Question

Consider the two-dimensional autonomous system $$ \begin{aligned} &x^{\prime}=f(x, y) \\ &y^{\prime}=g(x, y) \end{aligned} $$ Suppose there exist two functions \(K(x, y)\) and \(\mu(x, y)\) satisfying $$ \begin{aligned} \frac{\partial K(x, y)}{\partial x} &=-\mu(x, y) g(x, y) \\ \frac{\partial K(x, y)}{\partial y} &=\mu(x, y) f(x, y) \end{aligned} $$ Does the given autonomous system have a conserved quantity? If so, what is the conserved quantity?

Short answer

Question: Determine whether the given two-dimensional autonomous system with functions \(K(x, y)\) and \(\mu(x, y)\) has a conserved quantity and, if so, what that quantity is. Solution: The conserved quantity of the autonomous system, if it exists, is given by: $$Q(x, y) = \int \frac{1}{\mu(x, y)}\mathrm{d}K(x,y)$$ We need to check whether the conditions for the existence of \(\mu(x, y)\) are satisfied and if the integral exists. If these conditions are met, then the system has a conserved quantity and it is given by \(Q(x, y)\).

Step-by-step solution

Verify the conditions for the functions

First, let's see if the conditions for the functions \(K(x, y)\) and \(\mu(x, y)\) are satisfied. We are given two equations: $$\frac{\partial K(x, y)}{\partial x} = -\mu(x, y)g(x, y)$$ and $$\frac{\partial K(x, y)}{\partial y} = \mu(x, y)f(x, y)$$ We need to find out if \(\mu(x, y)\) exists. Let's use the total differential of two functions to achieve this: $$\mathrm{d} K(x, y) = \frac{\partial K(x, y)}{\partial x}\mathrm{d} x + \frac{\partial K(x, y)}{\partial y}\mathrm{d} y$$ Substitute the given conditions for the derivatives of \(K(x, y)\): $$\mathrm{d} K(x, y) = -\mu(x, y)g(x, y)\mathrm{d} x + \mu(x, y)f(x, y)\mathrm{d} y$$

Find the conserved quantity

If \(\mu(x, y)\) exists, we can remove it from the equation by dividing both sides by \(\mu(x, y)\): $$\frac{\mathrm{d} K(x, y)}{\mu(x, y)} = -g(x, y)\mathrm{d} x + f(x, y)\mathrm{d} y$$ Now, let's integrate both sides of the equation with respect to the variables: $$\int \frac{\mathrm{d} K(x, y)}{\mu(x, y)} = \int (-g(x, y)\mathrm{d} x + f(x, y)\mathrm{d} y)$$ Assuming that the integral exists, we can rewrite the equation as follows: $$\int \frac{1}{\mu(x, y)}\mathrm{d}K(x,y) = \int (-g(x, y)\mathrm{d} x + f(x, y)\mathrm{d} y)$$ The left side represents the conserved quantity. Let's denote it as \(Q(x, y)\): $$Q(x, y) = \int \frac{1}{\mu(x, y)}\mathrm{d}K(x,y)$$ If the integral exists, then the conserved quantity \(Q(x, y)\) exists for the given autonomous system. Thus, the conserved quantity of the autonomous system is: $$Q(x, y) = \int \frac{1}{\mu(x, y)}\mathrm{d}K(x,y)$$

Key concepts

Autonomous Systems

Autonomous systems are a special type of system in differential equations where the derivatives of the functions do not depend on the independent variable, usually time. Instead, these derivatives depend only on the functions themselves and their current state. This characteristic often simplifies the analysis of the systems.

For a two-dimensional autonomous system, the equations might take the form:

• \(x' = f(x, y)\),
• \(y' = g(x, y)\).

These functions describe how the variables \(x\) and \(y\) change with respect to time. The focus is not on the passage of time but on capturing how the system evolves based solely on its state, \(x\) and \(y\).

Because the time element is not explicit, properties such as conserved quantities can often be identified, giving deep insights into how the system behaves in specific conditions.

Functions in Differential Equations

In the context of differential equations, functions like \(f(x, y)\) and \(g(x, y)\) are indispensable. They define the system's rules and dictate how each variable evolves over time. In our autonomous system, these functions map the current state of the system to the rate of change of the variables.

This mapping enables the prediction of how variables will change, forming the foundation for modeling numerous phenomena in science and engineering. For different systems, these functions might represent rates of movement, growth, or other changes. Understanding their form and interaction is key to solving differential equations and predicting future states of the system.

Partial Derivatives

Partial derivatives are crucial when dealing with functions of multiple variables, like \(K(x, y)\). They describe how a function changes in a particular direction while keeping other variables constant.

In our autonomous system, partial derivatives show how the potential conserved quantity \(K(x, y)\) changes with respect to \(x\) and \(y\). These derivatives are given by the following equations:

• \(\frac{\partial K(x, y)}{\partial x} = -\mu(x, y) g(x, y),\)
• \(\frac{\partial K(x, y)}{\partial y} = \mu(x, y) f(x, y).\)

Understanding partial derivatives helps determine how changes in one variable affect the entire system, which is instrumental in seeking conserved quantities.

Integration in Differential Equations

Integration is a powerful tool in differential equations often used to find conserved quantities or other significant properties of systems. After identifying that a function or a part of the equation can integrate smoothly, it paves the way for deriving quantities that remain constant over time.

In our exercise, the integration process is used to find the conserved quantity \(Q(x, y)\):

• \(Q(x, y) = \int \frac{1}{\mu(x, y)}\mathrm{d}K(x,y)\).

Assuming the integral exists, this expression identifies a conserved quantity that helps reveal the system's inherent stability or predictability aspects.
By performing these integrations, you essentially integrate rates of change to identify which properties do not change, offering significant insights into the autonomous system's long-term behavior.