# Dirichlet boundary condtions in Nambu-Goto string action

The Nambu-Goto action for an open string with parameter domain $$[0,\tau_1]\times[0,\sigma_1]$$ is given by

$$$$S_{NG} = \int_{0}^{\tau_1} d\tau \int_{0}^{\sigma_1} \ d\sigma \ \mathcal{L}(\dot{X}, X') = \int_{0}^{\tau_1} d\tau \int_{0}^{\sigma_1} d\sigma \sqrt{(\dot{X} \cdot X')^2 - (\dot{X})^2(X')^2}$$$$

such that under a variation $$X^\mu \rightarrow X^\mu + \delta X^\mu$$ we have

$$$$\delta S_{NG} = \int_{0}^{\sigma_1} d\sigma \left[\delta X^\mu \mathcal{P}^\tau _\mu \right]_{0}^{\tau_1} + \int_{0}^{\tau_1} d\tau \left[ \delta X^\mu \mathcal{P}^\sigma _{\mu}\right]_{0}^{\sigma_1} + \int_{0}^{\tau_1} d\tau \int_{0}^{\sigma_1} d\sigma \ \delta X^\mu \partial_a \mathcal{P}^{a}_{\mu}$$$$

where $$\mathcal{P}^a _{\mu} := \frac{\partial \mathcal{L}}{\partial (\partial_a{X^\mu)}}, \ a= \tau, \sigma \ \text{and} \ \mu = 0,1,...,d.$$

The first boundary term can vanish if I impose B.C $$\delta X^\mu (0,\sigma) = \delta X^\mu (\tau_1, \sigma) = 0$$ for all $$\mu$$.

The second boundary term can vanish completely if I impose Neumann B.C : $$P^{\sigma}_{\mu}(\tau, 0) = P^{\sigma}_{\mu}(\tau, \sigma_1) = 0$$, and then if I use the action principle I can state that the EOM for this action are $$\partial_a \mathcal{P}^a_{\mu}= 0$$.

However, if I try to impose Dirichlet BC: $$\delta X^i (\tau, 0) = \delta X^i ( \tau ,\sigma_1) = 0$$ for $$i = 1,2,..., d$$ (because $$\delta X^0 (\tau,0) = 0$$ would imply that time would stop in string endpoints, which have no physical sense) I can no longer guarantee that the second boundary term vanishes. So Dirichlet BCs break the Poincaré invariance of the action. Even if the second Boundary term is not zero, is correct to state that the equations of motion for the string are $$\partial_a \mathcal{P}^a_{\mu} =0$$?

No, you must always force the second boundary term $$\int_{0}^{\tau_1} d\tau \left[ \delta X^\mu \mathcal{P}^\sigma _{\mu}\right]_{0}^{\sigma_1}$$ to vanish as well. In the context of $$D_p$$-branes for $$p \ge 0$$, this is done by enforcing the Neumann boundary condition for the variation in $$X^0 \dots X^q$$, and Dirichlet conditions for the remaining dimensions: $$X^{q+1}\dots X^p$$.
However, there also exists the $$D_{-1}$$ brane in which Dirichlet conditions are imposed for all the coordinates, and while the endpoints of the string are localised in time, this is not absurd as you seem to think - rather, they represent instanton solutions.