Math 2.0

Dynamical systems and ergodic theory Definition

Hamiltonian system

Builds on Autonomous systemFirst integral

Definition

Let n≥1n \ge 1, let UU be an open subset of ℝn×ℝn\mathbb{R}^n \times \mathbb{R}^n with coordinates (q,p)=(q1,…,qn,p1,…,pn)(q, p) = (q_1, \ldots, q_n, p_1, \ldots, p_n), and let H:U→ℝH : U \to \mathbb{R} be continuously differentiable. The Hamiltonian system with Hamiltonian HH is the autonomous system qi′=∂H∂pi(q,p),pi′=−∂H∂qi(q,p),i=1,…,n.q_i' = \frac{\partial H}{\partial p_i}(q, p), \qquad p_i' = -\frac{\partial H}{\partial q_i}(q, p), \qquad i = 1, \ldots, n.

Remarks

The function HH is a first integral when it is not constant on an open set: along a solution, ddtH=∑i(∂H∂qiqi′+∂H∂pipi′)=0\frac{d}{dt} H = \sum_i \left( \frac{\partial H}{\partial q_i} q_i' + \frac{\partial H}{\partial p_i} p_i' \right) = 0. For H(q,p)=12p2+V(q)H(q, p) = \frac{1}{2} p^2 + V(q) with n=1n = 1, the system is q′=pq' = p, p′=−V′(q)p' = -V'(q). This is Newton's law q′′=−V′(q)q'' = -V'(q) for a particle with potential energy VV. The harmonic oscillator has V(q)=12q2V(q) = \frac{1}{2} q^2, and the pendulum has V(q)=−cos⁡qV(q) = -\cos q in suitable units. A time-dependent Hamiltonian H(t,q,p)H(t, q, p) gives a non-autonomous Hamiltonian system, and then HH need not be conserved.

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Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionGrönwall's inequality, statement, no evidence levelGrönwall's inequalitystatementSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionAutonomous system, definition, no evidence levelAutonomous systemdefinitionInitial value problem, definition, no evidence levelInitial value problemdefinitionUniqueness of solutions for Lipschitz vector fields, statement, no evidence levelUniqueness of solutio…statementMaximal solution, definition, no evidence levelMaximal solutiondefinitionFlow of a vector field, definition, no evidence levelFlow of a vector fielddefinitionTrajectory, definition, no evidence levelTrajectorydefinitionFirst integral, definition, no evidence levelFirst integraldefinitionHamiltonian system, definition, no evidence levelHamiltonian systemdefinition
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  1. Differential equation concept
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  4. Ordinary differential equation definition
  5. Lipschitz condition definition
  6. Grönwall's inequality statement
  7. Solution of an ordinary differential equation definition
  8. Autonomous system definition
  9. Initial value problem definition
  10. Uniqueness of solutions for Lipschitz vector fields statement
  11. Maximal solution definition
  12. Flow of a vector field definition
  13. Trajectory definition
  14. First integral definition
  15. Hamiltonian system definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Hamiltonian_system, Overview
  2. Web https://en.wikipedia.org/wiki/Hamiltonian_system, Time-independent Hamiltonian systems
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MSC37J06, 70H05
DefinesHamiltonian system
UsesAutonomous system, First integral

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