Everything the others have been saying is great. I wanted to chime in because I also have a lazy horse (especially in an arena--she's got more pep on the trail, though she's still never in a hurry to get anywhere). My trainer doesn't want me to squeeze-squeeze-squeeze to get her to keep going. The horse will become dead to that cue, and you'll have to move to firmer and firmer cues if you do that. Instead, I need to teach my horse "responsibility."
The trainer helps me work with her on this at the lunge and riding, and at both the walk and trot (we're not worrying about the canter yet with my horse). If I ask her to walk, she needs to walk until I tell her otherwise. Same with trot. So when riding, if we're walking, I ask for a trot with my seat and a squeeze of my legs, and if she doesn't start trotting, I pop her with the whip. Not BEAT her with it, but a good whap that will cause her to re-think her decision. She trots. Great. No nagging, mission accomplished. We slow to a walk. I ask her to trot again, with seat and a squeeze of the legs. This time she decides to just go ahead and trot without the whip needing to get involved.
After a couple lessons of this, it usually isn't a problem to GET the trot anymore.
However, then we had to start working on maintaining the trot. The hardest part is that you have to let the horse make the mistake. If you feel them starting to slow down, and squeeze with your legs to make them speed up a bit, a green horse anyway won't necessarily associate those things, and you'll just have to start nagging them every time they slow down. So let them get all the way to a walk, then cue for the trot again, with the reinforcing pop of the whip if necessary. Repeat ad nauseum.
My trainer also works with us on posting properly. She's a centered riding instructor, so has all these great images she shares verbally for us to picture mentally while riding. The one that works for me and improves my posting the most is to not worry about pumping your butt up and down, but rather to think about pushing your knees (not even your feet, though that image works for some people) down toward the ground. Every time I think of it this way, my posting gets WAY more effortless, and less bouncy on the saddle, too. Who knows why. My instructor has also shown us, in clinics, the difference in posting by using an exercise ball. When you think "up down up down up down" while posting, you tend to be pushing the energy of the horse backward--the ball literally wants to squirt through your legs backwards, so if it's doing that to the ball, your horse is probably feeling like you WANT it to slow down. When you think "forward back" with your hips (it almost seems obscene doing these hip thrusts on an exercise ball, but it's much more natural on the real horse), and think of pushing your knees down, the ball has forward energy, and is much more likely to squirt forward between your knees instead of backward, so obviously the horse would have more forward energy, too. It's the weirdest thing, but really does seem to work.
Hopefully you have a trainer near you who can help, but try it out yourself--do a circle of your normal posting, then do a circle thinking up-down-up-down, then try a circle where you're thrusting your hips forward and back while posting, not worrying how high off the saddle you go, and think of pushing your knees down toward the ground. See if it helps!
Meanwhile, yeah, your legs WILL get tired and sore from posting--it does take muscles even if you're doing it right and the horse is mostly thrusting you upward/forward, but they shouldn't be AS sore if you're only posting, and not having to post AND squeeze to keep your horse moving.