# Dirichlet boundary conditions in space-time?

In the context of string theory, and world sheets the Dirichlet boundary conditions can be written as: $$\frac{\partial X^\mu(\tau,\sigma_1)}{\partial \tau}=0$$ where $\sigma_1$ is the value of the parameter $\sigma$ at the end of the 'string'. This however, seems to imply that $$\delta X^\mu(\tau,\sigma_1)=0$$ But I cannot see why, so please can you explain?

Here are my thoughts (which are wrong since I get the wrong outcome):

It is my assumption that $\delta X^\mu \equiv dX^\mu$ in this context although I could be wrong. This therefore means that: $$\delta X^\mu(\tau,\sigma_1) =\frac{\partial X^\mu(\tau,\sigma_1)}{\partial \tau}d\tau + \frac{\partial X^\mu(\tau,\sigma_1)}{\partial \sigma}d\sigma$$ So subbing in my first equation we get: $$\delta X^\mu (\tau,\sigma_1)=\frac{\partial X^\mu(\tau,\sigma_1)}{\partial \sigma}d\sigma$$ which is generally not equal to $0$. Thus my first equation does not necessary imply my second, as it should.

References:

1. A first course in string theory by Barton Zwiebach (2 e.d.) pg 114
2. http://www.damtp.cam.ac.uk/user/tong/string/three.pdf

This seems to be a simple matter of confusion regarding which variables are held constant. His notation $$\frac{\partial X}{\partial\tau}(\tau,\sigma_*)=0$$ is misleading. What he really means is $$\frac{\partial}{\partial\tau}\left(X(\sigma_*)\right)(\tau)=0$$ In other words, we fix $\sigma$ to be one of the end points and look at how it changes with respect to $\tau$. Your expansion above is correct, but you must take $\delta \sigma=0$ because we focus attention on one value of $\sigma$, namely an end point.
Your assumption that $$\delta X^{\mu} = dX^{\mu}$$ as well as the expansion is not correct. $$\delta X^{\mu}$$ is a variation and you can not rwright variation of a parameter (the functional derivative relates a change in a functional to a change in a function on which the functional depends).
$$$$\delta X^{\mu} = X^{'\mu}(\tau,\sigma^{*}) - X^{\mu}(\tau,\sigma^{*})$$$$ whereas $$dX^{\mu}$$ is an exterior derivative. $$$$dX^{\mu}(\tau, \sigma) = \frac{\partial X^{\mu}}{\partial\tau}d\tau + \frac{\partial X^{\mu}}{\partial\sigma}d\sigma$$$$ This means that the condition(1) that $$\frac{\partial X^{\mu}(\tau, \sigma^{*})}{\partial\tau} = 0$$ is only true when $$X^{\mu}(\tau,\sigma^{*}) = constant$$ that is for static D branes, So this is a sufficient condition not necessary. However, $$X^{\mu}(\tau,\sigma^{*}) = f(\tau)$$, which means that the D brane can be dynamical then only that $$\delta X^{\mu} = 0$$ is satisfied.
Dirichlet Boundary Condition is $$\delta X^{\mu}(\tau,\sigma^{*}) = 0$$ but for a special case when the brane is static, we can write it as $$\frac{\partial X^{\mu}(\tau, \sigma^{*})}{\partial\tau} = 0$$ .