{{{id=107| # trac 22801 on top of ... version() /// $\newcommand{\Bold}[1]{\mathbf{#1}}\verb|SageMath|\phantom{\verb!x!}\verb|version|\phantom{\verb!x!}\verb|8.1.rc3,|\phantom{\verb!x!}\verb|Release|\phantom{\verb!x!}\verb|Date:|\phantom{\verb!x!}\verb|2017-11-23|$ }}} {{{id=45| Parallelism().set(nproc=1) /// }}} {{{id=1| var('rho12,rho13,rho23', domain='real') assume(rho12>0, rho13>0, rho23>0) var('r12,r13,r23', domain='real') var('m1 m2 m3', domain='real') var('mu12,mu13,mu23', domain='real') assume(m1>0, m2>0, m3>0) #m1=1; m2=1; m3=1 # m3 = m2 #Extra constraint for 2 electrons # N.B.: If all {m1,m2,m3} are in SR and no two are equal, G.simplify_full() will seg-fault. # mu12 = 1/m1+1/m2; mu13 = 1/m1+1/m3; mu23 = 1/m2+1/m3; rho12 = r12^2; rho13 = r13^2; rho23 = r23^2 /// }}} {{{id=2| Ginv = matrix([[1/mu12, 1/m1*(rho12+rho13-rho23)/(2*r12*r13), 1/m2*(rho12+rho23-rho13)/(2*r12*r23)],\ [1/m1*(rho12+rho13-rho23)/(2*r12*r13), 1/mu13, 1/m3*(rho13+rho23-rho12)/(2*r13*r23)],\ [1/m2*(rho12+rho23-rho13)/(2*r12*r23), 1/m3*(rho13+rho23-rho12)/(2*r13*r23), 1/mu23]]) Ginv /// $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} \frac{1}{\frac{1}{m_{1}} + \frac{1}{m_{2}}} & \frac{r_{12}^{2} + r_{13}^{2} - r_{23}^{2}}{2 \, m_{1} r_{12} r_{13}} & \frac{r_{12}^{2} - r_{13}^{2} + r_{23}^{2}}{2 \, m_{2} r_{12} r_{23}} \\ \frac{r_{12}^{2} + r_{13}^{2} - r_{23}^{2}}{2 \, m_{1} r_{12} r_{13}} & \frac{1}{\frac{1}{m_{1}} + \frac{1}{m_{2}}} & -\frac{r_{12}^{2} - r_{13}^{2} - r_{23}^{2}}{2 \, m_{2} r_{13} r_{23}} \\ \frac{r_{12}^{2} - r_{13}^{2} + r_{23}^{2}}{2 \, m_{2} r_{12} r_{23}} & -\frac{r_{12}^{2} - r_{13}^{2} - r_{23}^{2}}{2 \, m_{2} r_{13} r_{23}} & \frac{1}{2} \, m_{2} \end{array}\right)$ }}}

Feed the following into the metric tensor: M.metric().set() seems to require the covariant form.

{{{id=3| G = Ginv.inverse(); #G.simplify_full() /// }}} {{{id=7| (G*Ginv).simplify_full(); (Ginv*G).simplify_full() # Check 2 /// $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)$ $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)$ }}}

Begin manifold creation

{{{id=38| M = Manifold(3,'R^3',field='real',start_index=1) # The following choice seems not to matter. The code always goes through manifolds/utilities.py where it calls simplify_trig(), which dives down the rat-hole. ### M.set_calculus_method('SR') # N.B. 'sympy' fails w/ nproc>1 (above) ### U = M.open_subset('U') /// }}} {{{id=39| Rho. = U.chart("r12:(0,+oo) r13:(0,+oo) r23:(0,+oo)") Rho.add_restrictions([r23 $\newcommand{\Bold}[1]{\mathbf{#1}}U$ $\newcommand{\Bold}[1]{\mathbf{#1}}\verb|<class|\phantom{\verb!x!}\verb|'sage.manifolds.differentiable.chart.RealDiffChart'>|$ }}} {{{id=108| # Check rij's r12.is_real(); r12.is_positive(); bool(rho12==r12^2) /// $\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$ $\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$ $\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$ }}} {{{id=109| # Pekeris coordinates Tau. = U.chart("u1:(0,+oo) u2:(0,+oo) u3:(0,+oo)") Rho_Tau = Rho.transition_map(Tau, ((r12+r13-r23)/2, (r12-r13+r23)/2, (-r12+r13+r23)/2)) Tau_Rho = Rho_Tau.inverse() U.set_default_chart(Tau) U.set_default_frame(Tau.frame()) Rho_Tau.display(); Tau_Rho.display(); /// $\newcommand{\Bold}[1]{\mathbf{#1}}\left\{\begin{array}{lcl} u_{1} & = & \frac{1}{2} \, r_{12} + \frac{1}{2} \, r_{13} - \frac{1}{2} \, r_{23} \\ u_{2} & = & \frac{1}{2} \, r_{12} - \frac{1}{2} \, r_{13} + \frac{1}{2} \, r_{23} \\ u_{3} & = & -\frac{1}{2} \, r_{12} + \frac{1}{2} \, r_{13} + \frac{1}{2} \, r_{23} \end{array}\right.$ $\newcommand{\Bold}[1]{\mathbf{#1}}\left\{\begin{array}{lcl} r_{12} & = & u_{1} + u_{2} \\ r_{13} & = & u_{1} + u_{3} \\ r_{23} & = & u_{2} + u_{3} \end{array}\right.$ }}} {{{id=113| U.atlas(); Rho.frame(); Tau.frame() /// $\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(U,(r_{12}, r_{13}, r_{23})\right), \left(U,(u_{1}, u_{2}, u_{3})\right)\right]$ $\newcommand{\Bold}[1]{\mathbf{#1}}\left(U, \left(\frac{\partial}{\partial r_{12} },\frac{\partial}{\partial r_{13} },\frac{\partial}{\partial r_{23} }\right)\right)$ $\newcommand{\Bold}[1]{\mathbf{#1}}\left(U, \left(\frac{\partial}{\partial u_{1} },\frac{\partial}{\partial u_{2} },\frac{\partial}{\partial u_{3} }\right)\right)$ }}} {{{id=46| ginv = g.inverse(); ginv.display() /// Traceback (most recent call last): File "", line 1, in File "_sage_input_15.py", line 10, in exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Z2ludiA9IGcuaW52ZXJzZSgpOyBnaW52LmRpc3BsYXkoKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in File "/tmp/tmpciHctE/___code___.py", line 2, in exec compile(u'ginv = g.inverse(); ginv.display()' + '\n', '', 'single') File "", line 1, in File "/home/rllozes/Development/sage/local/lib/python2.7/site-packages/sage/manifolds/differentiable/metric.py", line 692, in inverse self._inverse._restrictions[dom] = rst.inverse() # forces the File "/home/rllozes/Development/sage/local/lib/python2.7/site-packages/sage/manifolds/differentiable/metric.py", line 2281, in inverse val = chart.simplify(gmat_inv[i-si,j-si], method='SR') File "/home/rllozes/Development/sage/local/lib/python2.7/site-packages/sage/manifolds/calculus_method.py", line 236, in simplify return self._simplify_dict[method](expression) File "/home/rllozes/Development/sage/local/lib/python2.7/site-packages/sage/manifolds/utilities.py", line 597, in simplify_chain_real expr = expr.simplify_trig() File "sage/symbolic/expression.pyx", line 10015, in sage.symbolic.expression.Expression.simplify_trig (build/cythonized/sage/symbolic/expression.cpp:56590) File "/home/rllozes/Development/sage/local/lib/python2.7/site-packages/sage/interfaces/interface.py", line 657, in __call__ return self._obj.parent().function_call(self._name, [self._obj] + list(args), kwds) File "/home/rllozes/Development/sage/local/lib/python2.7/site-packages/sage/interfaces/interface.py", line 578, in function_call return self.new(s) File "/home/rllozes/Development/sage/local/lib/python2.7/site-packages/sage/interfaces/interface.py", line 347, in new return self(code) File "/home/rllozes/Development/sage/local/lib/python2.7/site-packages/sage/interfaces/interface.py", line 282, in __call__ return cls(self, x, name=name) File "/home/rllozes/Development/sage/local/lib/python2.7/site-packages/sage/interfaces/interface.py", line 697, in __init__ raise TypeError(x) TypeError: ECL says: THROW: The catch RAT-ERR is undefined. }}}