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Initial topology

2013-09-11T02:56:20Z

71.218.138.182:

{{About|the classical theory||Hamiltonian (disambiguation){{!}}Hamiltonian}}

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 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]].

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

:<math>\boldsymbol{r} = (\boldsymbol{q},\boldsymbol{p})</math>

and the evolution equation is given by the Hamilton's equations:

:<math>\begin{align}
& \frac{d\boldsymbol{p}}{dt} = -\frac{\partial H}{\partial \boldsymbol{q}}\\
& \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 the Hamilton's equations and the initial condition <math>\boldsymbol{r}(0) = \boldsymbol{r}_0\in\mathbb{R}^{2N}</math>.

==Time independent Hamiltonian system==

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/>

{| 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]], the [[harmonic oscillator]] or [[dynamical billiards]].

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

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

<math> H = \frac{p^2}{2m} + \frac{1}{2}k x^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}
\partial_\boldsymbol{q}H(\boldsymbol{q},\boldsymbol{p}) \\
\partial_\boldsymbol{p}H(\boldsymbol{q},\boldsymbol{p}) \\
\end{bmatrix}</math>

the evolution equation of the dynamical system can be written as

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

where

:<math>S_N =
\begin{bmatrix}
0 & I_N \\
-I_N & 0 \\
\end{bmatrix}</math>
and ''I''<sub>N</sub> the ''N''×''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]]:

{| class="wikitable1" width=300px
|-
|
{{show
|Liouville's theorem:
|
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/>

<math>\frac{d}{dt}\int_{S_t}d\boldsymbol{r} =
\int_{S_t}\frac{d\boldsymbol{r}}{dt}\cdot d\boldsymbol{S} =
\int_{S_t}\boldsymbol{F}\cdot d\boldsymbol{S} =
\int_{S_t}\nabla\cdot\boldsymbol{F} d\boldsymbol{r} = 0 </math>

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

==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]]

==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., & Babbitt, D. G. (2008). ''Hamiltonian systems and their integrability''. Providence, R.I: [[American Mathematical Society]]
* 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]]

Final topology

2013-09-11T02:56:08Z

71.218.138.182:

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]]

:''LF'' = δ(''x'').

Here ''F'' is ''a priori'' only assumed to be a [[Schwartz distribution]].

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]] — the most important case, directly linked to the possibility of using [[convolution]] to solve an [[arbitrary]] [[Sides of an equation|right hand side]] — was shown by [[Bernard Malgrange]] and [[Leon Ehrenpreis]].

==Example==
Consider the following differential equation ''Lf'' = sin(x) with

:<math> L=\frac{\partial^2}{\partial x^2} </math>.

The fundamental solutions can be obtained by solving ''LF'' = δ(''x''), explicitly,

:<math> \frac{\partial^2}{\partial x^2} F(x) = \delta(x) </math>.

Since for the [[Heaviside function]] ''H'' we have

:<math> \frac{\partial}{\partial x} H(x) = \delta(x) </math>.

there is a solution

:<math> \frac{\partial}{\partial x} F(x) = H(x) + C.</math>

Here ''C'' is an arbitrary constant introduced by the integration. For convenience, set ''C'' = − 1/2.

After integrating <math>\frac{\partial}{\partial x}F(x)</math> and taking the new integration constant as zero, we get

:<math> F(x) = x H(x) - \frac{1}{2}x = \frac{1}{2} |x| </math>

==Motivation==

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.

Fundamental solutions also play an important role in the numerical solution of partial differential equations by the [[boundary element method]].

===Application to the example===
Consider the operator L and the differential equation mentioned in the example.

:<math> \frac{\partial^2}{\partial x^2} f(x) = \sin(x) </math>

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>:

:<math> f(x) = \int_{-\infty}^{\infty} \frac{1}{2}|x - y|\sin(y)dy</math>

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.

===An example that more clearly works===

:<math> \frac{\partial^2}{\partial x^2} f(x) = I(x) </math>

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''.

===Proof that the convolution is a solution===
Denote the [[convolution]] of functions ''F'' and ''g'' as

:''F''*''g''.

Say we are trying to find the solution of

:''Lf'' = ''g''(''x'').

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

:''L''(''F''*''g'')=(''LF'')*''g'',

provided ''L'' has constant coefficients.

If ''F'' is the fundamental solution, the right side of the equation reduces to

:δ*''g''.

But since the delta function is an [[identity element]] for convolution, this is simply ''g''(''x''). Summing up,

:<math> L(F*g)=(LF)*g=\delta(x)*g(x)=\int_{-\infty}^{\infty} \delta (x-y) g(y) dy=g(x) </math>

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.

==Fundamental solutions for some partial differential equations==

===Laplace equation===
For the [[Laplace equation]],
:<math> [-\nabla^2] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')</math>
the fundamental solutions in two and three dimensions are

:<math> \Phi_{2D}(\mathbf{x},\mathbf{x}')=
-\frac{1}{2\pi}\ln|\mathbf{x}-\mathbf{x}'|,\quad \Phi_{3D}(\mathbf{x},\mathbf{x}')=
\frac{1}{4\pi|\mathbf{x}-\mathbf{x}'|} </math>

===Screened Poisson equation===
For the [[Screened Poisson equation]], where the parameter ''k'' is real and the fundamental solution a modified [[Bessel function]],

:<math> [-\nabla^2+k^2] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')</math>

the two and three dimensional [[Helmholtz equation]]s have the fundamental solutions

:<math> \Phi_{2D}(\mathbf{x},\mathbf{x}')=
\frac{1}{2\pi}K_0(k|\mathbf{x}-\mathbf{x}'|),\quad
\Phi_{3D}(\mathbf{x},\mathbf{x}')=
\frac{1}{4\pi|\mathbf{x}-\mathbf{x}'|}\exp(-k|\mathbf{x}-\mathbf{x}'|)
</math>

===Biharmonic equation===
For the [[Biharmonic equation]],

:<math> [-\nabla^4] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')</math>

the biharmonic equation has the fundamental solutions

:<math>\Phi_{2D}(\mathbf{x},\mathbf{x}')=
-\frac{|\mathbf{x}-\mathbf{x}'|^2}{8\pi}(\ln|\mathbf{x}-\mathbf{x}'| - 1),\quad
\Phi_{3D}(\mathbf{x},\mathbf{x}')=
\frac{|\mathbf{x}-\mathbf{x}'|}{8\pi}</math>

==Signal processing==
{{Main|Impulse response}}

In [[signal processing]], the analog of the fundamental solution of a differential equation is called the [[impulse response]] of a filter.

==See also==
* [[Green's function]]
* [[Impulse response]]
* [[Parametrix]]

==References==
*{{Springer|id=f/f042250|title=Fundamental solution}}

{{DEFAULTSORT:Fundamental Solution}}
[[Category:Partial differential equations]]
[[Category:Generalized functions]]