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Just what the title asks. What are the applications of it?

John Gally
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  • Dear John, it's a fascinating interest if your age is what you wrote down. ;-) The Hamiltonian approach is clearly a "tool of the adults" but I am sure that soon-to-be-teenagers may in principle get it, too. First, the "Hamiltonian" itself is a fancy name for energy. It's a function of positions and momenta expressed in joules that is conserved. The conservation is useful. You know energy at one moment, so you may quickly say it's the same at a different moment and use this fact to quickly calculate the speed or something else. – Luboš Motl Nov 17 '15 at 07:47
  • Second, the energy is just one number, so its knowledge isn't enough to calculate positions and speeds of many particles, for example. However, the Hamiltonian approach to mechanics is such that from the form of the energy itself, you may calculate how positions and momenta (or other "dynamical variables") are changing with time. The equations for the acceleration as a function of position etc. may be deduced from the form of the energy - from the form of the Hamiltonian. And it's the main point of the Hamiltonian mechanics - deduce everything about evolution from the formula for energy. – Luboš Motl Nov 17 '15 at 07:48
  • Third, while you could just write the equations directly, like F=ma, and never talk about the energy H when you do just mechanics and motion of planets etc., the quantity H becomes superuseful in the treatment of "statistical physics", the microscopic explanation of thermodynamics (phenomena with heat and temperature), as well as in quantum mechanics. In statistical physics, the probability that a particle gets to very high heights or speeds by random motion is expressed as exp(-H/kT) where H is the Hamiltonian/energy and T is the absolute temperature. – Luboš Motl Nov 17 '15 at 07:50
  • So the probabilities are always easily calculated from the form of the energy. Something similar is true in quantum mechanics - a more fancy modern form of mechanics where particles propagate according to "probability waves" around them and predictions are probabilistic. The equations for the "evolution in time" of the basic quantities in quantum mechanics, either the wave function or the operators, are only easily written in terms of the Hamiltonian. In the case of the wave function, it's not quite possible to avoid the Hamiltonian at all. – Luboš Motl Nov 17 '15 at 07:51
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    haha, thank you. Despite already studying a lot of higher algebra at my age, I'm still having difficulties communicating my mathematical idea to people around me and I almost have never been appreciated by anyone. – John Gally Nov 17 '15 at 11:06
  • @LubošMotl I'm sure you already know that Dimensio1n0 was only 13 when he first joined here, yet was answering questions on the QED Lagrangian! – Physiks lover Nov 27 '15 at 23:32
  • Hi @Physikslover - I wasn't quite aware of his age, it's fun if he was 13. – Luboš Motl Nov 29 '15 at 08:18
  • Well that means I'm outclassed then. Oh well I guess some people are just simply smarter than others then. – John Gally Nov 29 '15 at 14:59

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