Hamiltonian

The Hamiltonian is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H (q,p,t) = \dot{q_i}p_i -L(q,\dot{q},t)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_i} are the generalised coordinates, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{p_i} = - \frac{\partial H}{\partial q_i}}

and

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

Recommended 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.

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)