322 lines
9.2 KiB
R
322 lines
9.2 KiB
R
# Project 4 for the University of Tulsa's CS-7863 Sci-Stat Course
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# Higher Order Differential Equations and Shooting Method
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# Professor: Dr. McKinney, Spring 2023
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# Noah L. Schrick - 1492657
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## 1. Transform the second order ODE into a system of two first order ODEs
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# use ode45 or rk4sys to solve for the motion of the plane pendulum
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if (!require("pracma")) install.packages("pracma")
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library(pracma)
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tmin <- 0
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tmax <- 7
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theta0 <- pi/4
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omega0 <- 0
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y.init <- c(theta0, omega0)
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g <- 9.81
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L <- 1.0
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## Substitution Tricks
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# theta_1 = theta_prime
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# theta_2 = theta
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# theta_1_prime = theta_pp
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# theta_2_prime = theta_prime = theta_1
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## Equation Replacements
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ode_1.f <- function(t, y, k){
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theta <- y[1]
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omega <- y[2]
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g <- k[1]
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L <- k[2]
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omega_p <- -k[1]/k[2] * sin(theta)
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theta_p <- omega
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as.matrix(c(theta_p, omega_p))
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}
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p.params <- c(g, L)
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pend.sol <- ode45(f=function(t,y){ode_1.f(t,y,k=p.params)},
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y=y.init, t0=tmin, tfinal=tmax)
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# a. Plot
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plot(pend.sol$t,pend.sol$y[,1],type="l",col="blue",
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ylim=c(-2.3,2.3), xlab="time",ylab="amplitude")
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par(new=T)
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lines(pend.sol$t,pend.sol$y[,2],type="l",col="red")
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abline(h=0)
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legend("topleft", # coordinates, "topleft" etc
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c("angle","omega"), # label
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lty=c(1,1), # line
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lwd=c(2.5,2.5), # weight
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#cex=.8,
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bty="n", # no box
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col=c("blue","red") # color
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)
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# b. Overlay a plot of the small-angle approx
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plot(pend.sol$t,pend.sol$y[,1],type="l",col="blue",
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ylim=c(-2.3,2.3), xlab="time",ylab="amplitude")
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par(new=T)
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lines(seq(tmin,tmax,len=length(pend.sol$t)),
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theta0*cos(sqrt(g/L)*seq(tmin,tmax,len=length(pend.sol$t))),type="l",col="red")
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abline(h=0)
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legend("topleft", # coordinates, "topleft" etc
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c("angle","small angle approx"), # label
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lty=c(1,1), # line
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lwd=c(2.5,2.5), # weight
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#cex=.8,
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bty="n", # no box
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col=c("blue","red") # color
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)
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## 2. Use ode45 to solve the damped harmonic oscillator
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tmin <- 0
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tmax <- 10
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yo <- 1
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ybaro <- 0
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## Substitution Tricks
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# y1 = y_prime
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# y_2 = y
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# y_1_prime = y_pp
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# y_2_prime = y_prime = y_1
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# Eqs:
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# m*y_1_prime + c*y1 + k*y_2 = 0
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# y_1_prime = (-c*y_1 - k*y_2)/m
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# y_2_prime = y_1
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# a
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m <- 1
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c <- 1
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kvar <- 2
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dho.f <- function(t, y, kpass){
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yn <- y[1]
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ydot <- y[2]
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# c k m
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ypp <- (-kpass[2]*yn - kpass[3]*ydot)/kpass[1]
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as.matrix(c(ydot,ypp))
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}
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y.init <- c(yo, ybaro)
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dho.params <- c(m,c,kvar)
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dho.sol <- ode45(f=function(t,y){dho.f(t,y,kpass=dho.params)},
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y=y.init, t0=tmin, tfinal=tmax)
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# plot pos vs time
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pos <- (-m*dho.sol$y[,2] - c*dho.sol$y[,1])/kvar
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plot(dho.sol$t, pos ,type="o",col="blue",
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ylim=c(-1.0,1.0), xlab="time",ylab="amplitude")
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par(new=T)
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lines(dho.sol$t,dho.sol$y[,1],type="o",col="red")
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lines(dho.sol$t,dho.sol$y[,2],type="o",col="green")
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legend("topright", # coordinates, "topleft" etc
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c("y","yprime", "ydoubleprime"), # label
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lty=c(1,1), # line
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lwd=c(2.5,2.5), # weight
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#cex=.8,
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bty="n", # no box
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col=c("blue","red", "green") # color
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)
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# b
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m <- 1
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c <- 2
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k <- 1
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dho.params <- c(m,c,k)
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y.init <- c(yo, ybaro)
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dho.params <- c(m,c,kvar)
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dho.sol <- ode45(f=function(t,y){dho.f(t,y,kpass=dho.params)},
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y=y.init, t0=tmin, tfinal=tmax)
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# plot pos vs time
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pos <- (-m*dho.sol$y[,2] - c*dho.sol$y[,1])/kvar
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plot(dho.sol$t, pos ,type="o",col="blue",
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ylim=c(-1.0,1.0), xlab="time",ylab="amplitude")
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par(new=T)
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lines(dho.sol$t,dho.sol$y[,1],type="o",col="red")
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lines(dho.sol$t,dho.sol$y[,2],type="o",col="green")
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legend("topright", # coordinates, "topleft" etc
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c("y","yprime", "ydoubleprime"), # label
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lty=c(1,1), # line
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lwd=c(2.5,2.5), # weight
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#cex=.8,
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bty="n", # no box
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col=c("blue","red", "green") # color
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)
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## 3. Use ode45/rk4sys to solve for the traj of a projectile thrown vertically
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# a. IVP
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tmin <- 0
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tmax <- 2.04
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xo <- 0
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vo <- 10
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projectile.f <- function(t,y){
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g <- 9.81
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v <- y[2] # y1dot
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a <- -g # could be any f, y'' = f(t,y)
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matrix(c(v,a))
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}
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y.init <- c(xo, vo)
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ivp.sol <- ode45(f=function(t,y){projectile.f(t,y)},
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y=y.init, t0=tmin, tfinal=tmax)
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ivp.sol.rk4 <- rk4sys(projectile.f, 0, 2.04, c(0, 10), 100)
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plot(ivp.sol$t, ivp.sol$y[,1] ,type="o",col="blue",
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ylim = c(-0.5, 5.5), xlab="time",ylab="height")
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par(new=T)
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lines(ivp.sol.rk4$x, ivp.sol.rk4$y[,1] ,type="o",col="red",
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xlab="time",ylab="height")
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legend("topright", # coordinates, "topleft" etc
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c("ODE45, default","RK4Sys, 100 steps"), # label
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lty=c(1,1), # line
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lwd=c(2.5,2.5), # weight
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#cex=.8,
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bty="n", # no box
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col=c("blue","red", "green") # color
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)
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# b. BVP
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yo <- 0
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ymax <- 0
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proj.obj <- function(v0, y0=0, tfinal){
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# minimize w.r.t. v0
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proj.sol <- ode45(projectile.f,
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y=c(y0, v0), t0=0, tfinal=tfinal)
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final_index <- length(proj.sol$t)
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yf <- proj.sol$y[final_index,1] # want equal to right boundary
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log(abs(yf)) # minimize this
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}
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# user specifies tfinal and yfinal for BVP
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v_best <- optimize(proj.obj,
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interval=c(1,100), #bisect-esque interval
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tol=1e-10,
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y0=0, tfinal=10) # un-optimized obj params
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v_best$minimum # best v0
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best.sol <- rk4sys(projectile.f, a=0, b=10, y0=c(0, v_best$minimum),
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n=20) # 20 integration stepstmax
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plot(best.sol$x, best.sol$y[,1], type="l")
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computeHeights <- function(xo, vo, tmin){
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height <- xo + vo*tmin -0.5*9.81*tmin^2
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}
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height <- computeHeights(best.sol$x, best.sol$y[,1], seq(0,10,len=21))
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# c. Shooting method for damped oscillator with perturbation parameter
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yo <- 0
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y1 <- 1
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tmin <- 0
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tfinal <- 2
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pfirst <- 0.5
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psec <- 0.05
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dho.pert.f <- function(t, y, kpass){
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yn <- y[1]
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ydot <- y[2]
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ep <- kpass[1]
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# c m
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ypp <- (-(1+ep)*yn - ydot)/ep
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as.matrix(c(ydot,ypp))
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}
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proj.obj <- function(v0, y0=yo, tfinal){
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# minimize w.r.t. v0
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proj.sol <- ode45(dho.pert.f,
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y=c(yo, y1), kpass=pfirst, t0=tmin, tfinal=tfinal)
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final_index <- length(proj.sol$t)
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yf <- proj.sol$y[final_index,1] # want equal to right boundary
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log(abs(yf)) # minimize this
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}
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# user specifies tfinal and yfinal for BVP
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ep_best <- optimize(proj.obj,
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interval=c(1,100), #bisect-esque interval
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tol=1e-10,
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y0=0, tfinal=2) # un-optimized obj params
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ep_best$minimum # best v0
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best.sol <- rk4sys(dho.pert.f, a=0, b=2, y0=c(0, ep_best$minimum),
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n=20) # 20 integration stepstmax
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# analytical sol
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# yprime = -y
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## 4. Position of the earth and moon
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# a. Plotly
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G <- 6.673e-11 # m^3 kg-1 s^-2
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M_S <- 1.9891e30 # sun kg
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M_E <- 5.98e24 # earth kg
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M_m <- 7.32e22 # moon kg
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mu_sun <- G*M_S*(86400^2)/1e9 # km^3/days^2
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# t0: jan 1, 1999 00:00:00am
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x0_earth <- c(-27115219762.4, 132888652547.0, 57651255508.0)/1e3 # km
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v0_earth <- c(-29794.2199907, -4924.33333333,-2135.59540741)*(86400)/1e3
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# m/s -> km/day
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x0_moon <- c(-27083318944, 133232649728, 57770257344)/1e3 # km
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v0_moon <- c(-30864.2207031, -4835.03349304, -2042.89546204)*(86400)/1e3
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# m/s -> km/day
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sunearth.f <- function(t,y){
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G <- 6.673e-11 # m^3 kg^-1 s^-2
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M_S <- 1.9891e30
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M_E <- 5.98e24
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M_m <- 7.32e22 # not using right now
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mu_sun <- G*M_S*(86400^2)/1e9 # km^3/days^2
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x_earth <- y[1:3] # xyz position of earth wrt sun
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v_earth <- y[4:6] # xyz velocity of earth, also part of ydot
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r_earth <- sqrt(sum(x_earth^2)) # distance earth to sun
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acc_earth <- -mu_sun*x_earth/r_earth^3 # part of ydot
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ydot <- c(v_earth, acc_earth)
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return(matrix(ydot))
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}
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y.init.earth <- c(x0_earth, v0_earth)
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sunearth.sol <- rk4sys(f=sunearth.f, a=0, b=365.25, y0=y.init.earth, n=365)
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sunearth.df <- data.frame(x=sunearth.sol$y[,1],
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y=sunearth.sol$y[,2],
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z=sunearth.sol$y[,3])
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y.init.moon <- c(x0_moon, v0_moon)
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sunmoon.sol <- rk4sys(f=sunearth.f, a=0,b=365.25, y0=y.init.moon, n=365)
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sunmoon.df <- data.frame(x=sunmoon.sol$y[,1],
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y=sunmoon.sol$y[,2],
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z=sunmoon.sol$y[,3])
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if (!require("plotly")) install.packages("plotly")
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library(plotly)
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fig <- plot_ly(sunmoon.df, x = ~x, y = ~y, z = ~z, name="moon", type = "scatter3d", mode="markers")
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fig <- fig %>% layout(scene = list(xaxis = list(title = 'x'),
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yaxis = list(title = 'y'),
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zaxis = list(title = 'z')))
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fig <- add_trace(fig, x = 0, y = 0, z=0, mode="markers", color = I("red"), name="sun")
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fig <- add_trace(fig, x = sunearth.df[1]$x, y = sunearth.df[2]$y, z = sunearth.df[3]$z, mode="markers", color = I("green"), name="earth")
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fig
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mean(sqrt(rowSums(cbind(sunearth.sol$y[,1]^2,
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sunearth.sol$y[,2]^2,
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sunearth.sol$y[,3]^2))))
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# 150 million km
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mean(sqrt(rowSums(cbind((sunearth.sol$y[,1] - sunmoon.sol$y[,1])^2,
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(sunearth.sol$y[,2] - sunmoon.sol$y[,2])^2,
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(sunearth.sol$y[,3] - sunmoon.sol$y[,3])^2))))
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# Not correct: giving 60m km - should be ~380k
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# b. Find eclipses
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norm_vec <- function(x) sqrt(rowSums(cbind((x^2))))
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eclipses <- norm_vec(sunearth.sol$y) %*% norm_vec(sunmoon.sol$y)
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# c. keep orbiter at L2 Lagrange point for a year, ignoring the moon's effect
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