# model for the frogs/dolphins
# phimax is maximum angle of periodic driver
# w is frequency, k is magnetic force, mu is friction, a outer radius
# l is radius of frogs,m is mass
# x is angle, xp the derivative
phi=phimax*sin(w*t)
x0=a*sin(phi)
y0=-a*cos(phi)
# forces computed with MAPLE
t1 = sin(x) 
t2 = l*t1 
t3 = -t2-x0 
t4 = t3*t3 
t5 = cos(x) 
t6 = l*t5 
t7 = t6-y0 
t8 = t7*t7 
t10 = (t4+t8)^2
t19 = t2-x0 
t20 = t19*t19 
t21 = -t6-y0 
t22 = t21*t21  
t24 = (t20+t22)^2
t33 = -k/t10*(-2.0*t3*l*t5-2.0*t7*l*t1)-k/t24*(2.0*t19*l*t5+2.0*t21*l*t1)
# the dynamics
x'=xp
xp'=(-mu*xp-t33)/(2*m*l^2)
# some auxiliary stuff for graphing
x1=l*sin(x)
y1=-l*cos(x)
x2=-x1
y2=-y1
par w=1,phimax=1.5,mu=.05,k=.02,l=1,a=1.07,m=1
init x=1.5,xp=0
@ total=200,meth=qualrk,tol=1e-6
@ xhi=200,ylo=-50,yhi=50,bound=10000
done

# dolphin2.ani
line .5;.5;.5;0;$RED;4
line 0;0;1;0;$RED;4
circle   .5;.5;.4;$BLACK
fcircle .5+.4*sin(phi);.5-.4*cos(phi);.035;$BLACK
line .5-.35*sin(x);.5+.35*cos(x);.5+.35*sin(x);.5-.35*cos(x);$GREEN;4
fcircle .5-.35*sin(x);.5+.35*cos(x);.03;$BLUE
fcircle .5+.35*sin(x);.5-.35*cos(x);.03;$BLUE
end