Question

Use Theorem \(6.2\) to decide whether the given system is a Hamiltonian system. If it is, find a Hamiltonian function for the system. $$ \begin{aligned} &x^{\prime}=-\sin (2 x y)-x \\ &y^{\prime}=\sin (2 x y)+y \end{aligned} $$

Short answer

If so, find the Hamilton function. The given system of differential equations is a Hamiltonian system with the Hamiltonian function: $$ H(x, y) = -x^2 - xy + \frac{1}{2}y^2 + y\cos(2xy). $$

Step-by-step solution

Compare equations to Hamiltonian system equations

Compare the given system of equations: $$ \begin{aligned} x' &= -\sin(2xy) - x, \\ y' &= \sin(2xy) + y, \end{aligned} $$ to the equations for a Hamiltonian system: $$ \begin{aligned} x' &= \frac{\partial H}{\partial y}, \\ y' &= -\frac{\partial H}{\partial x}. \end{aligned} $$

Find partial derivatives of the potential Hamiltonian function

From the comparison of equations in Step 1, we can determine the partial derivatives of the potential Hamiltonian function: $$ \begin{aligned} \frac{\partial H}{\partial y} &= -\sin(2xy) - x, \\ -\frac{\partial H}{\partial x} &= \sin(2xy) + y. \end{aligned} $$ Now, let's integrate these partial derivatives to find H(x, y).

Integrate partial derivatives to determine H(x, y)

First, let's integrate the first equation with respect to y: $$ H(x, y) = \int (-\sin(2xy) - x)dy. $$ Similarly, let's integrate the second equation with respect to x: $$ H(x, y) = -\int (\sin(2xy) + y)dx. $$

Find H(x, y) using integration results

The results of the integrations are: $$ H(x, y) = x\cos(2xy) - xy + g(x), $$ and $$ H(x, y) = y\cos(2xy) + \frac{1}{2}y^2 + h(y). $$ Comparing these expressions for H(x, y), we can deduce that: $$ H(x, y) = -x^2 - xy + \frac{1}{2}y^2 + y\cos(2xy). $$

Conclusion

The given system is a Hamiltonian system since we were able to find a Hamiltonian function H(x, y). The Hamiltonian function for the system is: $$ H(x, y) = -x^2 - xy + \frac{1}{2}y^2 + y\cos(2xy). $$

Key concepts

Hamiltonian Function

Hamiltonian systems are a special class of dynamical systems characterized by a Hamiltonian function, denoted as \( H(x, y) \). This function represents the total energy of the system. In general, a Hamiltonian system consists of equations of motion derived from the principle of stationary action, with time derivatives expressed in terms of partial derivatives of \( H \). In this context, energy conservation plays a pivotal role, as the Hamiltonian function combines potential and kinetic energy terms.
Hamiltonian Mechanics offers an elegant framework with the equations defined as:

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

In a practical sense, finding a Hamiltonian function involves ensuring that your dynamical system fits this form. If successful, it means you’ve confirmed the system conserves some form of energy, making it a Hamiltonian system.

Partial Derivatives

When dealing with Hamiltonian systems, partial derivatives come into play as key tools. These derivatives measure how a multivariable function changes when one of its variables changes, keeping others constant. For a Hamiltonian function \( H(x, y) \), partial derivatives are used to express the system’s dynamics:

• \( \frac{\partial H}{\partial y} \) and \( \frac{\partial H}{\partial x} \)

These equations help us understand how the change in one variable affects the whole system. Calculating partial derivatives allows us to determine the influence of each variable on the energy represented by the Hamiltonian. Finding these derivatives is the first step in unraveling complex systems and understanding their behavior through the lens of energy dynamics.

Integration

Integration is a powerful mathematical tool used to find a function given its derivative. In Hamiltonian systems, integration plays a crucial role in constructing the Hamiltonian function \( H(x, y) \) from known partial derivatives. Once the partial derivatives \( \frac{\partial H}{\partial y} \) and \( -\frac{\partial H}{\partial x} \) have been identified, integration helps us to determine the full Hamiltonian function.
To perform integration:

• Integrate \( \frac{\partial H}{\partial y} \) with respect to \( y \)
• Integrate \( -\frac{\partial H}{\partial x} \) with respect to \( x \)

These steps create expressions for \( H \) from each derivative calculation. By combining results from both integrations, we form the complete Hamiltonian function revealing the energy landscape of the system.

Differential Equations

Differential equations are foundational in describing how a system evolves over time. In the context of Hamiltonian systems, differential equations come in pairs, derived from the partial derivatives of the Hamiltonian function \( H(x, y) \). Solving these equations yields insights into the system's trajectory through its phase space.
The specific form of differential equations we encounter:

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

These represent the rates of change of the system's coordinates. Solving these differential equations allows us to predict the future states of the system based solely on initial conditions, capturing the dynamic essence of Hamiltonian mechanics.