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. $$ \begin{aligned} &x^{\prime}=x-x^{2}+1 \\ &y^{\prime}=-y+2 x y+4 x \end{aligned} $$

Short answer

Additionally, what is the direction of motion for the solution point (1, 1)? Answer: Yes, the given system is a Hamiltonian system. The Hamiltonian function for the system is H(x, y) = (2y + 4)x + D(y) + C. The direction of motion for the solution point (1, 1) can be determined by graphing the phase-plane trajectory using computational software, and examining the arrows on the trajectory.

Step-by-step solution

Showing that the system is a Hamiltonian system using Theorem 6.2

Theorem 6.2 states that a system of differential equations $$ \begin{aligned} &x^{\prime}=P(x, y) \\ &y^{\prime}=Q(x, y) \end{aligned} $$ is a Hamiltonian system if there exists a scalar function H(x, y) such that $$ \begin{aligned} &P(x, y) = \frac{\partial H}{\partial y}\\ &Q(x, y) = -\frac{\partial H}{\partial x} \end{aligned} $$ Now, let us examine the given system: $$ \begin{aligned} &x^{\prime}=x-x^{2}+1 \\\ &y^{\prime}=-y+2 x y+4 x \end{aligned} $$ To show that it is a Hamiltonian system, we need to find H(x, y) such that the partial derivatives of H with respect to x and y are equal to the given system equations. First, we take the partial derivatives of the system with respect to x and y. $$ \begin{aligned} &\frac{\partial P}{\partial y} = 0 \\ &\frac{\partial Q}{\partial x} = 2y + 4 \end{aligned} $$ Now, we need to find the H(x, y) function such that $$ \begin{aligned} \frac{\partial H}{\partial y} &= 0 \\ -\frac{\partial H}{\partial x} &= 2y + 4 \end{aligned} $$

Finding a Hamiltonian function for the system

To find H(x, y), we can integrate the partial derivatives. Since \(\frac{\partial H}{\partial y} = 0\), we have $$ H(x, y) = G(x) + C $$ where G(x) is a function of x and C is an integration constant. Now, we need to find G(x) such that $$ -\frac{\partial H}{\partial x} = -\frac{\partial G}{\partial x} = 2y + 4 $$ Integrating the equation above with respect to x, we get $$ G(x) = (2y + 4)x + D(y) $$ where D(y) is another integration constant. Thus, the Hamiltonian function for the system is $$ H(x, y) = (2y + 4)x + D(y) + C $$

Graphing the phase-plane trajectory and determine the direction of motion

Using computational software, we can graph the phase-plane trajectory. For this exercise, it is recommended to use software such as Matlab, Python, or Mathematica to plot the phase-plane trajectory passing through the initial point (1, 1). Before plotting, please verify that you are using the correct Hamiltonian function that you found in Step 2, and input the initial condition (1, 1). Once you have graphed the phase-plane trajectory, you can determine the direction of motion by examining the arrows that the software places on the trajectory. The arrows should always be pointing in the direction of increasing time. For example, if the point (1, 1) is a part of a counterclockwise trajectory around a center, the arrows should point in the counterclockwise direction.

Key concepts

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are essential in modeling various dynamic systems where quantities change over time. In our exercise, we have a system of differential equations:

• \(x^{\prime}=x-x^{2}+1\)
• \(y^{\prime}=-y+2xy+4x\)

Both equations describe how the variables \(x\) and \(y\) change with time. Such systems often arise in physics, biology, and engineering to represent phenomena like population growth, motion of particles, or electrical circuits.
Understanding how to solve these equations helps in predicting the future behavior of the system. Solutions to differential equations can provide functions that define the system's state at any given point in time. The challenge lies in solving these equations to understand the intricacies of the dynamic interactions between variables.

Hamiltonian Function

A Hamiltonian function \(H(x, y)\) is a scalar function used to describe the total energy within a system. In a Hamiltonian system, the motion of a system can be predicted by finding this function, which is related to the system's potential and kinetic energy. The given system of differential equations becomes Hamiltonian when there exists a Hamiltonian function such that the equations can be expressed in terms of partial derivatives of \(H\).
In our exercise, to show that the system is Hamiltonian, we need to find \(H(x, y)\) where:

• \(\frac{\partial H}{\partial y} = 0\)
• \(-\frac{\partial H}{\partial x} = 2y + 4\)

The function can be found by integrating the partial derivatives relative to \(x\) and \(y\). Through this process, the Hamiltonian captures the essence of the system dynamics, allowing us to study the conservation of energy principle and stability of the system.

Phase-Plane Trajectory

A phase-plane trajectory is a graphical representation that shows all possible states of a system of differential equations as it evolves over time. Each point on this trajectory represents a specific state, and the path followed by these points illustrates the behavior of the system. For a two-dimensional system like ours with equations in \(x\) and \(y\), plotting the trajectory can help us understand the dynamics without solving the equations analytically.
Using computational software, one can visualize these trajectories and understand how solutions evolve from an initial condition, such as the point \((1,1)\) in the exercise. Arrows along the trajectory indicate the direction of motion as time progresses, providing insights into the nature of the solutions, such as whether they spiral inward, oscillate, or move toward a stable point. Graphical methods like these are powerful as they offer visual intuition into the complex behavior of differential systems.

Theorem 6.2

Theorem 6.2 provides criteria for identifying whether a given system of differential equations can be classified as a Hamiltonian system. According to the theorem, the system:

• \(x^{\prime} = P(x, y)\)
• \(y^{\prime} = Q(x, y)\)

is a Hamiltonian system if there exists a function \(H(x, y)\) satisfying:

• \(P(x, y) = \frac{\partial H}{\partial y}\)
• \(Q(x, y) = -\frac{\partial H}{\partial x}\)

In the context of the exercise, this theorem guides the identification of the appropriate Hamiltonian function \(H(x, y)\). It ensures that for our system equations, the flow represented can be traced back to a fundamental energetic basis described by \(H\). Using Theorem 6.2, we can verify that the given differential equations adhere to the principles of Hamiltonian dynamics, thus opening pathways to leveraging its properties for further analysis and insights.