Partial Hamiltonian formalism, multi-time dynamics and singular theories

We formulate singular classical theories without involving constraints. Applying the action principle for the action (27) we develop a partial (in the sense that not all velocities are transformed to momenta) Hamiltonian formalism in the initially reduced phase space (with the canonical coordinates $q_i,p_i$, where the number $n_p$ of momenta $p_i$, $i=1,\dot{\mathrm{.}}\mathrm{.},n_p$ (17) is arbitrary $n_p\le n$, where $n$ is the dimension of the configuration space), in terms of the partial Hamiltonian $H_0$ (18) and $(n-n_p)$ additional Hamiltonians $H_{\alpha }$, $\alpha =n_p+1,\dot{\mathrm{.}}\mathrm{.},n$ (20). We obtain $(n-n_p+1)$ Hamilton-Jacobi equations (25)-(26). The equations of motion are first order differential equations (33)-(34) with respect to $q_i,p_i$ and second order differential equations (35) for $q_{\alpha }$. If $H_0$, $H_{\alpha }$ do not depend on ${\dot{q}}_{\alpha }$ (42), then the second order differential equations (35) become algebraic equations (43) with respect to ${\dot{q}}_{\alpha }$. We interpret $q_{\alpha }$ as additional times by (45), and arrive at a multi-time dynamics. The above independence is satisfied in singular theories and $r_W\le n_p$ (58), where $r_W$ is the Hessian rank. If $n_p=r_W$, then there are no constraints. A classification of the singular theories is given by analyzing system (62) in terms of $F_{\alpha \beta }$ (63). If its rank is full, then we can solve the system (62); if not, some of ${\dot{q}}_{\alpha }$ remain arbitrary (sign of a gauge theory). We define new antisymmetric brackets (69) and (80) and present the equations of motion in the Hamilton-like form, (67)-(68) and (81)-(82) respectively. The origin of the Dirac constraints in our framework is shown: if we define extra momenta $p_{\alpha }$ by (86), then we obtain the standard primary constraints (87), and the new brackets transform to the Dirac bracket. Quantization is discussed.