Frog and Toad Are Friends
Once, there were two good friends, Frog and Toad. They loved to go for walks, to look at stars, and watch the birds. And when they were not together, they would write each other letters and poems.
dearest toad,
if you want to read my poem about the pelicans, here is a draft.
your friend,
frog
Dear Frog,
I wrote a poem about the pelicans too. In my poem, they fly away. Here is a draft.
Love,
Toad
dearest toad,
when the pelicans launched into orbit, I looked at you sideways, circling and circling, the motion of the birds described by some differential equation.
your friend,
frog
Dear Frog,
Remember when we searched for the ladle of the Big Dipper and the pelicans swooped under the night? In my poem, the birds are ellipses obscuring stars, planets, and clusters of galaxies.
Love,
Toad
dearest toad,
and on that night we lie belly up toward the moon on the hill and you fold your hands into variables on your lap. you glance at me. are we orbiting each other, covered in space dust?
x
frog
Frog,
We are rambling in an ellipse, unclosed and spiraling inward, rambling like Mercury as it precesses. To determine the path of our precession from that night, I set up a dimensionless N-body problem using Newton’s Laws to craft separable differential equations, both of us orbiting some distance from the sun.
Love,
Toad
toad,
and then we are sitting again on the hill, near the sun. we find feathers and eat wet sandwiches without iced tea and you read me a story. “I’m all alone!” you say, lost in the woods of dialogue. I sit at your knees, ask, do I find you again?
x
frog
Frog,
There is a pelican in my heart. It likes to sit on the hill, smooth its feathers, and listen to another pelican sweep its wings in a circular orbit. The pelicans, unlike us, never calculate answers to N-body problems. They already know. You will find me again, and we will go for a walk together, say, I love you. Tell me, is there a pelican in your heart that turns in circles?
Love,
Toad
Frog and Toad sit in the sun together after a walk. They read each other old letters and poems. “Toad,” says Frog, “there is a pelican in my heart, turning and turning in circles.” Toad curls their arm around Frog. They were both very happy.
◆
N-body problem setup
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
#@author: Toad.
#title: N-body problem setup, as promised. Love, Toad.
#Dear Frog,
#Enclosed you will find <instructions> and code
#for simulating an N-body problem
#patterned after that Feynman lecture
#from the red hardcover adjacent to our collection
#of stories. This assumes a closed orbit
#for all celestial bodies, whether they are gas giants,
#rocks, or part of some non-binary system.
#I have modeled the orbit of Jupiter and Mercury
#around the sun in dimensionless form. I have not considered
#relativistic effects and although
#Einstein would lecture, you understand.
#The pelicans inside our rib cages feel
#the math before they precess inward,
#beat their wings, say,
import numpy as np
import math
import matplotlib .pyplot as plt
from scipy.integrate import odeint
#The pelicans simulate Mercury and Jupiter in orbit
def merjup(mjlist, t):
tempm=(mjlist[0]**(2) + mjlist[2]**(2))**(1.5)
tempj=(mjlist[4]**(2) + mjlist[6]**(2))**(1.5)
tempmj=((mjlist[0]-mjlist[4])**(2) + (mjlist[2]-mjlist[6])**(2))**(1.5)
# mjlist = [ xm, vxm, ym, vym, xj, vxj, yj, vyj ]
#and they are defined the same, in a list
#of differential equations,
#you declared
#when you laid with your belly up at the sun
#wanting to launch toward the plasma itself.
#Frog, inside you, the pelican
#clutches a string of variables
#that spans out its wings and whirls
derivs = [ mjlist[1], (((-(4*math.pi**(2))*mjlist[0])/tempm)-(4*math.pi**(2)*(1.89813*10**(27))/(1.989*10**(30))*(mjlist[0]-mjlist[4])/tempmj)),
mjlist[3], (((-(4*math.pi**(2))*mjlist[2])/tempm)-(4*math.pi**(2)*(1.89813*10**(27))/(1.989*10**(30))*(mjlist[2]-mjlist[6])/tempmj)),
mjlist[5], (((-(4*math.pi**(2))*mjlist[4])/tempj)-(4*math.pi**(2)*(3.30104*10**(23))/(1.989*10**(30))*(mjlist[4]-mjlist[0])/tempmj)),
mjlist[7], (((-(4*math.pi**(2))*mjlist[6])/tempj)-(4*math.pi**(2)*(3.30104*10**(23))/(1.989*10**(30))*(mjlist[6]-mjlist[2])/tempmj))]
return derivs
#its purple tips in the ether. The pelican
#will orbit the solar system with a velocity long
#enough to measure distance in Astronomical Units
#—lines bending into a small, black ellipse—
#to plot the solution to this 3-body problem:
t = np.linspace(0.0, 100, 10001)
ic = (float(0.30749695009788227), 0.0, 0.0, float(12.4413513), float(4.9511259380102592), 0.0, 0.0, float(2.89126925))
n = odeint
soln = odeint(merjup, ic, t, rtol=1e-11, atol=1e-11)
#in which the answer is drawn but not understood:
plt.plot(soln[:,0], soln[:,2], color=’black’, linewidth=0.1)
plt.plot(soln[:,4], soln[:,6], color=’black’)
plt.scatter(0,0, color=’black’)
plt.title(‘Mercury and Jupiter’,
fontsize=20)
plt.xlabel(‘x’, fontsize=14)
plt.ylabel(‘y’)
plt.grid(color=’black’)
plt.show()
#Imagine a sun at the center of the birds
#as they beat their wings against their spines.
◆
“Frog and Toad Are Friends” is indebted to Arnold Lobel and G.B. In particular, Frog’s second letter and the phrase “belly up” is indebted to G.B. The first sentence of “Frog and Toad Are Friends” draws on a similar statement made by Frog in “The Story.” The reference to wet sandwiches and iced tea is a nod to “Alone.”
