The Hamiltonian

Phil, all the Philosophy in the world won't help if we don't speak-a de lingo, so to speak, of Physics. It begins with Calculus, yes, but things start to get interesting with Hamiltonian mechanics. I was never formally trained in this stuff, and I don't think you were either, so won't you join me and our boring old friend, Wikipedia? Let's begin with an illustration dear to your heart: :-)

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.

It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without recourse to Lagrangian mechanics using symplectic spaces (see Mathematical formalism, below). The Hamiltonian method differs from the Lagrangian method in that instead of expressing second-order differential constraints on an n-dimensional coordinate space(where n is the number of degrees of freedom of the system), it expresses first-order constraints on a 2n-dimensional phase space.^[1]

As with Lagrangian mechanics, Hamilton's equations provide a new and equivalent way of looking at classical mechanics. Generally, these equations do not provide a more convenient way of solving a particular problem. Rather, they provide deeper insights into both the general structure of classical mechanics and its connection to quantum mechanics as understood through Hamiltonian mechanics, as well as its connection to other areas of science.

Contents  [hide] 1 Simplified overview of uses 1.1 Basic physical interpretation 1.2 Using Hamilton's equations 1.3 Notes 2 Deriving Hamilton's equations 3 As a reformulation of Lagrangian mechanics 4 Geometry of Hamiltonian systems 5 Generalization to quantum mechanics through Poisson bracket 6 Mathematical formalism 7 Riemannian manifolds 8 Sub-Riemannian manifolds 9 Poisson algebras 10 Charged particle in an electromagnetic field 11 Relativistic charged particle in an electromagnetic field 12 See also 13 Notes and references 13.1 Notes 13.2 References 14 External links