Best source I ever had for this is a site I stumbled upon over a decade ago, but was too stupid to properly bookmark, so you'll have to settle for my memory. Bear in mind that for the next part little concrete proof exists, besides common sense, vague descriptions (since outdated by the time we have surviving letters/memoirs below general level), and careful extrapolation from LARPing as closest thing to "living archeology" for this kind of thing.
In any case, we do know for sure that humans simply cannot maintain high levels of physical effort (which fighting in full gear definitely counts as) for too long without risking total exhaustion, which is the last state you want to be in on a battlefield, as you will always need more energy later in the day. Battles also lasted for hours (sometimes even days), ergo, there must have been regular pauses of varying length, likely on the level of tactical units. i.e. century, more likely cohort, but not file (too small and vulnerable) or legion (larger than needed, temporarily removes all pressure from the enemy over a substantial part of the frontline).
Essentially, just before contact, your trot/jog charge (any faster is both bloody exhausting and extremely bad for unit cohesion), slows down a bit, likely even unconciously so, as neither man nor horse likes to impale himself on dangerous looking pointy sticks. So instead you start poking at each other from as far as the reach of your melee weapons allows. You are stressed, trying to avoid getting stabbed more than succeeding in stabbing the other guy, wearing weapons and armour, likely overdressed for the time of year, and thus almost guaranteed to be sweating like a pig. Hence not too much casualties at this point just yet, assuming equal equipment levels and the presence of shields/armour. This is the part where one would bring up the push of pike, or its ancient Greek counterpart, othismos, but there is no real concensus as to quite how literally the 'push' part is to be taken, though it is worth noting that deeper files definitely had an impact (ask the Spartans at Leuctra).
After 10-20 minutes, everyone will be in need of a breathing pause and wanting for a drink. As both sides consist of humans, this does not give an advantage to either side. Also, at the beginning of the battle, you are in no hurry to die for your country, and there is still plenty of time and opportunity to get those on the other side to do so for theirs. Even better, if you are lucky, the cohort/regiment/lochos next to yours will start routing its opponent, perhaps causing your opponent, now with exposed flank, to start running as well without you needing to do much more than just be present and look vaguely threatening.
So, with everyone both tired and not in a hurry to finish things, one or both sides simply slowly starts backing up a little (we're talking dozens of meters at most here) and the other side, grateful for the short reprieve, lets it happen (or maybe not quite yet, but if so you will quickly notice and 5-10min later they might be the ones to back up for a breather instead).
This gives the opportunity to evacuate a few wounded if both sides care about that kind of thing, swap out some broken spears for fresh ones, sending some wounded to the rear, have water pourches brought up from the rear etc... Then after some time you start all over again. Or perhaps a few individuals will come out between the lines and start taunting the enemy, or even try to take a stab at the enemy line (there are accounts of this in the Napoleonic wars, despite the danger of musket fire).