Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in or
create an account, your edits will be attributed to your username, along with other benefits.
The edit can be undone.
Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision |
Your text |
Line 1: |
Line 1: |
| The '''Hamiltonian''' <ref>[http://dx.doi.org/10.1098/rstl.1834.0017 William Rowan Hamilton "On a General Method in Dynamics; By Which the Study of the Motions of All Free Systems of Attracting or Repelling Points is Reduced to the Search and Differentiation of One Central Relation, or Characteristic Function", Philosophical Transactions of the Royal Society of London '''124''' pp. 247-308 (1834)]</ref><ref>[http://dx.doi.org/10.1098/rstl.1835.0009 William Rowan Hamilton "Second Essay on a General Method in Dynamics", Philosophical Transactions of the Royal Society of London '''125''' pp. 95-144 (1835)]</ref> is given by | | The '''Hamiltonian''' is given by |
|
| |
|
| :<math>H (q,p,t) = \dot{q_i}p_i -L(q,\dot{q},t)</math> | | :<math>H (q,p,t) = \dot{q_i}p_i -L(q,\dot{q},t)</math> |
Line 5: |
Line 5: |
| where <math>q_i</math> are the generalised coordinates, <math>p_i</math> are the canonical momentum, | | where <math>q_i</math> are the generalised coordinates, <math>p_i</math> are the canonical momentum, |
| and ''L'' is the [[Lagrangian]]. | | and ''L'' is the [[Lagrangian]]. |
| Using the Hamiltonian function, the equations of motion can be expressed in the so-called
| |
| canonical form:
| |
|
| |
| :<math>\dot{p_i} = - \frac{\partial H}{\partial q_i}</math>
| |
| and
| |
| :<math>\dot{q_i} = \frac{\partial H}{\partial p_i}</math>
| |
|
| |
| ==References==
| |
| <references/>
| |
| ;Related reading
| |
| *[http://www.aw-bc.com/catalog/academic/product/0,1144,0201657023,00.html Herbert Goldstein, Charles P. Poole, Jr. and John L. Safko "Classical Mechanics" (3rd edition) Addison-Wesley (2002)] Chapter 8: The Hamiltonian Equations of Motion.
| |
| [[category: classical mechanics]] | | [[category: classical mechanics]] |