> <\body> This document shows how to use reliable integration for the classical pendulum problem, within the interpreter. \; <\with|par-columns|2> |gr-frame|>|gr-geometry||gr-grid||gr-grid-old||1>|gr-edit-grid-aspect|||>|gr-edit-grid||gr-edit-grid-old||1>|gr-line-width|2ln|gr-dash-style|10|arrow-length|10ln|arrow-height|6ln|gr-arrow-end|\||>>||>>|||magnify|0.666666666666667|||>>||>|>>> <\with|par-mode|center> endulum vector field> <\eqnarray*> >||>>|>||>|>||*cos y>>|>||*sin y>>>> <\session|mathemagix|default> <\input> <|input> use "continewz" <\input> <|input> bit_precision := 128; significant_digits := 6; <\input> <|input> z == analytic (ball 0.0, ball 1.0); <\input> <|input> vector_field (v) == [ v[1], -v[2], v[1]*v[3], -v[1]*v[2] ]; <\input> <|input> initial_conditions == [ ball 0.0, ball 1.0, ball 0.0, ball 1.0 ]; <\input> <|input> pendulum == integrate_analytic (vector_field, initial_conditions); <\unfolded-io> <|unfolded-io> pendulum[0] <|unfolded-io> +1.00000*z+|)>*z-0.166667*z+|)>*z+0.0166667*z+|)>*z-0.00257937*z+|)>*z+4.43673e-4*z+O|)>> <\input> <|input> significant_digits := 0; <\unfolded-io> <|unfolded-io> pendulum[0][99] <|unfolded-io> <\unfolded-io> <|unfolded-io> pendulum[0] (0.0) <|unfolded-io> > <\unfolded-io> <|unfolded-io> pendulum[0] (0.1) <|unfolded-io> <\unfolded-io> <|unfolded-io> radius pendulum[0] (0.2) <|unfolded-io> <\input> <|input> significant_digits := 15; <\unfolded-io> <|unfolded-io> points == [ (0.25 * t, pendulum[0] (0.25 * t)) \|\| t in 0 to 20 ] <|unfolded-io> >||||||||||||||||||||>>|]>> <\unfolded-io> <|unfolded-io> include "graphix/simple_plot.mmx"; $draw_diagram ([ [Re center points[i,0], \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Re center points[i,1]] \| i in 0..rows points ]) <|unfolded-io> >|gr-geometry||||||||||||||||||||||>>||>>||>>||>>||>>||>>||>>||>>||>>|>|>||>>||>>||>>||>>||>>|>|>>>> . If you don't have this file, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.> <\initial> <\collection>