We were given the commandment to take the Lulav and Esrog so that we would have a reminder during Sukkos that the happiness we feel on the holiday is to be directed towards Hashem. The Sefer HaChinuch explains that on other holidays, we do not need such a reminder because of the various commandments associated with the day. One example is by Passover, because we have the commandments to eat Matzo and Maror, no further commandment is needed to assure that the focus of our happiness on the day is directed towards Hashem. The Sefer HaChinuch adds that by the holiday of Shmini Atzeres (which is the eighth day of Sukkos), although we have no special commandment on the day, we do not need anything to focus our happiness on Hashem. Why is this the case?
In Vayikra 23:36, the Torah tells us "...on the eight day, you will have a holy day...it is a day of detention ["Atzeres"]...." The Sefer HaChinuch writes that the Sages have told us that in reality, Shmini Atzeres is not the eighth day of Sukkos, but rather a separate holiday, which occurs at the end of the Sukkos holiday. What was the reason why Hashem commanded us to observe this day as a holiday? The Sefer HaChinuch explains that Sukkos is the last holiday of the Jewish calendar [if we consider the month of Nissan as the beginning of the year, and Pesach the first holiday]. By giving us this holiday of Shmini Atzeres, its as if Hashem was telling us "Please, stay with me one more day, as it is difficult for me to part with you." Hashem, out of his great love for the nation of Israel, finds it hard to "leave" his children at the end of the holidays. Therefore, He added on one more day, so we, the nation of Israel, can spend one more day with Hashem before our "departure."
Now that we understand the spirit behind Shmini Atzeres, we can also understand why we need no added commandment to assure that our joy stays properly focused. As the whole essence of the holiday is that we are spending a little more time with Hashem before we "leave" (as we will not be "returning" until Pesach), our mind will be focused on Hashem throughout the holiday. No added reminder is necessary, as there is nothing to distract us from remembering our close binds to Hashem on this day.