![]() Also, Hamiltonian mechanics, with the concept of phase space, lends itself really well for chaos theory, where we may want to analyze whether a complicated dynamical system is chaotic or not.This can be done using perturbation theory and the Hamiltonian formulation. An example of this is for computing relativistic corrections to planetary orbits, such as the perihelion precession.In cases where the equations of motion for a system are too complicated to solve analytically, perturbation theory can be used to find approximate solutions and this is often easiest done using the Hamiltonian formalism.For this, Hamiltonian mechanics provides some interesting insights, which we’ll look at later in this article. An example of this is for understanding the relation between conservative systems and the maximization of entropy. ![]() Therefore, many concepts in statistical mechanics require Hamiltonian mechanics to be understood properly. ![]() Statistical mechanics heavily relies on the notion of phase space, which is a central idea in Hamiltonian mechanics.Thermodynamics and statistical mechanics: There are also quite a lot of important applications and uses for Hamiltonian mechanics (which Newtonian or Lagrangian mechanics are not as well suited for), especially in other areas of physics.īelow, I’ve included a list of various applications of the Hamiltonian formulation to other areas of physics. In Hamiltonian mechanics, the way a system changes with time can be interpreted geometrically as a flow in phase space (which I’ll explain later) and this geometric notion has a lot of really deep consequences to the nature of physics in general. The real importance and beauty of Hamiltonian mechanics comes from its geometric structure, which simply does not exist in the Lagrangian formulation. However, the importance of Hamiltonian mechanics is not so obvious as it can essentially be used to solve mechanics problems just as well as Lagrangian mechanics. In many cases, the benefits of the Lagrangian formulation are quite clear (which you can read more about in this article). I also have this article that compares Lagrangian and Hamiltonian mechanics and covers the main differences, advantages and applications of both formulations.īefore we get started on the actual details of the Hamiltonian formulation, I think it’s important to make explicitly clear why exactly you would want to learn and even consider Hamiltonian mechanics. ![]() Hamiltonian mechanics does have a little bit to do with Lagrangian mechanics, so I’d recommend reading this introductory article first if you don’t know anything about the Lagrangian formulation. The context of this article is more about what Hamiltonian mechanics means in classical mechanics, although I will also give some insights about Hamiltonian mechanics and its significance to other areas in physics. Hamiltonian mechanics also has its own advantages and applications in more advanced physics, such as in quantum mechanics and quantum field theory. Hamiltonian mechanics is based on the Lagrangian formulation and is also equivalent to Newtonian mechanics.Įven though Newtonian, Lagrangian and Hamiltonian mechanics are all equivalent in principle, what really makes Hamiltonian mechanics unique is its geometric interpretation and the concept of phase space. One of these formulations is called Hamiltonian mechanics.Īs a general introduction, Hamiltonian mechanics is a formulation of classical mechanics in which the motion of a system is described through total energy by Hamilton’s equations of motion. In classical mechanics, there are quite many different formulations, which all have their unique purposes and advantages. ![]()
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