The Diff_Eq twitter feed had a link the other day to a fine question put on StackExchange.com: What’s the Point of Hamiltonian Mechanics? Hamiltonian Mechanics is a different way of expressing the laws of Newtonian mechanics from what you learn in high school, and what you learn from that F equals m a business, and it gets introduced in the Mechanics course you take early on as a physics major.
At this level of physics you’re mostly concerned with, well, the motion of falling balls, of masses hung on springs, of pendulums swinging back and forth, of satellites orbiting planets. This is all nice tangible stuff and you can work problems out pretty well if you know all the forces the moving things exert on one another, forming a lot of equations that tell you how the particles are accelerating, from which you can get how the velocities are changing, from which you can get how the positions are changing.
The Hamiltonian formation starts out looking like it’s making life harder, because instead of looking just at the positions of particles, it looks at both the positions and the momentums (which is the product of the mass and the velocity). However, instead of looking at the forces, particularly, you look at the energy in the system, which typically is going to be the kinetic energy plus the potential energy. The energy is a nice thing to look at, because it’s got some obvious physical meaning, and you should know how it changes over time, and because it’s just a number (a scalar, in the trade) instead of a vector, the way forces are.
And here’s a neat thing: the way the position changes over time is found by looking at how the energy would change if you made a little change in the momentum; and the way the momentum changes over time is found by looking at how the energy would change if you made a little change in the position. As that sentence suggests, that’s awfully pretty; there’s something aesthetically compelling about treating position and momentum so very similarly. (They’re not treated in exactly the same way, but it’s close enough.) And writing the mechanics problem this way, as position and momentum changing in time, means we can use tools that come from linear algebra and the study of matrices to answer big questions like whether the way the system moves is stable, which are hard to answer otherwise.
The questioner who started the StackExchange discussion pointed out that before they get to Hamiltonian mechanics, the course also introduced the Euler-Lagrange formation, which looks a lot like the Hamiltonian, and which was developed first, and gets introduced to students first; why not use that? Here I have to side with most of the commenters about the Hamiltonian turning out to be more useful when you go on to more advanced physics. The Euler-Lagrange form is neat, and particularly liberating because you get an incredible freedom in how you set up the coordinates describing the action of your system. But it doesn’t have that same symmetry in treating the position and momentum, and you don’t get the energy of the system built right into the equations you’re writing, and you can’t use the linear algebra and matrix tools that were introduced. Mostly, the good things that Euler-Lagrange forms give you, such as making it obvious when a particular coordinate doesn’t actually contribute to the behavior of the system, or letting you look at energy instead of forces, the Hamiltonian also gives you, and the Hamiltonian can be used to do more later on.