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Hamilton's equations of motion

 Definition/Summary Hamilton's equations of motion is a very general equation of a system evolving deterministically in phase space.

 Equations $$\left( {\begin{array}{*{20}{c}} {\dot q}\\ {\dot p} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&1\\ { - 1}&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\partial H/\partial q}\\ {\partial H/\partial p} \end{array}} \right)$$

 Scientists William Rowan Hamilton

 Recent forum threads on Hamilton's equations of motion

 Breakdown Physics > Classical Mechanics >> Lagrangian/Hamiltonian

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 Extended explanation A common misconception is that Hamilton's equations is only applicable in classical mechanics (celestial mechanics and elastic mechanics). Although this may have once been true, it has found applications everywhere in physics. It applies in almost every conservative system in classical (deterministic) physics. The meat of Hamilton's equations is given by the function H (the 'Hamiltonian'). This is a real function given by position variables and momenta variables with continuous second partial derivatives. In Newtonian gravity, the Hamiltonian may be given as $$H = \frac{{{p^2}}}{{2}} - \phi$$ Here, phi is the gravitational potential. Mass is taken to be unity. In classical electrodynamics, the Hamiltonian may be given as $$H = \frac{1}{2}{\left( {{\bf{p}} - e{\bf{A}}} \right)^2} + e\phi$$ Here, e is electric charge, phi is now the electric potential, and A is the magnetic vector potential. Mass is again taken to be unity. In general relativity, the expression of the Hamiltonian function gets far more complicated.

Commentary

 lpetrich @ 07:56 PM Jul18-12 I was half-thinking of starting an article on the Hamiltonian and how one can derive it from the Lagrangian. But should the derivation from the Lagrangian be here?

 Reigin @ 08:08 PM Jul6-12 And do you know, how to get that Hamiltonian?

 Redbelly98 @ 11:50 AM May21-12 Please post questions in the main forums at www.physicsforums.com, not in the library entry comments.

 BadriO @ 09:01 AM May21-12 HOW CAN WE EXPRESSE THE H WHEN THE DEPPENT WITH TIME WE KNOW THAT WILL CREAT E AND B PLEASE I WAITING ...... THANK YOU BEFOR

 r.ad @ 10:01 AM Apr7-12 Hamiltons and Lagrage's equations are very difficult to get. What is the best way or means to study them?

 sncum @ 05:22 AM Mar3-12 its nice information but not comprihensive