Since when do Moshe's hands make or break war?
Rashi (citing the Mishnah in Rosh Hashanah (29a): It was when Moshe raised his hands that the people raised their eyes heavenwards and rendered their hearts subservient to Hashem. That was when they were victorious in battle, and when he dropped his hands...
Rashbam: Moshe hand refers to the staff that he was holding; and it is human nature that when the army see their emblem raised, they fight harder and enter into a winning streak.
Targum Yonasan: When Moshe raised his hands in prayer, Yisrael were winning, and when he lowered them and stopped praying, then Amalek were winning.
Rashi, citing Rosh Hashanah, asked why the war depended on Moshe's hands. Is it difficult if Moshe's prayer helps to overpower Amalek?
Lev Eliyahu (p.96): This war was not based on physical strength, rather, Ruchniyus. Amalek sensed that Yisrael slackened from the Torah, and came 400 Parsa'os to Refidim through witchcraft. Therefore, Yisrael needed to consecrate their hearts to Hashem Sichon and Og were stronger, but those were physical wars; Moshe alone 1 was able to defeat them.