The time-dependent variational principle in the stationary action principle form is formulated with not only the normalization but also general constraint conditions for wavefunctions. The constraint conditions are classified into the first and the second classes as in Dirac's constrained classical-mechanics. The Lagrange multipliers for the second-class constraints are uniquely determined by the consistency conditions. The normalization condition, as the first-class constraint, is supplemented with an adequate gauge-fixing function. As one of the applications, the quantum dynamics in subspaces based on the TDVP is suggested.