Wednesday, May 18, 2011

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.

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3 comments:

Plato said...

Hi Steve,

Just a fast direct and hopefully I can shed light very quickly in terms of how one may see "space in the Lagrangian," as allotted to other areas of science?

Consider the three body problem and how this works amid interplanetary transport?

L5

See if this helps?

Best,

Phil Warnell said...

Hi Steven,

The choosing between the Hamiltonian approach and Lagrangian are as much a matter of philosophy as is anything else. That is in their choosing one starts to make decisions on how reality not only operates yet is structured fundamentally. For instance as Bohm was to reveal neither approach seemed to fit the situation when it came to the nature of the quanta and thus chose a hybrid Hamilton-Jacobi formulation. So in the end the math chosen most often is an expression of personal philosophy more so than something which necessitates the truth of one.

Regards

Phil

Steven Colyer said...

Thanks, Phil. :)