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<div>{{About|the classical theory||Hamiltonian (disambiguation){{!}}Hamiltonian}}<br />
<br />
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]].<br />
<br />
== Overview ==<br />
<br />
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]].<br />
<br />
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<br />
<br />
:<math>\boldsymbol{r} = (\boldsymbol{q},\boldsymbol{p})</math><br />
<br />
and the evolution equation is given by the Hamilton's equations:<br />
<br />
:<math>\begin{align}<br />
& \frac{d\boldsymbol{p}}{dt} = -\frac{\partial H}{\partial \boldsymbol{q}}\\<br />
& \frac{d\boldsymbol{q}}{dt} = +\frac{\partial H}{\partial \boldsymbol{p}}<br />
\end{align} </math>.<br />
<br />
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>.<br />
<br />
==Time independent Hamiltonian system==<br />
<br />
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/><br />
<br />
{| class="wikitable1" width=300px<br />
|-<br />
|<br />
{{show<br />
|derivation<br />
|<math>\frac{dH}{dt} = <br />
\frac{\partial H}{\partial \boldsymbol{p}} \cdot \frac{d \boldsymbol{p}}{dt} + <br />
\frac{\partial H}{\partial \boldsymbol{q}} \cdot \frac{d \boldsymbol{q}}{dt} + <br />
\frac{\partial H}{\partial t}</math><br />
<br />
<math>\frac{dH}{dt} = <br />
\frac{\partial H}{\partial \boldsymbol{p}} \cdot \left(-\frac{\partial H}{\partial \boldsymbol{q}}\right) + <br />
\frac{\partial H}{\partial \boldsymbol{q}} \cdot \frac{\partial H}{\partial \boldsymbol{p}} + <br />
0 = 0</math><br />
}}<br />
|}<br />
<br />
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]].<br />
<br />
===Example===<br />
{{main|Simple harmonic motion}}<br />
<br />
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<br />
<br />
<math> H = \frac{p^2}{2m} + \frac{1}{2}k x^2</math><br />
<br />
The Hamiltonian of this system does not depend on time and thus the energy of the system is conserved.<br />
<br />
== Symplectic structure ==<br />
One important property of a Hamiltonian dynamical system is that it has a symplectic structure.<ref name=ott/> Writing<br />
<br />
<math>\nabla_{\boldsymbol{r}} H(\boldsymbol{r}) = \begin{bmatrix}<br />
\partial_\boldsymbol{q}H(\boldsymbol{q},\boldsymbol{p}) \\<br />
\partial_\boldsymbol{p}H(\boldsymbol{q},\boldsymbol{p}) \\<br />
\end{bmatrix}</math><br />
<br />
the evolution equation of the dynamical system can be written as<br />
<br />
:<math>\frac{d\boldsymbol{r}}{dt} = S_N \cdot \nabla_{\boldsymbol{r}} H(\boldsymbol{r})</math><br />
<br />
where<br />
<br />
:<math>S_N =<br />
\begin{bmatrix}<br />
0 & I_N \\<br />
-I_N & 0 \\<br />
\end{bmatrix}</math><br />
and ''I''<sub>N</sub> the ''N''&times;''N'' [[identity matrix]].<br />
<br />
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]]:<br />
<br />
{| class="wikitable1" width=300px<br />
|-<br />
|<br />
{{show<br />
|Liouville's theorem:<br />
|<br />
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/><br />
<br />
<math>\frac{d}{dt}\int_{S_t}d\boldsymbol{r} = <br />
\int_{S_t}\frac{d\boldsymbol{r}}{dt}\cdot d\boldsymbol{S} = <br />
\int_{S_t}\boldsymbol{F}\cdot d\boldsymbol{S} = <br />
\int_{S_t}\nabla\cdot\boldsymbol{F} d\boldsymbol{r} = 0 </math><br />
<br />
where the third equality comes from the [[divergence theorem]].<br />
}}<br />
|}<br />
<br />
==Examples==<br />
*[[Dynamical billiards]]<br />
*[[Planetary system]]s, more specifically, the [[n-body problem]].<br />
*[[Canonical general relativity]]<br />
<br />
==See also==<br />
* [[Action-angle coordinates]]<br />
* [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]<br />
* [[Integrable system]]<br />
<br />
==References==<br />
{{Reflist}}<br />
<br />
==Further reading==<br />
* Almeida, A. M. (1992).'' Hamiltonian systems: Chaos and quantization''. Cambridge monographs on mathematical physics. Cambridge (u.a.: [[Cambridge Univ. Press]])<br />
* Audin, M., & Babbitt, D. G. (2008). ''Hamiltonian systems and their integrability''. Providence, R.I: [[American Mathematical Society]]<br />
* Dickey, L. A. (2003). ''Soliton equations and Hamiltonian systems''. Advanced series in mathematical physics, v. 26. River Edge, NJ: [[World Scientific]].<br />
*Treschev, D., & Zubelevich, O. (2010). ''Introduction to the perturbation theory of Hamiltonian systems''. Heidelberg: [[Springer Science+Business Media|Springer]]<br />
*[[George M. Zaslavsky|Zaslavsky, G. M.]] (2007). ''The physics of chaos in Hamiltonian systems''. London: [[Imperial College Press]].<br />
<br />
==External links==<br />
* {{scholarpedia|title=Hamiltonian Systems|urlname=Hamiltonian_Systems|curator=James Meiss}}<br />
<br />
{{DEFAULTSORT:Hamiltonian System}}<br />
[[Category:Hamiltonian mechanics]]</div>71.218.138.182https://en.formulasearchengine.com/index.php?title=Final_topology&diff=9383Final topology2013-09-11T02:56:08Z<p>71.218.138.182: </p>
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<div>In [[mathematics]], a '''fundamental solution''' for a linear [[partial differential operator]] ''L'' is a formulation in the language of [[Distribution (mathematics)|distribution theory]] of the older idea of a [[Green's function]]. In terms of the [[Dirac delta function]] δ(''x''), a fundamental solution ''F'' is the solution of the [[inhomogeneous equation]]<br />
<br />
:''LF'' = &delta;(''x'').<br />
<br />
Here ''F'' is ''a priori'' only assumed to be a [[Schwartz distribution]].<br />
<br />
This concept was long known for the [[Laplacian]] in two and three dimensions. It was investigated for all dimensions for the Laplacian by [[Marcel Riesz]]. The existence of a fundamental solution for any operator with [[constant coefficients]] &mdash; the most important case, directly linked to the possibility of using [[convolution]] to solve an [[arbitrary]] [[Sides of an equation|right hand side]] &mdash; was shown by [[Bernard Malgrange]] and [[Leon Ehrenpreis]].<br />
<br />
==Example==<br />
Consider the following differential equation ''Lf'' = sin(x) with<br />
<br />
:<math> L=\frac{\partial^2}{\partial x^2} </math>.<br />
<br />
The fundamental solutions can be obtained by solving ''LF'' = δ(''x''), explicitly,<br />
<br />
:<math> \frac{\partial^2}{\partial x^2} F(x) = \delta(x) </math>.<br />
<br />
Since for the [[Heaviside function]] ''H'' we have<br />
<br />
:<math> \frac{\partial}{\partial x} H(x) = \delta(x) </math>.<br />
<br />
there is a solution<br />
<br />
:<math> \frac{\partial}{\partial x} F(x) = H(x) + C.</math><br />
<br />
Here ''C'' is an arbitrary constant introduced by the integration. For convenience, set ''C'' = &minus; 1/2.<br />
<br />
After integrating <math>\frac{\partial}{\partial x}F(x)</math> and taking the new integration constant as zero, we get<br />
<br />
:<math> F(x) = x H(x) - \frac{1}{2}x = \frac{1}{2} |x| </math><br />
<br />
==Motivation==<br />
<br />
Once the fundamental solution is found, it is easy to find the desired solution of the original equation. In fact, this process is achieved by convolution.<br />
<br />
Fundamental solutions also play an important role in the numerical solution of partial differential equations by the [[boundary element method]].<br />
<br />
===Application to the example===<br />
Consider the operator L and the differential equation mentioned in the example.<br />
<br />
:<math> \frac{\partial^2}{\partial x^2} f(x) = \sin(x) </math><br />
<br />
We can find the solution of the original equation by convolving the right-hand side <math>\sin(x)</math> with the fundamental solution <math>F(x) = \frac{1}{2}|x|</math>:<br />
<br />
:<math> f(x) = \int_{-\infty}^{\infty} \frac{1}{2}|x - y|\sin(y)dy</math><br />
<br />
This shows that some care must be taken when working with functions which do not have enough regularity (e.g. compact support, <math>L^1</math> integrability) since, we know that the desired solution is <math>f(x) = -\sin x</math>, while the above integral diverges for all x. The two expressions for f are, however, equal as distributions.<br />
<br />
===An example that more clearly works===<br />
<br />
:<math> \frac{\partial^2}{\partial x^2} f(x) = I(x) </math><br />
<br />
where ''I'' is the characteristic (indicator) function of the unit interval ''[0,1]''. In that case, it can be readily verified that the convolution ''I*F'' with ''F(x)=|x|/2'' is a solution, i.e., has second derivative equal to ''I''.<br />
<br />
===Proof that the convolution is a solution===<br />
Denote the [[convolution]] of functions ''F'' and ''g'' as<br />
<br />
:''F''*''g''.<br />
<br />
Say we are trying to find the solution of<br />
<br />
:''Lf'' = ''g''(''x'').<br />
<br />
We want to prove that ''F''*''g'' is a solution of the previous equation, i.e. we want to prove that ''L(''F''*''g'')'' = ''g''(''x''). When applying the differential operator, ''L'', to the convolution it is known that<br />
<br />
:''L''(''F''*''g'')=(''LF'')*''g'',<br />
<br />
provided ''L'' has constant coefficients.<br />
<br />
If ''F'' is the fundamental solution, the right side of the equation reduces to<br />
<br />
:&delta;*''g''.<br />
<br />
But since the delta function is an [[identity element]] for convolution, this is simply ''g''(''x''). Summing up,<br />
<br />
:<math> L(F*g)=(LF)*g=\delta(x)*g(x)=\int_{-\infty}^{\infty} \delta (x-y) g(y) dy=g(x) </math><br />
<br />
Therefore, if ''F'' is the fundamental solution, the convolution ''F''*''g'' is one solution of ''Lf'' = ''g''(''x''). This does not mean that it is the only solution. Several solutions for different initial conditions can be found.<br />
<br />
==Fundamental solutions for some partial differential equations==<br />
<br />
===Laplace equation===<br />
For the [[Laplace equation]],<br />
:<math> [-\nabla^2] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')</math><br />
the fundamental solutions in two and three dimensions are<br />
<br />
:<math> \Phi_{2D}(\mathbf{x},\mathbf{x}')=<br />
-\frac{1}{2\pi}\ln|\mathbf{x}-\mathbf{x}'|,\quad \Phi_{3D}(\mathbf{x},\mathbf{x}')=<br />
\frac{1}{4\pi|\mathbf{x}-\mathbf{x}'|} </math><br />
<br />
===Screened Poisson equation===<br />
For the [[Screened Poisson equation]], where the parameter ''k'' is real and the fundamental solution a modified [[Bessel function]],<br />
<br />
:<math> [-\nabla^2+k^2] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')</math><br />
<br />
the two and three dimensional [[Helmholtz equation]]s have the fundamental solutions<br />
<br />
:<math> \Phi_{2D}(\mathbf{x},\mathbf{x}')=<br />
\frac{1}{2\pi}K_0(k|\mathbf{x}-\mathbf{x}'|),\quad<br />
\Phi_{3D}(\mathbf{x},\mathbf{x}')=<br />
\frac{1}{4\pi|\mathbf{x}-\mathbf{x}'|}\exp(-k|\mathbf{x}-\mathbf{x}'|)<br />
</math><br />
<br />
===Biharmonic equation===<br />
For the [[Biharmonic equation]],<br />
<br />
:<math> [-\nabla^4] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')</math><br />
<br />
the biharmonic equation has the fundamental solutions<br />
<br />
:<math>\Phi_{2D}(\mathbf{x},\mathbf{x}')=<br />
-\frac{|\mathbf{x}-\mathbf{x}'|^2}{8\pi}(\ln|\mathbf{x}-\mathbf{x}'| - 1),\quad<br />
\Phi_{3D}(\mathbf{x},\mathbf{x}')=<br />
\frac{|\mathbf{x}-\mathbf{x}'|}{8\pi}</math><br />
<br />
==Signal processing==<br />
{{Main|Impulse response}}<br />
<br />
In [[signal processing]], the analog of the fundamental solution of a differential equation is called the [[impulse response]] of a filter.<br />
<br />
==See also==<br />
* [[Green's function]]<br />
* [[Impulse response]]<br />
* [[Parametrix]]<br />
<br />
==References==<br />
*{{Springer|id=f/f042250|title=Fundamental solution}}<br />
<br />
{{DEFAULTSORT:Fundamental Solution}}<br />
[[Category:Partial differential equations]]<br />
[[Category:Generalized functions]]</div>71.218.138.182