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Chapter 6: Problem 17

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. $$ \begin{aligned} &x^{\prime}=2 y \cos x \\ &y^{\prime}=y^{2} \sin x \end{aligned} $$

Short Answer

Expert verified
#Answer# (a) The given system is Hamiltonian. (b) The Hamiltonian function is \(H(x, y) = y^2 \cos x\). (c) Use computational software to graph the phase-plane trajectory and determine the direction of motion for the solution point \((1,1)\).

Step by step solution

01

(a) Applying Theorem 6.2 #

Theorem 6.2 states that a planar system is Hamiltonian if it has the following form: $$ \begin{aligned} x' &= \frac{\partial H}{\partial y} \\ y' &= -\frac{\partial H}{\partial x} \end{aligned} $$ where \(H(x, y)\) is a scalar function, called the Hamiltonian function. Compare this with the given system: $$ \begin{aligned} x' &= 2y\cos x \\ y' &= y^2\sin x \end{aligned} $$ To see if the given system has the form specified by Theorem 6.2, we can set: $$ \begin{aligned} \frac{\partial H}{\partial y} &= 2y\cos x \\ -\frac{\partial H}{\partial x} &= y^2\sin x \end{aligned} $$ Now, we need to find the Hamiltonian function \(H(x, y)\). To do this, we can integrate both equations with respect to \(y\) and \(x\), respectively.
02

(b) Find the Hamiltonian function #

Integrate with respect to \(y\) and \(x\): $$ \begin{aligned} H(x, y) &= \int 2y\cos x \, dy \\ H(x, y) &= y^2 \cos x + C(x) \\ \end{aligned} $$ Now differentiate \(H(x, y)\) with respect to \(x\) and equate it to \(-y^2 \sin x\): $$ \begin{aligned} -\frac{\partial H}{\partial x} &= -y^2\sin x \\ -y^2 \sin x &= -y^2\sin x \\ \end{aligned} $$ The equation holds true meaning we found the correct form of the Hamiltonian function. So the Hamiltonian function is: $$ H(x, y) = y^2 \cos x + C(x) $$ However, \(C(x)\) doesn't contribute when computing \(x'\) and \(y'\) so we can drop it for the sake of simplicity, and obtain: $$ H(x, y) = y^2 \cos x $$
03

(c) Graph the phase-plane trajectory and determine the direction of motion #

You will need to use computational software to graph the phase-plane trajectory of the given system. Plot the trajectory passing through point \((1,1)\) and observe its direction. Some popular software options are MATLAB, Mathematica, or Python with the matplotlib library.

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Key Concepts

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

Hamiltonian Function
A fundamental concept in classical mechanics and mathematical physics is the Hamiltonian function, which is instrumental in describing a system's total energy鈥攃ombining both kinetic and potential energy. Think of it like a mathematical recipe that provides a snapshot of the entire system's energy landscape.

In the given exercise, the system's Hamiltonian function serves as a bridge to transforming a set of differential equations into a form that reveals the underlying mechanics. The integration step showed us that the Hamiltonian function for our system is \(H(x, y) = y^2 \times \text{cos}(x)\), which represents the energy of the system at any given point \((x, y)\) in space.

This perspective isn't just academic; by understanding the system through its Hamiltonian function, we gain powerful analytical tools. For instance, we can predict the possible states and evolution of the system over time without needing to solve the equations explicitly. It's like having a map of all the possible paths a system could take, just based on the energy terrain provided by the Hamiltonian.
Phase-Plane Trajectory
Phase-plane trajectory is a core concept used to visualize the behavior of dynamical systems in two dimensions. Imagine charting a course for a ship at sea: the phase-plane trajectory tells you not just where the ship is, but also where it's likely headed.

In our exercise, the phase-plane trajectory is the path traced out by the solution of the system of differential equations, which includes points like \((1, 1)\). By graphing these trajectories using computational software, we can see a picture of how the system evolves over time.

Understanding Directions

To gauge the direction of motion along these trajectories, pay attention to the arrows drawn on them in the graph. These arrows act like a compass, showing the direction in which the system will evolve from any given starting point. It's about dynamic forecasting, knowing the future state based on present conditions. And much like weather predictions, the phase-plane trajectories provide a forecast, not of weather, but of the system's states as they change through time.
Differential Equations
Differential equations are pivotal in translating real-world phenomena into the language of mathematics. They are to dynamics what grammar is to language: a set of rules explaining how things change.

In our context, the coupled differential equations \(x' = 2y\times\text{cos}(x)\) and \(y' = y^2\times\text{sin}(x)\) describe the rate of change of the variables 'x' and 'y' with respect to time. Solving these equations isn't just about finding 'x' and 'y'; it's about comprehending how the system's state (represented by 'x' and 'y') evolves as time marches on.

By following the step-by-step solution, we apply Theorem 6.2, matching the structure of our differential equations with the pattern of the Hamiltonian system. This allows us to unlock an analytical way to study our system's behavior, zooming out from the minutiae of calculation and diving into the broader narrative of the system's motion and equilibrium.

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Most popular questions from this chapter

Consider the linear system of Example 4 , $$ \mathbf{y}^{\prime}=\left[\begin{array}{ll} -4 & 5 \\ -5 & 4 \end{array}\right] \mathbf{y} $$ The coefficient matrix has eigenvalues \(\lambda_{1}=3 i, \lambda_{2}=-3 i\); the equilibrium point at the origin is a center. (a) Show that the linear system is a Hamiltonian system. Either use the results of Exercise 30 or apply the criterion directly to this example. (b) Derive the conservation law for this system. The result, \(\frac{5}{2} x^{2}-4 x y+\frac{5}{2} y^{2}=C>0\), defines a family of ellipses. These ellipses are the trajectories on which the solution point moves as time changes. (c) Plot the ellipses found in part (b) for \(C=\frac{1}{4}, \frac{1}{2}\), and 1. Indicate the direction in which the solution point moves on these ellipses.

In each exercise, (a) Rewrite the given \(n\)th order scalar initial value problem as \(\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y}), \mathbf{y}\left(t_{0}\right)=\mathbf{y}_{0}\), by defining \(y_{1}(t)=y(t), y_{2}(t)=y^{\prime}(t), \ldots, y_{n}(t)=y^{(n-1)}(t)\) and defining \(y_{1}(t)=y(t), y_{2}(t)=y(t), \ldots, y_{n}(t)=y^{\prime(n-t)}(t)\) \(\mathbf{y}(t)=\left[\begin{array}{c}y_{1}(t) \\ y_{2}(t) \\ \vdots \\\ y_{n}(t)\end{array}\right]\) (b) Compute the \(n^{2}\) partial derivatives \(\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{j}, i, j=1, \ldots, n\). (c) For the system obtained in part (a), determine where in \((n+1)\)-dimensional \(t \mathbf{y}\)-space the hypotheses of Theorem \(6.1\) are not satisfied. In other words, at what points \(\left(t, y_{1}, \ldots, y_{n}\right)\), if any, does at least one component function \(f_{i}\left(t, y_{1}, \ldots, y_{n}\right)\) and/or at least one partial derivative function \(\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{i}, i, j=1, \ldots, n\) fail to be continuous? What is the largest open rectangular region \(R\) where the hypotheses of Theorem \(6.1\) hold? $$ y^{\prime \prime \prime}+\frac{2 t^{1 / 3}}{(y-2)\left(y^{\prime \prime}+2\right)}=0, \quad y(0)=0, \quad y^{\prime}(0)=2, \quad y^{\prime \prime}(0)=2 $$

Consider a colony in which an infectious disease (such as the common cold) is present. The population consists of three "species" of individuals. Let \(s\) represent the susceptibles-healthy individuals capable of contracting the illness. Let \(i\) denote the infected individuals, and let \(r\) represent those who have recovered from the illness. Assume that those who have recovered from the illness are not permanently immunized but can become susceptible again. Also assume that the rate of infection is proportional to \(s i\), the product of the susceptible and infected populations. We obtain the model $$ \begin{aligned} &s^{\prime}=-\alpha s i+\gamma r \\ &i^{\prime}=\alpha s i-\beta i \\ &r^{\prime}=\beta i-\gamma r \end{aligned} $$ where \(\alpha, \beta\), and \(\gamma\) are positive constants. (a) Show that the system of equations (11) describes a population whose size remains constant in time. In particular, show that \(s(t)+i(t)+r(t)=N\), a constant. (b) Modify (11) to model a situation where those who recover from the disease are permanently immunized. Is \(s(t)+i(t)+r(t)\) constant in this case? (c) Suppose that those who recover from the disease are permanently immunized but that the disease is a serious one and some of the infected individuals perish. How does the system of equations you formulated in part (b) have to be further modified? Is \(s(t)+i(t)+r(t)\) constant in this case?

Locate the unique equilibrium point of the given nonhomogeneous system, and determine the stability properties of this equilibrium point. Is it asymptotically stable, stable but not asymptotically stable, or unstable? $$ \begin{aligned} &x^{\prime}=y+2 \\ &y^{\prime}=-x+1 \end{aligned} $$

In each exercise, (a) Rewrite the given \(n\)th order scalar initial value problem as \(\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y}), \mathbf{y}\left(t_{0}\right)=\mathbf{y}_{0}\), by defining \(y_{1}(t)=y(t), y_{2}(t)=y^{\prime}(t), \ldots, y_{n}(t)=y^{(n-1)}(t)\) and defining \(y_{1}(t)=y(t), y_{2}(t)=y(t), \ldots, y_{n}(t)=y^{\prime(n-t)}(t)\) \(\mathbf{y}(t)=\left[\begin{array}{c}y_{1}(t) \\ y_{2}(t) \\ \vdots \\\ y_{n}(t)\end{array}\right]\) (b) Compute the \(n^{2}\) partial derivatives \(\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{j}, i, j=1, \ldots, n\). (c) For the system obtained in part (a), determine where in \((n+1)\)-dimensional \(t \mathbf{y}\)-space the hypotheses of Theorem \(6.1\) are not satisfied. In other words, at what points \(\left(t, y_{1}, \ldots, y_{n}\right)\), if any, does at least one component function \(f_{i}\left(t, y_{1}, \ldots, y_{n}\right)\) and/or at least one partial derivative function \(\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{i}, i, j=1, \ldots, n\) fail to be continuous? What is the largest open rectangular region \(R\) where the hypotheses of Theorem \(6.1\) hold? $$ t y^{\prime \prime}+\frac{1}{1+y+2 y^{\prime}}=e^{-t}, \quad y(2)=2, \quad y^{\prime}(2)=1 $$
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