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SandBoxPoirier

Many Interacting Worlds - Hall, et al.

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P:=operator('P);

Type: BasicOperator?

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Q:=operator('Q);

Type: BasicOperator?

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g:=operator('g);

Type: BasicOperator?

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U:=operator('U);

Type: BasicOperator?

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r:=operator('r);

Type: BasicOperator?

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ℏ:=h;

Type: Variable(h)

Hall equations 15,19,20

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eq15:= r(q) = -D(U(q),q)

$\label{eq1}{r \left({q}\right)}= -{{U^{\prime}}\left({q}\right)}$

(1)

Type: Equation(Expression(Integer))

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eq19:= U(q) = 1/(2*m)*g(q)^2

$\label{eq2}{U \left({q}\right)}={{{g \left({q}\right)}^{2}}\over{2 \ m}}$

(2)

Type: Equation(Expression(Integer))

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eq20:= g(q) = ℏ/2 * 1/P(q)*D(P(q),q)

$\label{eq3}{g \left({q}\right)}={{h \ {{P^{\prime}}\left({q}\right)}}\over{2 \ {P \left({q}\right)}}}$

(3)

Type: Equation(Expression(Integer))

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eval(eq19,eq20)

$\label{eq4}{U \left({q}\right)}={{{{h}^{2}}\ {{{P^{\prime}}\left({q}\right)}^{2}}}\over{8 \ m \ {{P \left({q}\right)}^{2}}}}$

(4)

Type: Equation(Expression(Integer))

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RHall:=eval(eq15,D(lhs %,q)=D(rhs %,q))

$\label{eq5}\begin{array}{@{}l} \displaystyle {r \left({q}\right)}={{\left( \begin{array}{@{}l} \displaystyle -{{{h}^{2}}\ {P \left({q}\right)}\ {{P^{\prime}}\left({q}\right)}\ {{P^{\prime \prime}}\left({q}\right)}}+ \ \ \displaystyle {{{h}^{2}}\ {{{P^{\prime}}\left({q}\right)}^{3}}}$

(5)

Type: Equation(Expression(Integer))

Hall equations 6, 7, A3

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eq6:= r(q) = -D(Q(q),q)

$\label{eq6}{r \left({q}\right)}= -{{Q^{\prime}}\left({q}\right)}$

(6)

Type: Equation(Expression(Integer))

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eq7:= Q(q) = 1/sqrt(P(q))*-ℏ^2/2/m*D(sqrt(P(q)),[q,q])

$\label{eq7}{Q \left({q}\right)}={{{2 \ {{h}^{2}}\ {P \left({q}\right)}\ {{P^{\prime \prime}}\left({q}\right)}}-{{{h}^{2}}\ {{{P^{\prime}}\left({q}\right)}^{2}}}}\over{8 \ m \ {{P \left({q}\right)}^{2}}}}$

(7)

Type: Equation(Expression(Integer))

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Rforce:=eval(eq6,D(lhs eq7,q)=D(rhs eq7,q))

$\label{eq8}\begin{array}{@{}l} \displaystyle {r \left({q}\right)}={{\left( \begin{array}{@{}l} \displaystyle -{{{h}^{2}}\ {{P \left({q}\right)}^{2}}\ {{P^{\prime \prime \prime}}\left({q}\right)}}+ \ \ \displaystyle {2 \ {{h}^{2}}\ {P \left({q}\right)}\ {{P^{\prime}}\left({q}\right)}\ {{P^{\prime \prime}}\left({q}\right)}}- \ \ \displaystyle {{{h}^{2}}\ {{{P^{\prime}}\left({q}\right)}^{3}}}$

(8)

Type: Equation(Expression(Integer))

Gaussian

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PdfNorm(x)==1/2*sqrt(2)*exp(-1/2*x^2)/sqrt(%pi)

Type: Void

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eval(RHall,[P(q)=PdfNorm(q), _
            D(P(q),q)=D(PdfNorm(q),q), _
            D(P(q),[q,q])=D(PdfNorm(q),[q,q])])

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Compiling function PdfNorm with type Variable(q) -> Expression(
      Integer)

$\label{eq9}{r \left({q}\right)}= -{{{{h}^{2}}\ q}\over{4 \ m}}$

(9)

Type: Equation(Expression(Integer))

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eval(Rforce, [P(q)=PdfNorm(q), _
              D(P(q),q)=D(PdfNorm(q),q), _
              D(P(q),[q,q])=D(PdfNorm(q),[q,q]), _
              D(P(q),[q,q,q])=D(PdfNorm(q),[q,q,q])])

$\label{eq10}{r \left({q}\right)}= -{{{{h}^{2}}\ q}\over{4 \ m}}$

(10)

Type: Equation(Expression(Integer))

Cauchy

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PdfCauchy(x)==1/%pi/(1+x^2)

Type: Void

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eval(RHall,[P(q)=PdfCauchy(q), _
            D(P(q),q)=D(PdfCauchy(q),q), _
            D(P(q),[q,q])=D(PdfCauchy(q),[q,q])])

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Compiling function PdfCauchy with type Variable(q) -> Expression(
      Integer)

$\label{eq11}{r \left({q}\right)}={{{{{h}^{2}}\ {{q}^{3}}}-{{{h}^{2}}\ q}}\over{{m \ {{q}^{6}}}+{3 \ m \ {{q}^{4}}}+{3 \ m \ {{q}^{2}}}+ m}}$

(11)

Type: Equation(Expression(Integer))

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eval(Rforce,[P(q)=PdfCauchy(q), _
             D(P(q),q)=D(PdfCauchy(q),q), _
             D(P(q),[q,q])=D(PdfCauchy(q),[q,q]), _
             D(P(q),[q,q,q])=D(PdfCauchy(q),[q,q,q])])

$\label{eq12}{r \left({q}\right)}={{{2 \ {{h}^{2}}\ {{q}^{3}}}-{4 \ {{h}^{2}}\ q}}\over{{m \ {{q}^{6}}}+{3 \ m \ {{q}^{4}}}+{3 \ m \ {{q}^{2}}}+ m}}$

(12)

Type: Equation(Expression(Integer))

... --Bill Page, Tue, 21 Jul 2015 20:17:15 +0000 reply

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J(i,j)==matrix([[x[i,j]-x[i-1,j],y[i,j]-y[i-1,j]],[x[i,j]-x[j,j-1],y[i,j]-y[j,j-1]]])

Type: Void

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J(i,j)

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Compiling function J with type (Variable(i), Variable(j)) -> Matrix(
      Polynomial(Integer))

$\label{eq13}\left[ \begin{array}{cc} {{x_{i , \: j}}-{x_{{i - 1}, \: j}}}&{{y_{i , \: j}}-{y_{{i - 1}, \: j}}} \ {-{x_{j , \:{j - 1}}}+{x_{i , \: j}}}&{-{y_{j , \:{j - 1}}}+{y_{i , \: j}}}$

(13)

Type: Matrix(Polynomial(Integer))

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determinant(J(i,j))

$\label{eq14}\begin{array}{@{}l} \displaystyle {{\left(-{x_{i , \: j}}+{x_{{i - 1}, \: j}}\right)}\ {y_{j , \:{j - 1}}}}+{{\left({x_{j , \:{j - 1}}}-{x_{{i - 1}, \: j}}\right)}\ {y_{i , \: j}}}+ \ \ \displaystyle {{\left(-{x_{j , \:{j - 1}}}+{x_{i , \: j}}\right)}\ {y_{{i - 1}, \: j}}}$

(14)

Type: Polynomial(Integer)

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K(i,j)==inverse(J(i,j))

Type: Void

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K(i,j)

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Compiling function K with type (Variable(i), Variable(j)) -> Union(
      Matrix(Fraction(Polynomial(Integer))),"failed")

$\label{eq15}\left[ \begin{array}{cc} {{{y_{j , \:{j - 1}}}-{y_{i , \: j}}}\over{{{\left({x_{i , \: j}}-{x_{{i - 1}, \: j}}\right)}\ {y_{j , \:{j - 1}}}}+{{\left(-{x_{j , \:{j - 1}}}+{x_{{i - 1}, \: j}}\right)}\ {y_{i , \: j}}}+{{\left({x_{j , \:{j - 1}}}-{x_{i , \: j}}\right)}\ {y_{{i - 1}, \: j}}}}}&{{{y_{i , \: j}}-{y_{{i - 1}, \: j}}}\over{{{\left({x_{i , \: j}}-{x_{{i - 1}, \: j}}\right)}\ {y_{j , \:{j - 1}}}}+{{\left(-{x_{j , \:{j - 1}}}+{x_{{i - 1}, \: j}}\right)}\ {y_{i , \: j}}}+{{\left({x_{j , \:{j - 1}}}-{x_{i , \: j}}\right)}\ {y_{{i - 1}, \: j}}}}} \ {{-{x_{j , \:{j - 1}}}+{x_{i , \: j}}}\over{{{\left({x_{i , \: j}}-{x_{{i - 1}, \: j}}\right)}\ {y_{j , \:{j - 1}}}}+{{\left(-{x_{j , \:{j - 1}}}+{x_{{i - 1}, \: j}}\right)}\ {y_{i , \: j}}}+{{\left({x_{j , \:{j - 1}}}-{x_{i , \: j}}\right)}\ {y_{{i - 1}, \: j}}}}}&{{-{x_{i , \: j}}+{x_{{i - 1}, \: j}}}\over{{{\left({x_{i , \: j}}-{x_{{i - 1}, \: j}}\right)}\ {y_{j , \:{j - 1}}}}+{{\left(-{x_{j , \:{j - 1}}}+{x_{{i - 1}, \: j}}\right)}\ {y_{i , \: j}}}+{{\left({x_{j , \:{j - 1}}}-{x_{i , \: j}}\right)}\ {y_{{i - 1}, \: j}}}}}$

(15)

Type: Union(Matrix(Fraction(Polynomial(Integer))),...)

From Poirier to Hall --Bill Page, Fri, 31 Jul 2015 18:16:42 +0000 reply

SandBoxPoirier

Subject: Be Bold !!

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