,u)$, it is natural to expect that every first order “infinitesimal object” invariantly connected with our problem should be canonically (tensorially) expressed through the differential of the flow $e^{tX}$. To give this expression in our case, we first remark that the Hamiltonian vector field $\overrightarrow H_X$ coincides with the vector field $\operatorname{ad}_X$ on the cotangent bundle $T^*M$, which is uniquely defined by the equations
$$ \operatorname{ad}_Xa=Xa \quad \forall\,a\in C^\infty(M), \qquad \operatorname{ad}_XY=[X,Y] \quad \forall\,Y\in\operatorname{Vect}M. $$
Let $\mathcal L_X$ be the Lie derivative over the field $X$, $e^{t\mathcal L_X}=e_*^{tX}$, where $e_*^{tX}$ is the differential of the flow $e^{tX}$ on $M$.
According to the existing duality between the flows $e^{t\mathcal L_X}$ and $e^{t\operatorname{ad}_X}$ expressed by the identity
$$ e^{tX}\langle\omega,X\rangle=\langle e^{t\mathcal L_X}\omega,e^{t\operatorname{ad}_X}Y\rangle \quad \forall\,Y\in\operatorname{Vect}M, \quad \omega\in\Lambda^{(1)}(M), $$
we have
$$ e^{t\operatorname{ad}_X}=(e^{-t\mathcal L_X})^*=(e^{t\mathcal L_X})^{*{-1}}. $$
Hence the flow generated by the Hamiltonian vector field $\overrightarrow H_X=\operatorname{ad}_X$ is inverse to the conjugate of the differential $e_*^{tX}=e^{t\mathcal L_X}$, in particular, it is a bundle isomorphism of the cotangent bundle $T^*M$ over the flow $e^{tX}$ for $\forall\,t$, and the vector field $\overrightarrow H_X$ is a “Hamiltonian lift” of the vector field $X$.
Differentiating the above identity with respect to $t$ we establish the “infinitesimal” duality between $\mathcal L_X$ and $\operatorname{ad}_X$ (the generalized Leibnitz rule),
$$ X\langle\omega,Y\rangle=\langle\mathcal L_X\omega,Y\rangle+\langle\omega,\operatorname{ad}_XY\rangle \qquad \forall\,Y\in\operatorname{Vect}M, \quad \omega\in\Lambda^{(1)}(M). $$
The indicated relations completely identify the Hamiltonian vector field $\overrightarrow H_X$, hence the Hamiltonian format of the maximum principle.
4. Whereas the Lie derivative $\mathcal L_X$ and the flow it generates on the tangent bundle $TM$, the differential $e_*^{tX}=e^{t\mathcal L_X}$, are, in one or another way, the objects of everyday mathematical practice, the dual vector field to $\mathcal L_X$, the Hamiltonian lift $\overrightarrow H_X=\operatorname{ad}_X$ on the cotangent bundle $T^*M$, and the corresponding flow $e^{t\operatorname{ad}_X}$ were first introduced for computational purposes only in 1956 by L. S. Pontryagin under the name of “conjugate system”, and through the discovery of the maximum principle became since then a standard computational tool in engineering practice. Today, they are absolutely inevitable in optimization problems related to trajectory variations.
I think, it would be historically justified to baptize the vector field $\operatorname{ad}_X$, (considered precisely as a vector field on $T^*M$, and not as a derivation on the $C^\infty(M)$-module of vector fields on $M$), as the Pontryagin derivative.