# Hamiltonian system

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This article is about the classical theory. For other uses, see [Hamiltonian](/source/Hamiltonian_(disambiguation)).

A **Hamiltonian system** is a [dynamical system](/source/Dynamical_system) governed by [Hamilton's equations](/source/Hamilton's_equations). In [physics](/source/Physics), this dynamical system describes the evolution of a [physical system](/source/Physical_system) such as a [planetary system](/source/Planetary_system) or an [electron](/source/Electron) in an [electromagnetic field](/source/Electromagnetic_field). These systems can be studied in both [Hamiltonian mechanics](/source/Hamiltonian_mechanics) and [dynamical systems theory](/source/Dynamical_systems_theory).

## Overview

Informally, a Hamiltonian system is a mathematical formalism developed by [William Rowan Hamilton](/source/William_Rowan_Hamilton) to describe the [evolution equations](/source/Evolution_equation) of a physical system. The advantage of this description is that it gives important insights into the dynamics, even if the [initial value problem](/source/Initial_value_problem) cannot be solved analytically. One example is the [planetary movement of three bodies](/source/Three-body_problem): while there is no [closed-form solution](/source/Closed-form_solution) to the general problem, [Henri Poincaré](/source/Henri_Poincar%C3%A9) showed for the first time that it exhibits [deterministic chaos](/source/Deterministic_chaos).

Formally, a Hamiltonian system is a dynamical system characterised by the scalar function H(\boldsymbol{q},\boldsymbol{p},t), also known as the Hamiltonian.[1] The state of the system, \boldsymbol{r}, is described by the [generalized coordinates](/source/Generalized_coordinates) \boldsymbol{p} and \boldsymbol{q}, corresponding to generalized momentum and position respectively. Both \boldsymbol{p} and \boldsymbol{q} are real-valued vectors with the same dimension *N*. Thus, the state is completely described by the 2*N*-dimensional vector

- \boldsymbol{r} = (\boldsymbol{q},\boldsymbol{p})

and the evolution equations are given by [Hamilton's equations](/source/Hamilton's_equations):

- \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}

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

## Time-independent Hamiltonian systems

If the Hamiltonian is not explicitly time-dependent, i.e. if H(\boldsymbol{q},\boldsymbol{p},t) = H(\boldsymbol{q},\boldsymbol{p}), then the Hamiltonian does not vary with time at all:[1]

and thus the Hamiltonian is a [constant of motion](/source/Constant_of_motion), whose constant equals the total [energy](/source/Energy) of the system: H = E. Examples of such systems are the [undamped pendulum](/source/Pendulum), the [harmonic oscillator](/source/Harmonic_oscillator), and [dynamical billiards](/source/Dynamical_billiards).

### Example

Main article: [Simple harmonic motion](/source/Simple_harmonic_motion)

An example of a time-independent Hamiltonian system is the harmonic oscillator. Consider the system defined by the coordinates \boldsymbol{p} = m\dot{x} and \boldsymbol{q} = x. Then the Hamiltonian is given by

- H = \frac{p^2}{2m} + \frac{kq^2}{2}.

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](/source/Symplectic_structure).[1] Writing

- \nabla_{\boldsymbol{r}} H(\boldsymbol{r}) = \begin{bmatrix} \frac{\partial H(\boldsymbol{q},\boldsymbol{p})}{\partial \boldsymbol{q}} \\ \frac{\partial H(\boldsymbol{q},\boldsymbol{p})}{\partial \boldsymbol{p}} \\ \end{bmatrix}

the evolution equation of the dynamical system can be written as

- \frac{d\boldsymbol{r}}{dt} = M_N \nabla_{\boldsymbol{r}} H(\boldsymbol{r})

where

- M_N = \begin{bmatrix} 0 & I_N \\ -I_N & 0 \\ \end{bmatrix}

and *I**N* is the *N*×*N* [identity matrix](/source/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](/source/Liouville's_theorem_(Hamiltonian)), which states that on a Hamiltonian system, the phase-space volume of a closed surface is preserved under time evolution.[1]

- \begin{align} \frac{d}{dt}\oint_{\partial V} d\boldsymbol{r} &= \oint_{\partial V}\frac{d\boldsymbol{r}}{dt}\cdot d\hat{\boldsymbol{n}}_{\partial V} \\ &= \oint_{\partial V} \left(M_N \nabla_{\boldsymbol{r}} H(\boldsymbol{r})\right) \cdot d\hat{\boldsymbol{n}}_{\partial V} \\ &= \int_{V}\nabla_{\boldsymbol{r}}\cdot \left(M_N \nabla_{\boldsymbol{r}} H(\boldsymbol{r})\right) \, dV \\ &= \int_{V}\sum_{i=1}^N\sum_{j=1}^N\left(\frac{\partial^2 H}{\partial q_i \partial p_j} - \frac{\partial^2 H}{\partial p_i \partial q_j}\right) \, dV \\ &= 0 \end{align}

where the third equality comes from the [divergence theorem](/source/Divergence_theorem).

## Hamiltonian chaos

Certain Hamiltonian systems exhibit [chaotic behavior](/source/Chaos_theory). 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.

### Origins

The concept of chaos in Hamiltonian systems has its roots in the works of [Henri Poincaré](/source/Henri_Poincar%C3%A9), who in the late 19th century made pioneering contributions to the understanding of the [three-body problem](/source/Three-body_problem) in [celestial mechanics](/source/Celestial_mechanics). Poincaré showed that even a simple [gravitational system](/source/Newton's_law_of_universal_gravitation) 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](/source/Physics).[2]

### Characteristics

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. This is known as the butterfly effect.[3]

**Mixing**: Over time, the phases of the system become uniformly distributed in phase space.[4]

**Recurrence**: Though unpredictable, the system eventually revisits states that are arbitrarily close to its initial state, known as [Poincaré recurrence](/source/Poincar%C3%A9_recurrence_theorem).

Hamiltonian chaos is also associated with the presence of *chaotic invariants* such as the [Lyapunov exponent](/source/Lyapunov_exponent) and [Kolmogorov–Sinai entropy](/source/Kolmogorov%E2%80%93Sinai_entropy), which quantify the rate at which nearby trajectories diverge and the complexity of the system, respectively.[1]

### Applications

Hamiltonian chaos is prevalent in many areas of physics, particularly in classical mechanics and statistical mechanics. For instance, in [plasma physics](/source/Plasma_physics), the behavior of charged particles in a magnetic field can exhibit Hamiltonian chaos, which has implications for [nuclear fusion](/source/Nuclear_fusion) and [astrophysical plasmas](/source/Astrophysical_plasma). Moreover, in [quantum mechanics](/source/Quantum_mechanics), Hamiltonian chaos is studied through [quantum chaos](/source/Quantum_chaos), which seeks to understand the quantum analogs of classical chaotic behavior. Hamiltonian chaos also plays a role in [astrophysics](/source/Astrophysics), where it is used to study the dynamics of [star clusters](/source/Star_clusters) and the stability of [galactic](/source/Galaxy) structures.[5]

## Examples

- [Dynamical billiards](/source/Dynamical_billiards)
- [Planetary systems](/source/Planetary_system), more specifically, the [n-body problem](/source/N-body_problem).
- [Canonical general relativity](/source/Canonical_general_relativity)

## See also

- [Action-angle coordinates](/source/Action-angle_coordinates)
- [Liouville's theorem](/source/Liouville's_theorem_(Hamiltonian))
- [Integrable system](/source/Integrable_system)
- [Symplectic manifold](/source/Symplectic_manifold)
- [Kolmogorov–Arnold–Moser theorem](/source/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem)

- [Poincaré recurrence theorem](/source/Poincar%C3%A9_recurrence_theorem)
- [Lyapunov exponent](/source/Lyapunov_exponent)
- [Three-body problem](/source/Three-body_problem)
- [Ergodic theory](/source/Ergodic_theory)

## References

1. Ott, Edward (1994). *Chaos in Dynamical Systems*. Cambridge University Press.

1. Poincaré, Henri. "New Methods of Celestial Mechanics." (1892)

1. Lorenz, Edward N. (1963-03-01). ["Deterministic Nonperiodic Flow"](https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_020_0130_dnf_2_0_co_2.xml). *Journal of the Atmospheric Sciences*. **20** (2): 130–141. [doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2](https://doi.org/10.1175/1520-0469(1963)020%3C0130:DNF%3E2.0.CO;2). [ISSN 0022-4928](https://www.worldcat.org/issn/0022-4928)

1. Kornfel'd, Isaak P.; Fomin, Sergej V.; Sinaj, Jakov G. (1982). *Ergodic Theory*. Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics. New York, NY Heidelberg Berlin: Springer. ISBN 978-1-4615-6929-9.

1. Regev, Oded (2009), ["Astrophysics, Chaos and Complexity in"](https://doi.org/10.1007/978-0-387-30440-3_26), *Encyclopedia of Complexity and Systems Science*, Meyers, Robert A. (ed.), New York, NY: Springer, pp. 381–399, [doi:10.1007/978-0-387-30440-3_26](https://doi.org/10.1007/978-0-387-30440-3_26). ISBN 978-0-387-30440-3, retrieved 2023-06-25

## Further reading

- Almeida, A. M. (1992).*Hamiltonian systems: Chaos and quantization*. Cambridge monographs on mathematical physics. Cambridge (u.a.: [Cambridge Univ. Press](/source/Cambridge_Univ._Press))
- Audin, M., (2008). *Hamiltonian systems and their integrability*. Providence, R.I: [American Mathematical Society](/source/American_Mathematical_Society), ISBN 978-0-8218-4413-7
- Dickey, L. A. (2003). *Soliton equations and Hamiltonian systems*. Advanced series in mathematical physics, v. 26. River Edge, NJ: [World Scientific](/source/World_Scientific).
- Treschev, D., & Zubelevich, O. (2010). *Introduction to the perturbation theory of Hamiltonian systems*. Heidelberg: [Springer](/source/Springer_Science%2BBusiness_Media)
- [Zaslavsky, G. M.](/source/George_M._Zaslavsky) (2007). *The physics of chaos in Hamiltonian systems*. London: [Imperial College Press](/source/Imperial_College_Press).

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