  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 $p_{i}$, $i=1,\...,n_{p}$
(17) is arbitrary $n_{p}\leq n$, where $n$ is the dimension of the
configuration space), in terms of the partial Hamiltonian $H_{0}$ (18) and
$(n-n_{p})$ additional Hamiltonians $H_{\alpha}$, $\alpha=n_{p}+1,\...,n$ (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 $r_{W}\leq n_{p}$ (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.
