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A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can be studied in both Hamiltonian mechanics and dynamical systems theory.

Overview

Informally, a Hamiltonian system is a mathematical formalism developed by William Rowan Hamilton to describe the evolution equations of a physical system. The advantage of this description is that it gives important insights into the dynamics, even if the initial value problem cannot be solved analytically. One example is the planetary movement of three bodies: while there is no closed-form solution to the general problem, Henri Poincaré showed for the first time that it exhibits deterministic chaos.

Formally, a Hamiltonian system is a dynamical system characterised by the scalar function <math>H(\boldsymbol{q},\boldsymbol{p},t)</math>, also known as the Hamiltonian. The state of the system, <math>\boldsymbol{r}</math>, is described by the generalized coordinates <math>\boldsymbol{p}</math> and <math>\boldsymbol{q}</math>, corresponding to generalized momentum and position respectively. Both <math>\boldsymbol{p}</math> and <math>\boldsymbol{q}</math> are real-valued vectors with the same dimension&nbsp;N. Thus, the state is completely described by the 2N-dimensional vector

:<math>\boldsymbol{r} = (\boldsymbol{q},\boldsymbol{p})</math>

and the evolution equations are given by Hamilton's equations:

:<math>\begin{align}

& \frac{d\boldsymbol{p{dt} = -\frac{\partial H}{\partial \boldsymbol{q, \\[5pt]

& \frac{d\boldsymbol{q{dt} = +\frac{\partial H}{\partial \boldsymbol{p.

\end{align} </math>

The trajectory <math>\boldsymbol{r}(t)</math> is the solution of the initial value problem defined by Hamilton's equations and the initial condition <math>\boldsymbol{r}(t = 0) = \boldsymbol{r}_0\in\mathbb{R}^{2N}</math>.

Time-independent Hamiltonian systems

If the Hamiltonian is not explicitly time-dependent, i.e. if <math>H(\boldsymbol{q},\boldsymbol{p},t) = H(\boldsymbol{q},\boldsymbol{p})</math>, then the Hamiltonian does not vary with time at all:

Characteristics

Hamiltonian chaos is characterized by the following features:

Mixing: Over time, the phases of the system become uniformly distributed in phase space.

Recurrence: Though unpredictable, the system eventually revisits states that are arbitrarily close to its initial state, known as Poincaré recurrence.

Hamiltonian chaos is also associated with the presence of chaotic invariants such as the Lyapunov exponent and Kolmogorov–Sinai entropy, which quantify the rate at which nearby trajectories diverge and the complexity of the system, respectively.

Examples

  • Dynamical billiards
  • Planetary systems, more specifically, the n-body problem.
  • Canonical general relativity

See also

  • Action-angle coordinates
  • Liouville's theorem
  • Integrable system
  • Symplectic manifold
  • Kolmogorov–Arnold–Moser theorem
  • Poincaré recurrence theorem
  • Lyapunov exponent
  • Three-body problem
  • Ergodic theory

References

Further reading

  • Almeida, A. M. (1992). Hamiltonian systems: Chaos and quantization. Cambridge monographs on mathematical physics. Cambridge (u.a.: Cambridge Univ. Press)
  • Audin, M., (2008). Hamiltonian systems and their integrability. Providence, R.I: American Mathematical Society,
  • Dickey, L. A. (2003). Soliton equations and Hamiltonian systems. Advanced series in mathematical physics, v. 26. River Edge, NJ: World Scientific.
  • Treschev, D., & Zubelevich, O. (2010). Introduction to the perturbation theory of Hamiltonian systems. Heidelberg: Springer
  • Zaslavsky, G. M. (2007). The physics of chaos in Hamiltonian systems. London: Imperial College Press.