The canonical formalism

In order to formulate the Hamiltonian formalism, it is necessary to iimpose some restrictions on the choice of $\phi_{m}$. This is because $H = {\dot{q}}_{n}p_{n}-L$ is not uniquely determined as a function of the $q$ and $p$, because of the constraint relations $\phi_{m}$ $\approx$ 0, which are identities when the momenta are expressed as functions of $q_{m}$ and $\dot{q}_{m}$ via $p_{m}=\partial L/\partial \dot{q}_{m}$. The Hamiltonian is thus only well defined on the constraint manifold. This ambiguity implies that the formalism remains invariant under

${\hspace{5cm}}$ $H$ ${\hspace{5mm}}$and ${\hspace{5mm}}$ $H$ + $c^m\phi_{m}$,

where the Lagrange multiplicators $c^m$ are some arbitrary functions. The phase space action is then

$$S = \int_{t_{1}}^{t_{2}}(p_{n}\dot{q}_{n}-H- c^{m}\phi_{m})=0$$ (42)

and the preservation in time of the constraints $\phi_{m}$ is expressed by

$${\dot{\phi}_{m}} = \{\phi_{m},H\} + c^{n}\{\phi_{m},\phi_{n}\} \approx 0,$$ (43)

which unless $\{\phi_{m},\phi_{n}\} \approx 0$ constitutes a restriction on the possible $c_{n}$. If there are also secondary constraints yet another set of restriction parameters must be iimposed.