Editing Hamiltonian system

{{Short description|Dynamical system governed by Hamilton's equations}}
{{About|the classical theory||Hamiltonian (disambiguation){{!}}Hamiltonian}}

{{more footnotes|date=November 2018}}
[[File:Hamiltonian_and_Equation_of_motion.jpg|thumb]]
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|Hamilton]] to describe the [[evolution equation|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 [[Three-body problem|planetary movement of three bodies]]: while there is no [[closed-form solution]] to the general problem, [[Henri Poincaré|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.<ref name=ott>{{cite book|last=Ott|authorlink=Edward Ott|first=Edward|title=Chaos in Dynamical Systems|year=1994|publisher=Cambridge University Press}}</ref> 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 2''N''-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:<ref name=ott/>

{| class="wikitable1" width=300px
|-
|
{{show
|derivation
|
: <math>\frac{dH}{dt} = 
\frac{\partial H}{\partial \boldsymbol{p}} \cdot \frac{d \boldsymbol{p}}{dt} + 
\frac{\partial H}{\partial \boldsymbol{q}} \cdot \frac{d \boldsymbol{q}}{dt} + 
\frac{\partial H}{\partial t}</math>

: <math>\frac{dH}{dt} = 
\frac{\partial H}{\partial \boldsymbol{p}} \cdot \left(-\frac{\partial H}{\partial \boldsymbol{q}}\right) + 
\frac{\partial H}{\partial \boldsymbol{q}} \cdot  \frac{\partial H}{\partial \boldsymbol{p}} + 
0 = 0</math>
}}
|}

and thus the Hamiltonian is a [[constant of motion]], whose constant equals the total [[energy]] of the system: <math>H = E</math>. Examples of such systems are the [[pendulum|undamped pendulum]], the [[harmonic oscillator]], and [[dynamical billiards]].

===Example===
{{main|Simple harmonic motion}}

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

: <math> H = \frac{p^2}{2m} + \frac{kq^2}{2}.</math>

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]].<ref name=ott/> Writing

: <math>\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}</math>

the evolution equation of the dynamical system can be written as

:<math>\frac{d\boldsymbol{r}}{dt} = M_N \nabla_{\boldsymbol{r}} H(\boldsymbol{r})</math>

where

:<math>M_N =
\begin{bmatrix}
0 & I_N \\
-I_N & 0 \\
\end{bmatrix}</math>
and ''I''<sub>''N''</sub> is the ''N''&times;''N'' [[identity matrix]].

One important consequence of this property is that an infinitesimal phase-space volume is preserved.<ref name=ott/> A corollary of this is [[Liouville's theorem (Hamiltonian)|Liouville's theorem]], which states that on a Hamiltonian system, the phase-space volume of a closed surface is preserved under time evolution.<ref name=ott/>

:<math>\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}</math>

where the third equality comes from the [[divergence theorem]].

== Hamiltonian chaos ==
Certain Hamiltonian systems exhibit [[Chaos theory|chaotic behavior]]. 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]], 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 [[Newton's law of universal gravitation|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 [[Physics|physical systems]].<ref name="Poincare2">Poincaré, Henri. "New Methods of Celestial Mechanics." (1892)</ref>

=== Characteristics ===
Hamiltonian chaos is characterized by the following features:<ref name="ott" />

'''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.<ref>{{Cite journal |last=Lorenz |first=Edward N. |date=1963-03-01 |title=Deterministic Nonperiodic Flow |url=https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_020_0130_dnf_2_0_co_2.xml |journal=Journal of the Atmospheric Sciences |language=EN |volume=20 |issue=2 |pages=130–141 |doi=10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 |issn=0022-4928|doi-access=free }}</ref>

'''Mixing''': Over time, the phases of the system become uniformly distributed in phase space.<ref>{{Cite book |last=Kornfel'd |first=Isaak P. |title=Ergodic Theory |last2=Fomin |first2=Sergej V. |last3=Sinaj |first3=Jakov G. |date=1982 |publisher=Springer |isbn=978-1-4615-6929-9 |series=Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics |location=New York, NY Heidelberg Berlin}}</ref>

'''Recurrence''': Though unpredictable, the system eventually revisits states that are arbitrarily close to its initial state, known as [[Poincaré recurrence theorem|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.<ref name="ott" />

=== Applications ===
Hamiltonian chaos is prevalent in many areas of physics, particularly in classical mechanics and statistical mechanics. 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 plasma|astrophysical plasmas]]. Moreover, in [[quantum mechanics]], Hamiltonian chaos is studied through [[quantum chaos]], which seeks to understand the quantum analogs of classical chaotic behavior. Hamiltonian chaos also plays a role in [[astrophysics]], where it is used to study the dynamics of [[star clusters]] and the stability of [[Galaxy|galactic]] structures.<ref>{{Citation |last=Regev |first=Oded |title=Astrophysics, Chaos and Complexity in |date=2009 |url=https://doi.org/10.1007/978-0-387-30440-3_26 |work=Encyclopedia of Complexity and Systems Science |pages=381–399 |editor-last=Meyers |editor-first=Robert A. |access-date=2023-06-25 |place=New York, NY |publisher=Springer |language=en |doi=10.1007/978-0-387-30440-3_26 |isbn=978-0-387-30440-3|url-access=subscription }}</ref>

==Examples==
*[[Dynamical billiards]]
*[[Planetary system]]s, more specifically, the [[n-body problem]].
*[[Canonical general relativity]]

==See also==
* [[Action-angle coordinates]]
* [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]
* [[Integrable system]]
* [[Symplectic manifold]]
* [[Kolmogorov–Arnold–Moser theorem]]

* [[Poincaré recurrence theorem]]
* [[Lyapunov exponent]]
* [[Three-body problem]]
* [[Ergodic theory]]

==References==
{{Reflist}}

==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| 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]].
*Treschev, D., & Zubelevich, O. (2010). ''Introduction to the perturbation theory of Hamiltonian systems''. Heidelberg: [[Springer Science+Business Media|Springer]]
*[[George M. Zaslavsky|Zaslavsky, G. M.]] (2007). ''The physics of chaos in Hamiltonian systems''. London: [[Imperial College Press]].

==External links==
* {{scholarpedia|title=Hamiltonian Systems|urlname=Hamiltonian_Systems|curator=James Meiss}}

{{DEFAULTSORT:Hamiltonian System}}
[[Category:Hamiltonian mechanics]]