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Chapter 8: Problem 11

Write a program to solve the differential equation $$ \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}-\left(\frac{\mathrm{d} x}{\mathrm{~d} t}\right)^{2}+x+5=0 $$ using the leapfrog method. Solve from \(t=0\) to \(t=50\) in steps of \(h=0.001\) with initial condition \(x=1\) and \(\mathrm{d} x / \mathrm{d} t=0\). Make a plot of your solution showing \(x\) as a function of \(t\).

Short Answer

Expert verified
Use the Leapfrog method in Python to solve the differential equation numerically and plot the results with Matplotlib.

Step by step solution

01

- Understand the Leapfrog Method

The leapfrog method is a numerical technique widely used for solving differential equations. For a second-order differential equation such as \(\frac{d^2 x}{dt^2} = f(x, \frac{dx}{dt}, t)\), it involves updating positions and velocities in a staggered manner.
02

- Discretize the Differential Equation

Rewrite the differential equation \(\frac{d^2 x}{dt^2}-\bigg(\frac{dx}{dt}\bigg)^2 + x + 5 = 0\). Let \(\frac{dx}{dt} = v\) and \(\frac{d^2 x}{dt^2} = a = v \frac{dv}{dx}\).
03

- Leapfrog Algorithm Steps

Initialize the initial conditions: \(x(0) = 1\), \(v(0) = 0\). The leapfrog steps are: 1. Compute initial acceleration \(a_0 = v_0 \frac{dv}{dt} = -vx - 5 + v^2\). 2. Update x by half a time step: \(x_{0.5} = x_0 + \frac{h}{2}v_0\). 3. Perform the main loop for each time step \(t \).
04

- Implementing the Leapfrog Method in Python

Write the following Python code:```pythonimport numpy as npimport matplotlib.pyplot as plt# Parametersh = 0.001T = 50N = int(T/h) + 1# Initialize arraysx = np.zeros(N)v = np.zeros(N)t = np.linspace(0, T, N)# Initial conditionsx[0] = 1v[0] = 0# Leapfrog methodfor i in range(N-1): if i == 0: a0 = v[0]**2 - x[0] - 5 x_half = x[0] + 0.5 * h * v[0] v[1] = v[0] + h * a0 x[1] = x_half + 0.5 * h * v[1] else: a = v[i]**2 - x[i] - 5 v[i+1] = v[i] + h * a x[i+1] = x[i] + h * v[i+1]# Plot the solutionplt.plot(t, x)plt.xlabel('t')plt.ylabel('x(t)')plt.title('Solution using Leapfrog Method')plt.show()```
05

- Plot the Solution

Use Matplotlib to plot \(x(t)\). The code provided already includes this step. Run the script, and observe the plot generated, showing \(x\) as a function of \(t\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Leapfrog Method
The leapfrog method is an effective numerical technique used to solve differential equations, particularly second-order ones. It gets its name from the way it 'leaps' over the intermediate steps. The method works by staggering the updates to the position and velocity variables.

For a second-order differential equation like \(\frac{d^2 x}{dt^2} = f(x, \frac{dx}{dt}, t)\), the leapfrog method updates the position and velocity at different points in the time step:
  • First, the position is updated by half a time step.
  • Then, the velocity is updated for the full time step.
  • Finally, the position is updated again by another half time step.
This staggered approach improves the stability and accuracy of the method, making it particularly useful for long-time simulations. In the given problem, the leapfrog method is applied to an equation involving \(\frac{d^2 x}{dt^2} - (\frac{dx}{dt})^2 + x + 5 = 0\). The variables, initial conditions, and acceleration must all be handled with the specific leapfrog steps to accurately resolve the motion over time.
Numerical Methods
Numerical methods are techniques used to approximate solutions to mathematical problems that may not have exact solutions. These methods play a crucial role in fields such as physics, engineering, and finance.

Here’s a quick rundown of why numerical methods are important:
  • They handle complex problems where analytical solutions are impractical or impossible.
  • They are adaptable, allowing for custom handling of initial conditions and boundary values.
  • They can be implemented in computational tools to handle large datasets and extensive calculations.
The leapfrog method, which falls under numerical methods, is specifically used for solving differential equations by iterating over small time steps. This helps in effectively modeling the evolution of systems over time without resorting to simplified analytical models.
Python Programming
Python programming is an excellent choice for implementing numerical methods, including the leapfrog method. Python’s simplicity and readability make it ideal for educational purposes, while its powerful libraries support complex mathematical and graphical tasks.

In the provided solution, Python code is used to solve the differential equation using the leapfrog method. Key aspects include:
  • **NumPy**: This library offers support for large multi-dimensional arrays and matrices, along with a collection of mathematical functions to operate on these arrays.
  • **Matplotlib**: It's a plotting library used in Python for data visualization. In this case, Matplotlib is utilized to plot the solution of the differential equation.
  • **Initial Setup**: Arrays for position, velocity, and time are initialized. Initial conditions are set, and parameters such as step size are defined.
  • **Loop Implementation**: The loop runs through each time step, updating positions and velocities according to the leapfrog method's staggered scheme.
  • **Visualization**: Finally, the results are plotted to visualize how the position changes over time.
By leveraging Python's capabilities, complex numerical methods can be implemented efficiently and effectively, making it accessible even for beginners.

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Most popular questions from this chapter

One of the most celebrated sets of differential equations in physics is the Lorenz equations: $$ \frac{\mathrm{d} x}{\mathrm{~d} t}=\sigma(y-x), \quad \frac{\mathrm{d} y}{\mathrm{~d} t}=r x-y-x z, \quad \frac{\mathrm{d} z}{\mathrm{~d} t}=x y-b z, $$ where \(\sigma, r\), and \(b\) are constants. (The names \(\sigma, r\), and \(b\) are odd, but traditional-they are always used in these equations for historical reasons.) These equations were first studied by Edward Lorenz in 1963 , who derived them from a simplified model of weather patterns. The reason for their fame is that they were one of the first incontrovertible examples of deterministic chaos, the occurrence of apparently random motion even though there is no randomness built into the equations. We encountered a different example of chaos in the logistic map of Exercise \(3.6\). a) Write a program to solve the Lorenz equations for the case \(\sigma=10, r=28\), and \(b=\frac{8}{3}\) in the range from \(t=0\) to \(t=50\) with initial conditions \((x, y, z)=\) \((0,1,0)\). Have your program make a plot of \(y\) as a function of time. Note the unpredictable nature of the motion. (Hint: If you base your program on previous ones, be careful. This problem has parameters \(r\) and \(b\) with the same names as variables in previous programs-make sure to give your variables new names, or use different names for the parameters, to avoid introducing errors into your code.) b) Modify your program to produce a plot of z against \(x\). You should see a picture of the famous "strange attractor" of the Lorenz equations, a lop-sided butterflyshaped plot that never repeats itself.

Many elementary mechanics problems deal with the physics of objects moving or flying through the air, but they almost always ignore friction and air resistance to make the equations solvable. If we're using a computer, however, we don't need solvable equations. Consider, for instance, a spherical cannonball shot from a cannon standing on level ground. The air resistance on a moving sphere is a force in the opposite direction to the motion with magnitude $$ F=\frac{1}{2} \pi R^{2} \rho C v^{2}, $$ where \(R\) is the sphere's radius, \(\rho\) is the density of air, \(v\) is the velocity, and \(C\) is the so-called coefficient of drag (a property of the shape of the moving object, in this case a sphere). a) Starting from Newton's second law, \(F=m a\), show that the equations of motion for the position \((x, y)\) of the cannonball are $$ \ddot{x}=-\frac{\pi R^{2} \rho C}{2 m} \dot{x} \sqrt{\dot{x}^{2}+\dot{y}^{2}}, \quad \ddot{y}=-g-\frac{\pi R^{2} \rho C}{2 m} \dot{y} \sqrt{\dot{x}^{2}+\dot{y}^{2}}, $$ where \(m\) is the mass of the cannonball, \(g\) is the acceleration due to gravity, and \(\dot{x}\) and \(x\) are the first and second derivatives of \(x\) with respect to time. b) Change these two second-order equations into four first-order equations using the methods you have learned, then write a program that solves the equations for a cannonball of mass \(1 \mathrm{~kg}\) and radius \(8 \mathrm{~cm}\), shot at \(30^{\circ}\) to the horizontal with initial velocity \(100 \mathrm{~m} \mathrm{~s}^{-1}\). The density of air is \(\rho=1.22 \mathrm{~kg} \mathrm{~m}^{-3}\) and the coefficient of drag for a sphere is \(C=0.47\). Make a plot of the trajectory of the cannonball (i.e., a graph of \(y\) as a function of \(x\) ). c) When one ignores air resistance, the distance traveled by a projectile does not depend on the mass of the projectile. In real life, however, mass certainly does make a difference. Use your program to estimate the total distance traveled (over horizontal ground) by the cannonball above, and then experiment with the program to determine whether the cannonball travels further if it is heavier or lighter. You could, for instance, plot a series of trajectories for cannonballs of different masses, or you could make a graph of distance traveled as a function of mass. Describe briefly what you discover.

Consider the one-dimensional, time-independent Schródinger equation in a harmonic (i.e., quadratic) potential \(V(x)=V_{0} x^{2} / a^{2}\), where \(V_{0}\) and \(a\) are constants. a) Write down the Schrödinger equation for this problem and convert it from a second-order equation to two first-order ones, as in Example 8.9. Write a program, or modify the one from Example 8.9, to find the energies of the ground state and the first two excited states for these equations when \(m\) is the electron mass, \(V_{0}=50 \mathrm{eV}\), and \(a=10^{-11} \mathrm{~m}\). Note that in theory the wavefunction goes all the way out to \(x=\pm \infty\), but you can get good answers by using a large but finite interval. Try using \(x=-10 a\) to \(+10 a\), with the wavefunction \(\psi=0\) at both boundaries. (In effect, you are putting the harmonic oscillator in a box with impenetrable walls.) The wavefunction is real everywhere, so you don't need to use complex variables, and you can use evenly spaced points for the solution-there is no need to use an adaptive method for this problem. The quantum harmonic oscillator is known to have energy states that are equally spaced. Check that this is true, to the precision of your calculation, for your answers. (Hint: The ground state has energy in the range 100 to \(200 \mathrm{eV}\).) b) Now modify your program to calculate the same three energies for the anharmonic oscillator with \(V(x)=V_{0} x^{4} / a^{4}\), with the same parameter values. c) Modify your program further to calculate the properly normalized wavefunctions of the anharmonic oscillator for the three states and make a plot of them, all on the same axes, as a function of \(x\) over a modest range near the origin-say \(x=-5 a\) to \(x=5 a\). To normalize the wavefunctions you will have to calculate the value of the integral \(\int_{-\infty}^{\infty}|\psi(x)|^{2} \mathrm{~d} x\) and then rescale \(\psi\) appropriately to ensure that the area under the square of each of the wavefunctions is 1. Either the trapezoidal rule or Simpson's rule will give you a reasonable value for the integral. Note, however, that you may find a few very large values at the end of the array holding the wavefunction. Where do these large values come from? Are they real, or spurious? One simple way to deal with the large values is to make use of the fact that the system is symmetric about its midpoint and calculate the integral of the wavefunction over only the left-hand half of the system, then double the result. This neatly misses out the large values. The methods described in this section allow us to calculate solutions of the Schrödinger equation in one dimension, but the real world, of course, is threedimensional. In three dimensions the Schrödinger equation becomes a partial differential equation, whose solution requires a different set of techniques. Partial differential equations are the topic of the next chapter.

This exercise asks you to calculate the orbits of two of the planets using the BulirschStoer method. The method gives results significantly more accurate than the Verlet method used to calculate the Earth's orbit in Exercise 8.12. The equations of motion for the position \(x, y\) of a planet in its orbital plane are the same as those for any orbiting body and are derived in Exercise \(8.10\) on page 361: $$ \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}=-G M \frac{x}{r^{3}}, \quad \frac{\mathrm{d}^{2} y}{\mathrm{~d} t^{2}}=-G M \frac{y}{r^{3}}, $$ where \(\mathrm{G}=6.6738 \times 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}\) is Newton's gravitational constant, \(M=1.9891 \times\) \(10^{30} \mathrm{~kg}\) is the mass of the Sun, and \(r=\sqrt{x^{2}+y^{2}}\). Let us first solve these equations for the orbit of the Earth, duplicating the results of Exercise 8.12, though with greater accuracy. The Earth's orbit is not perfectly circular, but rather slightly elliptical. When it is at its closest approach to the Sun, its perihelion, it is moving precisely tangentially (i.e., perpendicular to the line between itself and the Sun) and it has distance \(1.4710 \times 10^{11} \mathrm{~m}\) from the Sun and linear velocity \(3.0287 \times\) \(10^{4} \mathrm{~ms}^{-1}\). a) Write a program, or modify the one from Example 8.7, to calculate the orbit of the Earth using the Bulirsch-Stoer method to a positional accuracy of \(1 \mathrm{~km}\) per year. Divide the orbit into intervals of length \(H=1\) week and then calculate the solution for each interval using the combined modified midpoint/Richardson extrapolation method described in this section. Make a plot of the orbit, showing at least one complete revolution about the Sun. b) Modify your program to calculate the orbit of the dwarf planet Pluto. The distance between the Sun and Pluto at perihelion is \(4.4368 \times 10^{12} \mathrm{~m}\) and the linear velocity is \(6.1218 \times 10^{3} \mathrm{~ms}^{-1}\). Choose a suitable value for \(H\) to make your calculation run in reasonable time, while once again giving a solution accurate to \(1 \mathrm{~km}\) per year. You should find that the orbit of Pluto is significantly elliptical-much more so than the orbit of the Earth. Pluto is a Kuiper belt object, similar to a comet, and (unlike true planets) it's typical for such objects to have quite elliptical orbits.

The Lotka-Volterra equations are a mathematical model of predator-prey interactions between biological species. Let two variables \(x\) and \(y\) be proportional to the size of the populations of two species, traditionally called "rabbits" (the prey) and "foxes" (the predators). You could think of \(x\) and \(y\) as being the population in thousands, say, so that \(x=2\) means there are 2000 rabbits. Strictly the only allowed values of \(x\) and \(y\) would then be multiples of \(0.001\), since you can only have whole numbers of rabbits or foxes. But \(0.001\) is a pretty close spacing of values, so it's a decent approximation to treat \(x\) and \(y\) as continuous real numbers so long as neither gets very close to zero. In the Lotka-Volterra model the rabbits reproduce at a rate proportional to their population, but are eaten by the foxes at a rate proportional to both their own population and the population of foxes: $$ \frac{\mathrm{d} x}{\mathrm{~d} t}=\alpha x-\beta x y, $$ where \(\alpha\) and \(\beta\) are constants. At the same time the foxes reproduce at a rate proportional to the rate at which they eat rabbits-because they need food to grow and reproducebut also die of old age at a rate proportional to their own population: $$ \frac{d y}{d t}=\gamma x y-\delta y $$ where \(\gamma\) and \(\delta\) are also constants. a) Write a program to solve these equations using the fourth-order Runge-Kutta method for the case \(\alpha=1, \beta=\gamma=0.5\), and \(\delta=2\), starting from the initial condition \(x=y=2\). Have the program make a graph showing both \(x\) and \(y\) as a function of time on the same axes from \(t=0\) to \(t=30\). (Hint: Notice that the differential equations in this case do not depend explicitly on time \(t\)-in vector notation, the right-hand side of each equation is a function \(f(\mathbf{r})\) with no \(t\) dependence. You may nonetheless find it convenient to define a Python function \(f(r, t)\) including the time variable, so that your program takes the same form as programs given earlier in this chapter. You don't have to do it that way, but it can avoid some confusion. Several of the following exercises have a similar lack of explicit time-dependence.) b) Describe in words what is going on in the system, in terms of rabbits and foxes.
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