Hamiltonian system
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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.
Contents
Overview
Informally, a Hamiltonian system is a mathematical formalism developed by Hamilton to describe the evolution equations of a physical system. The advantage of this description is that it gives important insight about the dynamics, even if the initial value problem cannot be solved analytically. One example is the planetary movement of three bodies: even if there is no simple solution to the general problem, Poincaré showed for the first time that it exhibits deterministic chaos.
Formally, a Hamiltonian system is a dynamical system completely described by the scalar function , the Hamiltonian.^{[1]} The state of the system, , is described by the generalized coordinates 'momentum' and 'position' where both and are vectors with the same dimension N. So, the system is completely described by the 2N dimensional vector
and the evolution equation is given by the Hamilton's equations:
 .
The trajectory is the solution of the initial value problem defined by the Hamilton's equations and the initial condition .
Time independent Hamiltonian system
If the Hamiltonian is not explicitly time dependent, i.e. if , then the Hamiltonian does not vary with time at all:^{[1]}
derivation

and thus the Hamiltonian is a constant of motion, whose constant equals the total energy of the system, . Examples of such systems are the pendulum, the harmonic oscillator or dynamical billiards.
Example
One example of time independent Hamiltonian system is the harmonic oscillator. Consider the system defined by the coordinates and whose Hamiltonian is given by
The Hamiltonian of this system does not depend on time and thus the energy of the system is conserved.
Symplectic structure
One important property of a Hamiltonian dynamical system is that it has a symplectic structure.^{[1]} Writing
the evolution equation of the dynamical system can be written as
where
and I_{N} the N×N identity matrix.
One important consequence of this property is that an infinitesimal phasespace volume is preserved.^{[1]} A corollary of this is Liouville's theorem, which states that on a Hamiltonian system, the phasespace volume of a closed surface is preserved under time evolution.^{[1]}
where the third equality comes from the divergence theorem.
Examples
 Dynamical billiards
 Planetary systems, more specifically, the nbody problem.
 Canonical general relativity
See also
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, ISBN 9780821844137
 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.
External links
 James Meiss (ed.). "Hamiltonian Systems". Scholarpedia.