Everything about Hamiltonian Mechanics totally explained
Hamiltonian mechanics is a re-formulation of
classical mechanics that was introduced in 1833 by Irish mathematician
William Rowan Hamilton. It arose from
Lagrangian mechanics, another re-formulation of classical mechanics, introduced by
Joseph Louis Lagrange in 1788. It can however be formulated without recourse to Lagrangian mechanics, using
symplectic spaces. See the section on its mathematical formulation for this. The Hamiltonian method differs from the Lagrangian method in that instead of expressing second-order differential constraints on an n-dimensional
coordinate space, it expresses first-order constraints on a 2n-dimensional
phase space(External Link
).
As with Lagrangian mechanics, Hamilton's equations provide a new and equivalent way of looking at classical mechanics. Generally, these equations don't 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.
Simplified overview of uses
For a closed system the sum of the kinetic and potential energy in the system is represented by a set of
differential equations known as the
Hamilton equations for that system. Hamiltonians can be used to describe such simple systems as a bouncing ball, a pendulum or an oscillating spring in which energy changes from kinetic to potential and back again over time. Hamiltonians can also be employed to model the energy of other more complex dynamic systems such as planetary orbits and in quantum mechanics.
The Hamilton equations are generally written as follows:
» . Substitute for the velocities using the results in step (3).
Apply Hamilton's equations.
==
Further Information
Get more info on 'Hamiltonian Mechanics'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://hamiltonian_mechanics.totallyexplained.com">Hamiltonian mechanics Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
