Focus on potential fields

[Analog_Engine.git] / scripts / Gravity-2-dimensional.LACE

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+# Gravity–2-dimensional.LACE
+
+# The small mass m moves within the gravitational potential of a mass M.
+# The situation is analysed in polar coordinates.
+
+# r'' = r * phi'^2 - gamma*M/r^2
+# phi'' = -2* r' / r * phi'
+
+include CompoundFunctions.LACE   # idivide
+
+coefficient(1): +1 -> r0' # -1 to move initially towards M, +1 to  move away from it
+coefficient(2): -1 -> -r0 # initial distance of m to M
+coefficient(3): +1 -> phi0' 
+coefficient(4): -1 -> -phi0 
+coefficient(5): +1 -> gamma*M
+
+iintegrate r*phi'^2, -gamma*M/r^2 -> -r' # input is r''
+   IC: r0'
+iintegrate -r' -> r
+  IC: -r0
+
+iintegrate -r'/r*phi', -r'/r*phi' -> -phi' # input is phi''
+  IC: phi0'
+iintegrate -phi' -> phi
+  IC: -phi0
+  
+multiply r, -phi' -> -r*phi'
+multiply -r*phi', -phi' -> r*phi'^2
+multiply r, r -> r^2
+idivide gamma*M, r^2 -> -gamma*M/r^2
+
+idivide -r', -r*phi' -> r'/r*phi'
+invert r'/r*phi' -> -r'/r*phi'
+  
+output(x): r      # distance
+output(y): phi    # angle