hong said:
Well, in that case, I guess there's no reason not to allow you to drop a quarterstaff after a failed trip attempt. 
And yet the core rules don't allow it. Otherwise they would specifically mention it as they do for every single weapon that it notes as being used in a trip attempt.
If it was intended to be something you could do with every weapon, it would be mentioned in the trip description. Instead the ability to drop the weapon is only mentioned for specific weapons. Logically this means that only those weapons can be dropped on a failed trip attempt, and this very strongly indicates that only those weapons can be used to trip people, under the core rules.
Look at the flail and guisarme descriptions on page 100 and 101 of the PHB. It specifically states for the flail: "You can use this weapon to make trip attacks". For the guisarme: "because of the guisarme's curved blade, you can also use it to make trip attacks."
If any weapon can be used to make a trip attack, why is it specifically spelled out for these weapons (and for every other trip weapon)?
You can sweep someone's legs out from under them with any polearm. Or you can just give them a solid shove in the chest, and knock them down that way. Or whatever; many fighting styles teach techniques for knocking a person down.
In the real world, yes. In D&D, it seems to require a weapon with sharp curve or hook (guisarme), or a flexible weapon (chain, whip, flail) that can wrap around your limb.
It's a bad assumption then.
I'm not saying it isn't. I'm just saying that's what the core rules are using.
And I'm saying that by the core rules, there's nothing to stop you from tripping with any weapon, and certain weapons let you do certain things as part of that.
I'm pretty sure that is not the case.
You may disagree with the core rules on this issue, and I'm not saying your wrong to disagree. This appears to be one of those areas where they oversimplified things. Since you more knowledgable about tripping techniques, the holes in the rules (for this area) are more obvious to you.
I feel the same way when I try to make a fighter who can use "real" fencing techniques. (At best you can do a crude approximation, the D&D combat system is too abstract to allow you to realistically simulate the specific maneuvers used in real fencing.)