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Carl McBride (talk | contribs) (New page: The '''Hamiltonian''' is given by :<math>H (q,p,t) = \dot{q_i}p_i -L(q,\dot{q},t)</math> where <math>q_i</math> are the generalised coordinates, <math>p_i</math> are the canonical mome...) |
Carl McBride (talk | contribs) m (Changed references to Cite format) |
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The '''Hamiltonian''' is given by | 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 | ||
:<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> | ||
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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]] | ||
Latest revision as of 15:11, 18 May 2011
The Hamiltonian [1][2] is given by
where are the generalised coordinates, are the canonical momentum, and L is the Lagrangian. Using the Hamiltonian function, the equations of motion can be expressed in the so-called canonical form:
and
References[edit]
- ↑ 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)
- ↑ William Rowan Hamilton "Second Essay on a General Method in Dynamics", Philosophical Transactions of the Royal Society of London 125 pp. 95-144 (1835)
- Related reading
- Herbert Goldstein, Charles P. Poole, Jr. and John L. Safko "Classical Mechanics" (3rd edition) Addison-Wesley (2002) Chapter 8: The Hamiltonian Equations of Motion.