What were you trying to do when maneuvering your tactical divisions of men in battle?
If it is to "gain an advantageous position relative to your enemy," particularly what problems / factors were you trying to alleviate upon your tactical maneuvers?
When looking at classic battles, why do the roles of divisions of men differ by their degree of mobility?
Would tactical maneuvers even exist if the divisions of men possessed equal speed?
Are the variations of tactical maneuvers few and fairly simple?
Flanking, target, terrain, and disorder were the four main goals when one ancient army unit maneuvered against another. But achieving any of those was pretty hard: formation infantry is hard to move, and battles are chaotic. Winning armies were not so much the ones with complicated maneuvers, as the ones with enough experience and leadership to pull off the right basic maneuvers for the battlefield they were on.
Flanking: most army formations are strong to the front, not to the side or rear. Whether it's a Macedonian phalanx's hedgehog of spears or a Roman cohort's line of swords and shields, it's much stronger in front than on the side. (Later, in the Renaissance, you start to see infantry 'squares', but those have their own challenges.) On the side, soldiers don't have a solid wall of interlocking weapons and shields to protect them, and turning an entire formation to the side is much harder than turning a single individual. So if you can flank an enemy, if you can get your unit's front cutting into the enemy's unit's side, that's a big advantage.
Target: sometimes a particular enemy unit is especially valuable or vulnerable, and you'd like to get your strongest forces up against them. Alexander the Great won some battles by breaking through with his best troops to the enemy king, making the king retreat -- and then the entire enemy army, seeing their king run, also broke and ran. More common were battles where one of the enemy's units was less skilled or more nervous, and the commander might target that unit in hopes that those soldiers would break and run and undermine their entire force.
Terrain: sometimes the enemy can be trapped into fighting in a narrow space where they can't use their numbers, or lured into fighting going uphill against you or with the wind or sun in their eyes.
Disorder: once the battle starts, everybody's neat formations begin to break up, and one unit or another may start to lose courage or strength and retreat. At that moment, the commander might send their unused reserve units to shore up their weak points or hammer into the enemy's weak points. (Alternatively, a unit may fake a retreat to bait the enemy into a disordered pursuit, then turn on them or lead them into ambush. But this tactic seems to have been more common in cavalry-rich armies like the Mongols than the Greek and Roman forces.) When an army's line is broken apart, the individual units are far more vulnerable to flanking (or outright envelopment); when one army unit is driven into retreat, the other units become much more likely themselves to flee.
Because of the need to keep in formation, all these maneuvers, as simple as they sound, were terribly hard for amateur troops and generals to pull off on the field of battle. As Clausewitz says, "Everything is very simple in war, but the simplest thing is difficult."
Non-formation units or mounted units could move much faster, at the price of being less able to defend a patch of ground under pressure. Different battlefields and armies could offer radically different chances for good tactics, but only a good general could tell what options their forces could actually pull off. Above all, more experienced soldiers under more experienced officers could do far more maneuvering with the same or similar equipment than amateur soldiers under amateur officers.