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| {{About|the classical theory||Hamiltonian (disambiguation){{!}}Hamiltonian}}
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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]].
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| == Overview ==
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| Informally, a Hamiltonian system is a mathematical formalism developed by [[William Rowan Hamilton|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, [[Henri Poincaré|Poincaré]] showed for the first time that it exhibits [[deterministic chaos]].
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| Formally, a Hamiltonian system is a dynamical system completely described by the scalar function <math>H(\boldsymbol{q},\boldsymbol{p},t)</math>, the Hamiltonian.<ref name=ott>{{cite book|last=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]] 'momentum' <math>\boldsymbol{p}</math> and 'position' <math>\boldsymbol{q}</math> where both <math>\boldsymbol{p}</math> and <math>\boldsymbol{q}</math> are vectors with the same dimension N. So, the system is completely described by the 2N dimensional vector
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| :<math>\boldsymbol{r} = (\boldsymbol{q},\boldsymbol{p})</math>
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| and the evolution equation is given by the Hamilton's equations:
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| :<math>\begin{align}
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| & \frac{d\boldsymbol{p}}{dt} = -\frac{\partial H}{\partial \boldsymbol{q}}\\
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| & \frac{d\boldsymbol{q}}{dt} = +\frac{\partial H}{\partial \boldsymbol{p}}
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| \end{align} </math>.
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| The trajectory <math>\boldsymbol{r}(t)</math> is the solution of the [[initial value problem]] defined by the Hamilton's equations and the initial condition <math>\boldsymbol{r}(0) = \boldsymbol{r}_0\in\mathbb{R}^{2N}</math>.
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| ==Time independent Hamiltonian system==
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| If the Hamiltonian is not time dependent, i.e. if <math>H(\boldsymbol{q},\boldsymbol{p},t) = H(\boldsymbol{q},\boldsymbol{p})</math>, the Hamiltonian does not vary with time:<ref name=ott/>
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| {| class="wikitable1" width=300px
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| {{show
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| |derivation
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| |<math>\frac{dH}{dt} =
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| \frac{\partial H}{\partial \boldsymbol{p}} \cdot \frac{d \boldsymbol{p}}{dt} +
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| \frac{\partial H}{\partial \boldsymbol{q}} \cdot \frac{d \boldsymbol{q}}{dt} +
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| \frac{\partial H}{\partial t}</math>
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| <math>\frac{dH}{dt} =
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| \frac{\partial H}{\partial \boldsymbol{p}} \cdot \left(-\frac{\partial H}{\partial \boldsymbol{q}}\right) +
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| \frac{\partial H}{\partial \boldsymbol{q}} \cdot \frac{\partial H}{\partial \boldsymbol{p}} +
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| 0 = 0</math>
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| }}
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| |}
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| 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]], the [[harmonic oscillator]] or [[dynamical billiards]].
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| ===Example===
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| {{main|Simple harmonic motion}}
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| One example of time independent Hamiltonian system is the harmonic oscillator. Consider the system defined by the coordinates <math>\boldsymbol{p} = p</math> and <math>\boldsymbol{q} = x</math> whose Hamiltonian is given by
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| <math> H = \frac{p^2}{2m} + \frac{1}{2}k x^2</math>
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| The Hamiltonian of this system does not depend on time and thus the energy of the system is conserved.
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| == Symplectic structure ==
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| One important property of a Hamiltonian dynamical system is that it has a symplectic structure.<ref name=ott/> Writing
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| <math>\nabla_{\boldsymbol{r}} H(\boldsymbol{r}) = \begin{bmatrix}
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| \partial_\boldsymbol{q}H(\boldsymbol{q},\boldsymbol{p}) \\
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| \partial_\boldsymbol{p}H(\boldsymbol{q},\boldsymbol{p}) \\
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| \end{bmatrix}</math>
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|
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| the evolution equation of the dynamical system can be written as
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| :<math>\frac{d\boldsymbol{r}}{dt} = S_N \cdot \nabla_{\boldsymbol{r}} H(\boldsymbol{r})</math>
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| where
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| :<math>S_N =
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| \begin{bmatrix}
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| 0 & I_N \\
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| -I_N & 0 \\
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| \end{bmatrix}</math>
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| and ''I''<sub>N</sub> the ''N''×''N'' [[identity matrix]].
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| 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]]:
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| {| class="wikitable1" width=300px
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| {{show
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| |Liouville's theorem:
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| Liouville's theorem states that on a Hamiltonian system, the phase-space volume of a closed surface is preserved under time evolution.<ref name=ott/>
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| <math>\frac{d}{dt}\int_{S_t}d\boldsymbol{r} =
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| \int_{S_t}\frac{d\boldsymbol{r}}{dt}\cdot d\boldsymbol{S} =
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| \int_{S_t}\boldsymbol{F}\cdot d\boldsymbol{S} =
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| \int_{S_t}\nabla\cdot\boldsymbol{F} d\boldsymbol{r} = 0 </math>
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| where the third equality comes from the [[divergence theorem]].
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| }}
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| |}
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| ==Examples==
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| *[[Dynamical billiards]]
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| *[[Planetary system]]s, more specifically, the [[n-body problem]].
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| *[[Canonical general relativity]]
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| ==See also==
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| * [[Action-angle coordinates]]
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| * [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]
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| * [[Integrable system]]
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| * Almeida, A. M. (1992).'' Hamiltonian systems: Chaos and quantization''. Cambridge monographs on mathematical physics. Cambridge (u.a.: [[Cambridge Univ. Press]])
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| * Audin, M., & Babbitt, D. G. (2008). ''Hamiltonian systems and their integrability''. Providence, R.I: [[American Mathematical Society]]
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| * Dickey, L. A. (2003). ''Soliton equations and Hamiltonian systems''. Advanced series in mathematical physics, v. 26. River Edge, NJ: [[World Scientific]].
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| *Treschev, D., & Zubelevich, O. (2010). ''Introduction to the perturbation theory of Hamiltonian systems''. Heidelberg: [[Springer Science+Business Media|Springer]]
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| *[[George M. Zaslavsky|Zaslavsky, G. M.]] (2007). ''The physics of chaos in Hamiltonian systems''. London: [[Imperial College Press]].
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| ==External links==
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| * {{scholarpedia|title=Hamiltonian Systems|urlname=Hamiltonian_Systems|curator=James Meiss}}
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| {{DEFAULTSORT:Hamiltonian System}}
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| [[Category:Hamiltonian mechanics]]
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I am Oscar and I totally dig that title. Body building is what my family members and I appreciate. For many years I've been working as a payroll clerk. California is our birth location.
Here is my web blog: home std test (Find Out More)