formulasearchengine - User contributions [en]

Convex hull algorithms

2013-12-23T14:29:45Z

132.76.61.22: /* Simple polygon */

In [[classical mechanics]], a '''Liouville dynamical system''' is an exactly soluble [[dynamical system]] in which the [[kinetic energy]] ''T'' and [[potential energy]] ''V'' can be expressed in terms of the ''s'' [[generalized coordinate]]s ''q'' as follows:<ref name="liouville_1849">{{cite journal | last = Liouville | year = 1849 | title = Mémoire sur l'intégration des équations différentielles du mouvement d'un nombre quelconque de points matériels | journal = Journal de Mathématiques Pures et Appliquées | volume = 14 | pages = 257–299 | url = http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16393&Deb=263&Fin=305&E=PDF}}</ref>

:<math>
T = \frac{1}{2} \left\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) \right\}
\left\{ v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} \right\}
</math>

:<math>
V = \frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) }
</math>

The solution of this system consists of a set of separably integrable equations

:<math>
\frac{\sqrt{2}}{Y}\, dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} =
\frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots =
\frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}}
</math>

where ''E = T + V'' is the conserved energy and the <math>\gamma_{s}</math> are constants. As described below, the variables have been changed from ''qs'' to φs, and the functions ''us'' and ''ws'' substituted by their counterparts ''χs'' and ''ωs''. This solution has numerous applications, such as the orbit of a small planet about two fixed stars under the influence of [[gravitation|Newtonian gravity]]. The Liouville dynamical system is one of several things named after [[Joseph Liouville]], an eminent French mathematician.

==Example of bicentric orbits==

In [[classical mechanics]], [[Euler's three-body problem]] describes the motion of a particle in a plane under the influence of two fixed centers, each of which attract the particle with an [[inverse-square law|inverse-square force]] such as [[gravitation|Newtonian gravity]] or [[Coulomb's law]]. Examples of the bicenter problem include a [[planet]] moving around two slowly moving [[star]]s, or an [[electron]] moving in the [[electric field]] of two positively charged [[atomic nucleus|nuclei]], such as the first [[ion]] of the hydrogen molecule H2, namely the [[Dihydrogen cation|hydrogen molecular ion]] or H2+. The strength of the two attractions need not be equal; thus, the two stars may have different masses or the nuclei two different charges.

===Solution===

Let the fixed centers of attraction be located along the ''x''-axis at ±''a''. The potential energy of the moving particle is given by

:<math>
V(x, y) = \frac{-\mu_{1}}{\sqrt{\left( x - a \right)^{2} + y^{2}}} - \frac{\mu_{2}}{\sqrt{\left( x + a \right)^{2} + y^{2}}} .
</math>

The two centers of attraction can be considered as the foci of a set of ellipses. If either center were absent, the particle would move on one of these ellipses, as a solution of the [[Kepler problem]]. Therefore, according to [[Bonnet's theorem]], the same ellipses are the solutions for the bicenter problem.

Introducing [[elliptic coordinates]],

:<math>
x = a \cosh \xi \cos \eta,
</math>

:<math>
y = a \sinh \xi \sin \eta,
</math>

the potential energy can be written as

:<math>
V(\xi, \eta) = \frac{-\mu_{1}}{a\left( \cosh \xi - \cos \eta \right)} - \frac{\mu_{2}}{a\left( \cosh \xi + \cos \eta \right)}
= \frac{-\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right)}{a\left( \cosh^{2} \xi - \cos^{2} \eta \right)},</math>

and the kinetic energy as

:<math>
T = \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right) \left( \dot{\xi}^{2} + \dot{\eta}^{2} \right).
</math>

This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function ''Y'' equals

:<math>
Y = \cosh^{2} \xi - \cos^{2} \eta
</math>

and the function ''W'' equals

:<math>
W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right)
</math>

Using the general solution for a Liouville dynamical system below, one obtains

:<math>
\frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\xi}^{2} = E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma
</math>

:<math>
\frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\eta}^{2} = -E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma
</math>

Introducing a parameter ''u'' by the formula

:<math>
du = \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} =
\frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}},
</math>

gives the [[parametric solution]]

:<math>
u = \int \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} =
\int \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}.
</math>

Since these are [[elliptic integral]]s, the coordinates ξ and η can be expressed as elliptic functions of ''u''.

===Constant of motion===

The bicentric problem has a constant of motion, namely,

:<math>
r_{1}^{2} r_{2}^{2} \left( \frac{d\theta_{1}}{dt} \right) \left( \frac{d\theta_{2}}{dt} \right) -
2c \left[ \mu_{1} \cos \theta_{1} + \mu_{2} \cos \theta_{2} \right],
</math>

from which the problem can be solved using the method of the last multiplier.

==Derivation==

===New variables===

To eliminate the ''v'' functions, the variables are changed to an equivalent set

:<math>
\varphi_{r} = \int dq_{r} \sqrt{v_{r}(q_{r})},
</math>

giving the relation

:<math>
v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} =
\dot{\varphi}_{1}^{2} + \dot{\varphi}_{2}^{2} + \cdots + \dot{\varphi}_{s}^{2} = F,
</math>

which defines a new variable ''F''. Using the new variables, the u and w functions can be expressed by equivalent functions χ and ω. Denoting the sum of the χ functions by ''Y'',

:<math>
Y = \chi_{1}(\varphi_{1}) + \chi_{2}(\varphi_{2}) + \cdots + \chi_{s}(\varphi_{s}),
</math>

the kinetic energy can be written as

:<math>
T = \frac{1}{2} Y F.
</math>

Similarly, denoting the sum of the ω functions by ''W''

:<math>
W = \omega_{1}(\varphi_{1}) + \omega_{2}(\varphi_{2}) + \cdots + \omega_{s}(\varphi_{s}),
</math>

the potential energy ''V'' can be written as

:<math>
V = \frac{W}{Y}.
</math>

===Lagrange equation===

The Lagrange equation for the ''r''th variable <math>\varphi_{r}</math> is

:<math>
\frac{d}{dt} \left( \frac{\partial T}{\partial \dot{\varphi}_{r}} \right) =
\frac{d}{dt} \left( Y \dot{\varphi}_{r} \right) = \frac{1}{2} F \frac{\partial Y}{\partial \varphi_{r}}
-\frac{\partial V}{\partial \varphi_{r}}.
</math>

Multiplying both sides by <math>2 Y \dot{\varphi}_{r}</math>, re-arranging, and exploiting the relation 2''T = YF'' yields the equation

:<math>
2 Y \dot{\varphi}_{r} \frac{d}{dt} \left(Y \dot{\varphi}_{r}\right) =
2T\dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 Y \dot{\varphi}_{r} \frac{\partial V}{\partial \varphi_{r}} =
2 \dot{\varphi}_{r} \frac{\partial}{\partial \varphi_{r}} \left[ (E-V) Y \right],
</math>

which may be written as

:<math>
\frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) =
2 E \dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 \dot{\varphi}_{r} \frac{\partial W}{\partial \varphi_{r}} =
2E \dot{\varphi}_{r} \frac{d\chi_{r} }{d\varphi_{r}} - 2 \dot{\varphi}_{r} \frac{d\omega_{r}}{d\varphi_{r}},
</math>

where ''E = T + V'' is the (conserved) total energy. It follows that

:<math>
\frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) =
2\frac{d}{dt} \left( E \chi_{r} - \omega_{r} \right),
</math>

which may be integrated once to yield

:<math>
\frac{1}{2} Y^{2} \dot{\varphi}_{r}^{2} = E \chi_{r} - \omega_{r} + \gamma_{r},
</math>

where the <math>\gamma_{r}</math> are constants of integration subject to the energy conservation

:<math>
\sum_{r=1}^{s} \gamma_{r} = 0.
</math>

Inverting, taking the square root and separating the variables yields a set of separably integrable equations:

:<math>
\frac{\sqrt{2}}{Y} dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} =
\frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots =
\frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}}.
</math>

==References==

{{reflist|1}}

==Further reading==

* {{cite book | last = Whittaker | first = ET | year = 1937 | title = A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies | edition = 4th | publisher = Dover Publications | location = New York | id = ISBN }}

[[Category:Classical mechanics]]

Spinodal decomposition

2013-07-17T07:43:18Z

132.76.61.22: /* Diffusion equation */

In [[mathematics]], the '''Babuška–Lax–Milgram theorem''' is a generalization of the famous [[Lax–Milgram theorem]], which gives conditions under which a [[bilinear form]] can be "inverted" to show the existence and uniqueness of a [[weak solution]] to a given [[boundary value problem]]. The result is named after the [[mathematician]]s [[Ivo Babuška]], [[Peter Lax]] and [[Arthur Milgram]].

==Background==
In the modern, [[functional analysis|functional-analytic]] approach to the study of [[partial differential equations]], one does not attempt to solve a given partial differential equation directly, but by using the structure of the [[vector space]] of possible solutions, e.g. a [[Sobolev space]] ''W''''k'',''p''. Abstractly, consider two [[real number|real]] [[normed space]]s ''U'' and ''V'' with their [[continuous dual space]]s ''U''∗ and ''V''∗ respectively. In many applications, ''U'' is the space of possible solutions; given some [[partial differential operator]] Λ : ''U'' → ''V''∗ and a specified element ''f'' ∈ ''V''∗, the objective is to find a ''u'' ∈ ''U'' such that

:<math>\Lambda u = f.\ </math>

However, in the [[weak formulation]], this equation is only required to hold when "tested" against all other possible elements of ''V''. This "testing" is accomplished by means of a bilinear function ''B'' : ''U'' × ''V'' → '''R''' which encodes the differential operator Λ; a ''weak solution'' to the problem is to find a ''u'' ∈ ''U'' such that

:<math>B(u, v) = \langle f, v \rangle \mbox{ for all } v \in V.</math>

The achievement of Lax and Milgram in their 1954 result was to specify sufficient conditions for this weak formulation to have a unique solution that depends [[continuous function|continuously]] upon the specified datum ''f'' ∈ ''V''∗: it suffices that ''U'' = ''V'' is a [[Hilbert space]], that ''B'' is continuous, and that ''B'' is strongly [[coercive function|coercive]], i.e.

:<math>| B(u, u) | \geq c \| u \|^{2}</math>

for some constant ''c'' > 0 and all ''u'' ∈ ''U''.

For example, in the solution of the [[Poisson equation]] on a [[bounded set|bounded]], [[open set|open]] domain Ω ⊂ '''R'''''n'',

:<math>\begin{cases} - \Delta u(x) = f(x), & x \in \Omega; \\ u(x) = 0, & x \in \partial \Omega; \end{cases}</math>

the space ''U'' could be taken to be the Sobolev space ''H''01(Ω) with dual ''H''−1(Ω); the former is a subspace of the [[Lp space|''L''''p'' space]] ''V'' = ''L''2(Ω); the bilinear form ''B'' associated to −Δ is the ''L''2(Ω) [[inner product]] of the derivatives:

:<math>B(u, v) = \int_{\Omega} \nabla u(x) \cdot \nabla v(x) \, \mathrm{d} x.</math>

Hence, the weak formulation of the Poisson equation, given ''f'' ∈ ''L''2(Ω), is to find ''u''''f'' such that

:<math>\int_{\Omega} \nabla u_{f}(x) \cdot \nabla v(x) \, \mathrm{d} x = \int_{\Omega} f(x) v(x) \, \mathrm{d} x \mbox{ for all } v \in H_{0}^{1} (\Omega).</math>

==Statement of the theorem==

In 1971, Babuška provided the following generalization of Lax and Milgram's earlier result, which begins by dispensing with the requirement that ''U'' and ''V'' be the same space. Let ''U'' and ''V'' be two real Hilbert spaces and let ''B'' : ''U'' × ''V'' → '''R''' be a continuous bilinear functional. Suppose also that ''B'' is weakly coercive: for some constant ''c'' > 0 and all ''u'' ∈ ''U'',

:<math>\sup_{\| v \| = 1} | B(u, v) | \geq c \| u \|</math>

and, for 0 ≠ ''v'' ∈ ''V'', for some positive constant k,

:<math>\sup_{\| u \| = 1} | B(u, v) | \geq k \| v \|</math>

Then, for all ''f'' ∈ ''V''∗, there exists a unique solution ''u'' = ''u''''f'' ∈ ''U'' to the weak problem

:<math>B(u_{f}, v) = \langle f, v \rangle \mbox{ for all } v \in V.</math>

Moreover, the solution depends continuously on the given datum:

:<math>\| u_{f} \| \leq \frac{1}{c} \| f \|.</math>

==See also==

* [[Lions–Lax–Milgram theorem]]

==References==

* {{cite journal
| last = Babuška
| first = Ivo
| authorlink = Ivo Babuška
| title = Error-bounds for finite element method
| journal = [[Numerische Mathematik]]
| volume = 16
| year = 1970/1971
| pages = 322–333
| issn = 0029-599X
| doi = 10.1007/BF02165003
| mr = 0288971
}}
* {{cite book
| last = Lax
| first = Peter D.
| authorlink = Peter Lax
| coauthors = Milgram, Arthur N.
| chapter = Parabolic equations
| title = Contributions to the theory of partial differential equations
| series = Annals of Mathematics Studies, no. 33
| pages = 167–190
| publisher = Princeton University Press
| location = Princeton, N. J.
| year = 1954
| mr = 0067317
}}

==External links==
* {{springer
| title = Babuška–Lax–Milgram theorem
| id = B/b110020
| last = Roşca
| first = Ioan
}}

{{DEFAULTSORT:Babuska-Lax-Milgram theorem}}
[[Category:Theorems in analysis]]
[[Category:Partial differential equations]]

STO-nG basis sets

2013-06-14T12:56:06Z

132.76.61.22: corrected ref. year

'''Huber's equation''', first derived by a Polish engineer [[Tytus Maksymilian Huber]], is a basic formula in elastic material [[tension (physics)|tension]] calculations, an equivalent of the [[equation of state]], but applying to solids. In most simple expression and commonly in use it looks like this:

<math>
\sigma_{red}=\sqrt{({\sigma}^2) + 3({\tau}^2)}
</math>

where <math>\sigma</math> is the [[tensile stress]], and <math>\tau</math> is the [[shear stress]], measured in newtons per square meter (N/m², also called [[pascal (unit)|pascal]]s, Pa), while <math>\sigma_{red}</math> - called a reduced tension, is the resultant tension of the material.

Very useful in calculating the span width of the bridges like [[Golden Gate Bridge]] or [[Verrazano-Narrows Bridge]], their beam cross-sections, etc.

==See also==
* [[Yield surface]]
* [[Stress energy tensor]]
* [[Tensile stress]]

{{classicalmechanics-stub}}

[[Category:Physical quantities]]
[[Category:Structural analysis]]