ellenjxu / diffeq

MATH 206 final project visualizations

Differential equations

Problem (polar destroyer):

(ds/dt)^2 	=	 (dx/dt)^2 + (dy/dt)^2
		= 	r^2 (dθ/dt)^2 + (dr/dt)^2
(√3/ t) dt 	= 	d θ
(√3 ln t) + c= 	θ 
√3 ln (⅕) + c	 = 	0
c		 = 	-√3 ln (⅕)

θ(t)		 =  	(√3 ln t)  - √3 ln (⅕)
recall: r/15 = t

θ(r) 		= 	 (√3 ln r/15)  - √3 ln (⅕)
θ(r) 		= 	√3 ln (r/3)
r		= 	3e^(θ/√3)

Time:

max θ = 2 π
max t of curve = ⅕(e^(2π/√3)) 
upper bound of T):  ⅕(e^(2π/√3))  + ⅕ 

Visualize solution (logarithmic curve) of DE for problem of two bodies in 2D space. Final project for MATH 206.

Logarithmic spiral has many special properties: Self-similarity (i.e. zooming out makes rotation of itself) Turnings of a logarithmic spiral increase in geometric progression Constant angle between an intersecting circle centred at the origin and a logarithmic spiral

Particularly, these properties imply that when you increase the radius (ex. circle), there will be a constant angle of rotation no matter what size the circle is.

Same pattern as hawks circling to catch prey, or hurricane!