For myself and the thousands of other people that simply golf for fun, one of the most frustrating parts of the game is putting. It seems like it should be so easy to line up a putt and hit it in the hole, but it is truly one of the hardest and most important part of the game. Not only is it important, it's also far different from any other aspect of the game, and this can be seen in the physics of the putt.
For starters, basically any other good shot in golf is hit in a projectile motion. The ball is hit high in the air and comes down much further down the fairway or the green or in the rough, depending on how well the ball is struck. Putting is different because the ball generally doesn't leave the ground, although it could for a split second on much longer putts with more force applied to it. The reason the putter doesn't hit the ball into the air the same way that all the other clubs in golf do is because the putter head is a nearly flat surface where it makes contact with the ball, although they usually have a very slight angle to them, while the heads of any other club are angled a great amount in order to get the necessary loft on the ball.
When the golf ball is rolling, it is a sphere rolling on a flat surface, which means it is impacted by the coefficient of static friction rather than kinetic friction. In golf, there are many different types of greens. On some the ball travels very fast, some it travels very slow, some have many different slopes scattered over it and some are flat. The fast greens obviously have a lower coefficient of friction acting on the ball, and the opposite for a slow green. A majority of putting greens are not completely flat in order to make putting more difficult. This forces a golfer to voluntarily hit the ball up a slope in order to get it to go on the proper path to the hole.
Once the ball reaches the hole, the ball must be traveling at a low enough velocity in order for it to be angled properly once it goes over the hole in order to go in. If the ball is still traveling at a high velocity once it reaches the hole, the ball will likely only slightly go in the hole and hit the back lip and pop into the air, or roll right over the hole as if it wasn't even there. If it is traveling slow enough, it will be angled properly in order to fall into the bottom of the hole.
Even looking at the physics of a putt it doesn't seem too difficult, but this doesn't account for how hard it can be to line up the ball with the right path for the hole, and how challenging it is to apply the proper force to the ball in order get it moving the proper velocity. It's pretty darn hard.