Hamiltonian

The Hamiltonian is given by

$H(q,p,t)={\dot {q_{i}}}p_{i}-L(q,{\dot {q}},t)$

where $q_{i}$ are the generalised coordinates, $p_{i}$ 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:

${\dot {p_{i}}}=-{\frac {\partial H}{\partial q_{i}}}$

and

${\dot {q_{i}}}={\frac {\partial H}{\partial p_{i}}}$

References

1. 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)
2. William Rowan Hamilton "Second Essay on a General Method in Dynamics", Philosophical Transactions of the Royal Society of London 125 pp. 95-144 (1835)