Hamiltonian 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, Poincaré showed for the first time that it exhibits deterministic chaos.

are real-valued vectors with the same dimension N. Thus, the state is completely described by the 2N-dimensional vector and the evolution equations are given by Hamilton's equations: The trajectory

Examples of such systems are the undamped pendulum, the harmonic oscillator, and dynamical billiards.

An example of a time-independent Hamiltonian system is the harmonic oscillator.

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 IN is the N×N identity matrix.

One important consequence of this property is that an infinitesimal phase-space volume is preserved.

[1] 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.

When the evolution of a Hamiltonian system is highly sensitive to initial conditions, and the motion appears random and erratic, the system is said to exhibit Hamiltonian chaos.

The concept of chaos in Hamiltonian systems has its roots in the works of Henri Poincaré, who in the late 19th century made pioneering contributions to the understanding of the three-body problem in celestial mechanics.

Poincaré showed that even a simple gravitational system of three bodies could exhibit complex behavior that could not be predicted over the long term.

His work is considered to be one of the earliest explorations of chaotic behavior in physical systems.

[2] Hamiltonian chaos is characterized by the following features:[1] Sensitivity to Initial Conditions: A hallmark of chaotic systems, small differences in initial conditions can lead to vastly different trajectories.

[4] 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.

For instance, in plasma physics, the behavior of charged particles in a magnetic field can exhibit Hamiltonian chaos, which has implications for nuclear fusion and astrophysical plasmas.

Hamiltonian chaos also plays a role in astrophysics, where it is used to study the dynamics of star clusters and the stability of galactic structures.