Question

For the given system, (a) Use Theorem \(6.2\) to show that the system is a Hamiltonian system. (b) Find a Hamiltonian function for the system. (c) Use computational software to graph the phase-plane trajectory passing through \((1,1)\). Also, indicate the direction of motion for the solution point. $$ \begin{aligned} &x^{\prime}=-8 y \\ &y^{\prime}=2 x \end{aligned} $$

Short answer

Answer: H = -x^2 - 4y^2 + C₀. 2. What is the direction of motion at the point (1,1)? Answer: At point (1,1), the direction of motion is upward and left.

Step-by-step solution

Show that the system is Hamiltonian using Theorem 6.2

Theorem 6.2 states that if a system of equations is of the form: $$ \begin{aligned} x^{\prime}=\frac{\partial H}{\partial y}, \\ y^{\prime}=-\frac{\partial H}{\partial x}, \\ \end{aligned} $$ then the system is Hamiltonian. Compare the given system to Theorem 6.2: $$ \begin{aligned} x^{\prime}=-8 y, \\ y^{\prime}=2 x. \\ \end{aligned} $$ Therefore, we have: $$ \begin{aligned} \frac{\partial H}{\partial y} = -8y \\ -\frac{\partial H}{\partial x} = 2x. \\ \end{aligned} $$ Since the given system matches the conditions in Theorem 6.2, we conclude that it is a Hamiltonian system.

Find the Hamiltonian function

To find the Hamiltonian function, we need to integrate the partial derivatives obtained in Step 1: $$ \begin{aligned} H = \int \frac{\partial H}{\partial y} dy = \int (-8y) dy = -4y^2 + C(x), \\ H = \int -\frac{\partial H}{\partial x} dx = - \int (2x) dx = -x^2 + C'(y). \\ \end{aligned} $$ Since the Hamiltonian function is unique up to a constant, we can equate these two expressions: $$ -4y^2 + C(x) = -x^2 + C'(y). $$ Let the constant term in Hamiltonian function be \(C_{0}\). Then, $$ H = -x^2 - 4y^2 + C_{0}. $$

Plot the phase-plane trajectory and indicate the direction of motion

To plot the trajectories, we will use a computational software (such as Python or MATLAB). The specific implementation might differ based on the chosen software. Here, we will provide a general guideline: 1. Define the system of differential equations (ODE). 2. Set the initial conditions (x(0) = 1, y(0) = 1). 3. Solve the system of ODE using the appropriate solver for the chosen software. 4. Plot the phase-plane trajectory. The direction of motion can be determined by evaluating the signs of \(x^{\prime}\) and \(y^{\prime}\). In this case, the direction of motion depends on whether the evaluation is in the positive or negative quadrant for \(x\) and \(y\). At point (1,1), we have \(x^{\prime} = -8(1) < 0\) and \(y^{\prime} = 2(1) > 0\), indicating an upward and left direction of motion at point (1,1).

Key concepts

Differential Equations

Understanding differential equations is crucial to analyzing a wide variety of natural phenomena ranging from physics to economics. Simply put, a differential equation is a mathematical equation that relates a function with its derivatives. In the context of dynamical systems, they can describe how a particular quantity changes over time or space.

In our exercise, we have a pair of first-order differential equations representing the rate at which one variable changes in relation to the other. Specifically, the use of x' and y' signify the derivatives of these variables with respect to time. These kinds of systems can often be visualized in terms of their phase-plane trajectory, representing the state of the system as a point whose coordinates are given by the values of x and y at any given time.

Phase-Plane Trajectory

A phase-plane trajectory is a graphical representation that helps visualize the behavior of systems of differential equations, such as our Hamiltonian system. Each point in the phase plane corresponds to a state of the system with coordinates (x, y), while the trajectory depicts the path followed by these points over time.

The trajectory is a crucial tool to visualize equilibrium points where the system is stable or unstable and periodic orbits which may exist in a dynamical system. The slope at each point of this trajectory is determined by the differential equations governing the system. Interpreting the trajectories can provide valuable insights into the underlying mechanics of the system being studied, such as whether the motion is in a closed orbit, which implies conservation properties, or if the system spirals out, indicating instability.

Computational Software for Graphing

Computational software such as MATLAB, Mathematica, and Python with libraries like Matplotlib or SciPy offer powerful tools for graphing complex equations and systems.

For our exercise, graphing software can numerically solve the system of differential equations and then plot the phase-plane trajectory. This involves creating a set of initial conditions, as given by the exercise (1,1) in our case, and using numerical solvers like the Euler or Runge-Kutta methods to approximate the system's behavior over time. Graphically, the software will render a trajectory revealing how variables interact and evolve, which is a practical way of recognizing patterns or predicting system behavior that may not be evident from the equations alone. For students, this visual representation aids comprehension and reinforces theoretical findings.

Theorem Application

Theorems provide the theoretical underpinnings necessary for understanding and proving particular properties of mathematical entities. In reference to the exercise, Theorem 6.2 plays a key role in confirming that a system of differential equations is Hamiltonian, which broadly means that it can be described by a Hamiltonian function, a concept related to energy conservation in physics.

To apply a theorem like Theorem 6.2, which provides the conditions for a Hamiltonian system, one needs to compare the given system with the criteria stated in the theorem. If the system meets these criteria, then certain conclusions can be drawn, and additional properties can be leveraged, such as the ease of finding conserved quantities. Using such a theorem not only helps validate the type of system at hand but also guides subsequent steps, like finding the Hamiltonian function itself; a crucial step in the analysis and solving of the system.