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.