Hamiltonian equations for agreeable Lagrangians

Let $L$ be a Lagrangian whose constraint set $\mathscr{C}$ is closed, and let $h_{\mathscr{C}}$ be the restriction to $\mathscr{C}$ of the Hamiltonian determined by $L$. The Lagrangian is agreeable if the restriction of $h_{\mathscr{C}}$ to every $\mathfrak{X}_{\mathcal{F}}$-orbit $O$ is smooth. For a first class function $f$, the reduced Poisson bracket is defined orbit by orbit by

$$\{f_{\mid \mathscr{C}},h_{\mathscr{C}}\}_{\mid O}=\{f_{\mid O},h_{\mathscr{C}\mid O}\}=-X_{f\mid O}h_{\mid O}.$$

Hamiltonian equations for agreeable Lagrangians. For an agreeable Lagrangian $L$, the Hamiltonian equations of motion for first class functions are given by

$$\dot{f}_{\mid \mathscr{C}}=\{f_{\mid \mathscr{C}},h_{\mathscr{C}}\},$$

where the Poisson bracket on the right-hand side is defined orbit by orbit as above.

The agreeability condition permits the reduced equations to retain the formal Poisson-bracket form even when the Hamiltonian has no smooth extension to the full cotangent bundle. The source states the equations in a conjecture environment, but the supplied text does not indicate whether they have been proved or refuted.