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Many Interacting Worlds - Hall, et al.

fricas
P:=operator('P);
fricas
Q:=operator('Q);
fricas
g:=operator('g);
fricas
U:=operator('U);
fricas
r:=operator('r);
fricas
ℏ:=h;
Type: Variable(h)

Hall equations 15,19,20

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

\label{eq1}{r \left({q}\right)}= -{{U_{\ }^{,}}\left({q}\right)}(1)
Type: Equation(Expression(Integer))
fricas
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))
fricas
eq20:= g(q) = ℏ/2 * 1/P(q)*D(P(q),q)

\label{eq3}{g \left({q}\right)}={{h \ {{P_{\ }^{,}}\left({q}\right)}}\over{2 \ {P \left({q}\right)}}}(3)
Type: Equation(Expression(Integer))
fricas
eval(eq19,eq20)

\label{eq4}{U \left({q}\right)}={{{{h}^{2}}\ {{{P_{\ }^{,}}\left({q}\right)}^{2}}}\over{8 \  m \ {{P \left({q}\right)}^{2}}}}(4)
Type: Equation(Expression(Integer))
fricas
RHall:=eval(eq15,D(lhs %,q)=D(rhs %,q))

\label{eq5}{r \left({q}\right)}={{-{{{h}^{2}}\ {P \left({q}\right)}\ {{P_{\ }^{,}}\left({q}\right)}\ {{P_{\ }^{, ,}}\left({q}\right)}}+{{{h}^{2}}\ {{{P_{\ }^{,}}\left({q}\right)}^{3}}}}\over{4 \  m \ {{P \left({q}\right)}^{3}}}}(5)
Type: Equation(Expression(Integer))

Hall equations 6, 7, A3

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

\label{eq6}{r \left({q}\right)}= -{{Q_{\ }^{,}}\left({q}\right)}(6)
Type: Equation(Expression(Integer))
fricas
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_{\ }^{, ,}}\left({q}\right)}}-{{{h}^{2}}\ {{{P_{\ }^{,}}\left({q}\right)}^{2}}}}\over{8 \  m \ {{P \left({q}\right)}^{2}}}}(7)
Type: Equation(Expression(Integer))
fricas
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_{\ }^{, , ,}}\left({q}\right)}}+ 
\
\
\displaystyle
{2 \ {{h}^{2}}\ {P \left({q}\right)}\ {{P_{\ }^{,}}\left({q}\right)}\ {{P_{\ }^{, ,}}\left({q}\right)}}-{{{h}^{2}}\ {{{P_{\ }^{,}}\left({q}\right)}^{3}}}
(8)
Type: Equation(Expression(Integer))

Gaussian

fricas
PdfNorm(x)==1/2*sqrt(2)*exp(-1/2*x^2)/sqrt(%pi)
Type: Void
fricas
eval(RHall,[P(q)=PdfNorm(q), _
            D(P(q),q)=D(PdfNorm(q),q), _
            D(P(q),[q,q])=D(PdfNorm(q),[q,q])])
fricas
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))
fricas
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

fricas
PdfCauchy(x)==1/%pi/(1+x^2)
Type: Void
fricas
eval(RHall,[P(q)=PdfCauchy(q), _
            D(P(q),q)=D(PdfCauchy(q),q), _
            D(P(q),[q,q])=D(PdfCauchy(q),[q,q])])
fricas
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))
fricas
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))

fricas
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
fricas
J(i,j)
fricas
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))
fricas
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)
fricas
K(i,j)==inverse(J(i,j))
Type: Void
fricas
K(i,j)
fricas
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




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