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.
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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
and the evolution equation is given by the Hamilton's equations:
The trajectory
Time independent Hamiltonian system
If the Hamiltonian is not time dependent, i.e. if
and thus the Hamiltonian is a constant of motion, whose constant equals the total energy of the system,
Example
One example of time independent Hamiltonian system is the harmonic oscillator. Consider the system defined by the coordinates
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. Writing
the evolution equation of the dynamical system can be written as
where
and IN the N×N identity matrix.
One important consequence of this property is that an infinitesimal phase-space volume is preserved. A corollary of this is Liouville's theorem, which states that on a Hamiltonian system, the phase-space volume of a closed surface is preserved under time evolution.
where the third equality comes from the divergence theorem.