Question

Let \(f(u)\) and \(g(u)\) be defined and continuously differentiable on the interval \(-\infty

Short answer

Question: Determine if the given system is Hamiltonian, and if so, find the associated Hamiltonian function expressed in terms of the antiderivatives F(u) and G(u). Given system: $$ \begin{aligned} x^{\prime} &= 3 f(y) - 2xy, \\ y^{\prime} &= g(x) + y^2 + 1. \end{aligned} $$ Answer: The given system is Hamiltonian, and the associated Hamiltonian function is: $$ H(x, y) = - \int 3f(y) - 2xy \,dy + \int g(x) + y^2 + 1 \,dx. $$

Step-by-step solution

Recall Theorem 6.2

Theorem 6.2 (Hamiltonian Systems) states that a system of the form $$ \begin{aligned} x^{\prime} &= -\frac{\partial H}{\partial y}, \\ y^{\prime} &= \frac{\partial H}{\partial x}, \end{aligned} $$ where \(H(x, y)\) is the Hamiltonian function, is a Hamiltonian system.

Match Theorem 6.2 with the given system

We are given the system: $$ \begin{aligned} x^{\prime} &= 3 f(y) - 2xy, \\ y^{\prime} &= g(x) + y^2 + 1. \end{aligned} $$ We want to match this with the form in Theorem 6.2. We will try to find expressions for \(\frac{\partial H}{\partial x}\) and \(\frac{\partial H}{\partial y}\) that allow us to identify a Hamiltonian function for the system.

Relate x' and y' to Hamiltonian function's derivatives

For the x-component in Theorem 6.2, we have $$ x^{\prime} = -\frac{\partial H}{\partial y}. $$ Comparing this to our given x-component, we have $$ 3 f(y) - 2xy = -\frac{\partial H}{\partial y}. $$ For the y-component in Theorem 6.2, we have $$ y^{\prime} = \frac{\partial H}{\partial x}. $$ Comparing this to our given y-component, we have $$ g(x) + y^2 + 1 = \frac{\partial H}{\partial x}. $$

Integrate H's partial derivatives

Now we can integrate each partial derivative expression to find the Hamiltonian function. For \(\frac{\partial H}{\partial y}\): $$ H(x, y) = - \int 3f(y) - 2xy \,dy + G(x). $$ For \(\frac{\partial H}{\partial x}\): $$ H(x, y) = \int g(x) + y^2 + 1\,dx + F(y). $$

Find H(x, y) by comparing integrations

Since both integrations above were for the same Hamiltonian function \(H(x, y)\), we can compare them to determine the overall Hamiltonian function. We have: $$ H(x, y) = - \int 3f(y) - 2xy \,dy + G(x) =\int g(x) + y^2 + 1 \,dx + F(y). $$ Therefore, the Hamiltonian function for this system is: $$ H(x, y) = - \int 3f(y) - 2xy \,dy + \int g(x) + y^2 + 1 \,dx. $$ This function is expressed in terms of the antiderivatives \(F(u)\) and \(G(u)\), as requested. We have shown that the given system is Hamiltonian and we found the associated Hamiltonian function.

Key concepts

Antiderivatives

Antiderivatives play a crucial role in solving Hamiltonian systems, particularly when it comes to finding the Hamiltonian function itself. In simple terms, an antiderivative of a function is another function whose derivative is the original function. For example, if you have a function \( f(u) \), an antiderivative of this function, often denoted as \( F(u) \), satisfies the relationship: \( F'(u) = f(u) \). This means if you differentiate \( F(u) \), you will get \( f(u) \) back.

In the context of Hamiltonian systems, antiderivatives are used to build the Hamiltonian function from the partial derivatives given in the system. The problem requires integrating the expressions obtained by comparing the system equations to find these functions. In our example, the Hamiltonian function \( H(x, y) \) is derived by integrating the derivatives \( \frac{\partial H}{\partial x} \) and \( \frac{\partial H}{\partial y} \) to reconstruct the function \( H \).

Recall that integrals are essentially the opposite of derivatives. By integrating the expressions for the partial derivatives of \( H(x, y) \), we use antiderivatives to find the equation for the Hamiltonian system. This process demonstrates the connection between antiderivatives and Hamiltonian functions, providing a meaningful solution that ties back to the original differential system.

Differential Equations

Differential equations are equations involving derivatives of a function or functions. They showcase how these functions change, and solving these equations involves finding the unknown function that satisfies them. In a Hamiltonian system, we typically deal with a set of differential equations that describe a physical system's energy dynamic.

The system of equations given in the example, where we have \( x' = 3f(y) - 2xy \) and \( y' = g(x) + y^2 + 1 \), represents such a scenario. Solving these equations involves finding functions \( x(t) \) and \( y(t) \) that satisfy these relationships over time \( t \).

Hamiltonian systems are a special class of differential equations used particularly in the context of classical mechanics and mathematical physics, where they describe the energy conservation properties of a system. The goal is to find a Hamiltonian function, \( H(x, y) \), which, when differentiated properly, provides the original set of differential equations. The Hamiltonian encodes the information of a system's total energy, and when the system is Hamiltonian, it implies that this energy is conserved and provides further insights into the system's behavior.

Theorem 6.2

Theorem 6.2 provides the mathematical framework for identifying and verifying Hamiltonian systems. It states that a system of differential equations is Hamiltonian if it can be represented in the form:

• \( x' = -\frac{\partial H}{\partial y} \)
• \( y' = \frac{\partial H}{\partial x} \)

where \( H(x, y) \) is the Hamiltonian function. Using Theorem 6.2, we can determine whether a given system of differential equations is Hamiltonian and find the corresponding Hamiltonian function.

In our example, we re-write the given system equations to match the form in Theorem 6.2. By doing this, we explicitly find the partial derivatives of an unknown function \( H(x, y) \). Integrating these partial derivative expressions then allows us to determine the exact form of \( H(x, y) \).

Theorem 6.2 acts as the guideline to not only validate a Hamiltonian system but also to systematically extract the Hamiltonian function which represents the total energy of the system. This not only assists in understanding the system's behavior but also ensures energy conservation in a physical context. This theorem is fundamental when analyzing systems in physics and engineering.