Variation

The $\Delta$-variation is a variation in which the varied path over which an integral is evaluated may end at different times than the correct path, and there may be variation in the coordinates at the endpoints.

The $\delta$-variation is a variation in which the varied path in configuration space terminates at the endpoints representing the system configuration at the same time $t_1$ and $t_2$ as the correct path; i.e., the varied path always returns to the same endpoints in configuration space, so

$$\delta q_i(t_1) = \delta q_i(t_2) = 0.$$

See also Calculus of Variations, Variation of Argument, Variation of Parameters