Difference between revisions of "Hamiltonian Dynamics"
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| − | + | '''Hamiltonian [[dynamics]]''' is a formulation of [[mechanics]] that can be a useful alternative to [[Isaac Newton|Newton's]] formulation. It is closely related to [[Lagrangian Dynamics|Lagrangian dynamics]], and makes use of the [[Lagrangian]] and [[Hamiltonian]] functions. From a mathematical viewpoint, problems in dynamics can sometimes be simpler to solve when written in the Hamiltonian formulation. The formulation is also important because it allows deep connections between [[Classical mechanics|classical]] and [[Quantum mechanics|quantum]] mechanics to be made. | |
| + | |||
| + | ==Hamilton's Equations== | ||
| + | Hamilton's equations are a set of 2n first order differential equations, which relate the co-ordinates <math>q_i</math> and the [[Momentum|generalized momenta]]: | ||
| + | |||
| + | <math> | ||
| + | \frac{dp_i}{dt}=-\frac{\partial H}{\partial q_i} | ||
| + | </math> | ||
| + | |||
| + | <math> | ||
| + | \frac{dq_i}{dt}=\frac{\partial H}{\partial p_i} | ||
| + | </math> | ||
| + | |||
| + | [[Category:Mechanics]] | ||
Latest revision as of 16:18, July 5, 2019
Hamiltonian dynamics is a formulation of mechanics that can be a useful alternative to Newton's formulation. It is closely related to Lagrangian dynamics, and makes use of the Lagrangian and Hamiltonian functions. From a mathematical viewpoint, problems in dynamics can sometimes be simpler to solve when written in the Hamiltonian formulation. The formulation is also important because it allows deep connections between classical and quantum mechanics to be made.
Hamilton's Equations
Hamilton's equations are a set of 2n first order differential equations, which relate the co-ordinates 
and the generalized momenta:
