Potentials and Motion

[Analog_Engine.git] / scripts / Pendulum.LACE

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+# Pendulum: Comparing Harmonic Oscillator with 4th order Taylor approximation
+
+### Taylor 4 approximation: phi'' = - g/r * (phi - 1/6 * phi^3)
+# fortunately, the components of orders 0, 2 and 4 are zero
+
+coefficient.1     -> g/r
+coefficient.2     -> 1/6   # set to 0,167 or 0 (for harmonic solution)
+coefficient.3(-1) -> -phi0
+
+iintegrate phi'' -> -phi'
+iintegrate -phi' -> phi
+  IC: -phi0
+
+multiply phi, phi   -> phi^2
+multiply phi^2, phi -> phi^3
+invert phi^3 -> -phi^3
+
+cmultiply 1/6, -phi^3 -> -1/6*phi^3
+isum phi, -1/6*phi^3 -> -(phi-1/6*phi^3)
+cmultiply -(phi-1/6*phi^3), g/r -> -g/r*(phi-1/6*phi^3)
+assign -g/r*(phi-1/6*phi^3) -> phi''
+
+output phi -> out.x
+
+### Harmonic oscillator: phi'' = - g/r * phi
+coefficient.5     -> g/r_h    # identical to g/r
+coefficient.7(-1) -> -phi0_h  # identical to -phi0
+
+iintegrate phi_h'' -> -phi_h'
+iintegrate -phi_h' -> phi_h
+  IC: -phi0_h
+invert phi_h -> -phi_h
+
+cmultiply -phi_h, g/r_h -> -g/r*phi_h
+assign -g/r*phi_h -> phi_h''
+
+output phi_h -> out.y