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Differential equations are the heart of many dynamical systems in mathematics, modeling how things change over time. In a differential equation, we focus on understanding functions and their rates of change. The problem at hand revolves around a system of differential equations given by:

• \( x' = y \)
• \( y' = -x + x^3 \)

Differential equations can be of various types, and they play a crucial role in predicting behavior in engineering, physics, and beyond. The primary task when working with such equations is to identify the relationships between the rates of change of the system's variables. In this case, the equations represent the evolution of a system over time. Understanding the behavior of solutions to these equations provides insight into the system dynamics, potentially indicating stability or identifying periodic nature. Here, the goal is to determine if these differential equations represent a Hamiltonian system.

The Hamiltonian function is a powerful concept in classical mechanics, offering a comprehensive energy-based view of dynamical systems. It serves as an energy function combining potential and kinetic energies to describe the total energy of the system. A system of differential equations is Hamiltonian if it can be expressed as:

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

Where \( H = H(x, y) \) is the Hamiltonian function. Identifying the appropriate Hamiltonian not only classifies the system but also simplifies studying its behavior, as it provides a conservation law studied over time.For our problem, we start by deriving the possible form of \( H \) by equating and integrating equations derived from the system's conditions. By fulfilling these conditions, you establish a connection between the given parts of the system and the potential Hamiltonian, leading to a differential equation that provides insights into its energy states.

Partial derivatives involve differentiating a function with respect to one of its variables while keeping other variables constant. They are fundamental in formulating and solving Hamiltonian systems, as they enable us to express dynamic systems in terms of derivatives with respect to different variables. For our system, you use:

• Partial derivative of \( H \) with respect to \( y \): \( \frac{\partial H}{\partial y} = y \)
• Partial derivative of \( H \) with respect to \( x \): \( -\frac{\partial H}{\partial x} = -x + x^3 \)

By setting up these equations, the information allows us to identify or verify the exact nature of the function \( H(x, y) \). Calculating partial derivatives accurately is crucial for correctly interpreting the solutions of a differential system and ensuring the system's Hamiltonian nature.

Integral calculus complements differential calculus, focusing on accumulation functions such as areas under curves. When dealing with Hamiltonian systems, once we identify expressions for the partial derivatives as done in previous sections, we often need to integrate these expressions to derive the Hamiltonian's functional form.In this exercise, the task involves integrating the partial derivative with respect to \( x \):\[\int (x - x^3) \, dx = \frac{x^2}{2} - \frac{x^4}{4}\]This integral is central to forming the Hamiltonian:\( H(x, y) = \frac{x^2}{2} - \frac{x^4}{4} + \frac{1}{2} y^2 + C \)where \( C \) is an integration constant.Understanding and performing these integrals correctly is crucial for accurately constructing the Hamiltonian function. Practicing these techniques enhances skills in solving complex dynamic systems and interpreting the natural phenomena they model.