1. The problem involves solving a system of differential equations with given parameters $\kappa = 0.27$, $b = 10$, $\sigma = \frac{5}{2}$, and $h = 1.2$. The function $g[t]$ is defined as $$ g[t] = \frac{\exp\left(\frac{b h}{\kappa} \sin(\psi[t]) + \frac{\sigma}{\kappa} \cos^2(\phi[t] - \psi[t])\right)}{\int_{-\frac{1}{2}}^{\frac{1}{2}} \exp\left(\frac{b h}{\kappa} \sin(\psi[t]) + \frac{\sigma}{\kappa} \cos^2(\phi[t] - \psi[t])\right) dt} $$ 2. The system to solve is: $$ \begin{cases} \left(\cos^2(\phi[t]) + \frac{3}{2} \sin^2(\phi[t])\right) \phi''[t] + \frac{1}{4} \sin(2\phi[t]) (\phi'[t])^2 - \frac{1}{2} h^2 \sin(2\phi[t]) - \sigma g[t] \sin(2(\phi[t] - \psi[t])) = 0 \\ b h \cos(\psi[t]) + \sigma \sin(2(\phi[t] - \psi[t])) = 0 \\ \phi[0] = 1.57, \quad \phi'[0] = 0 \end{cases} $$ 3. This is a coupled nonlinear system involving $\phi[t]$ and $\psi[t]$. The second equation is algebraic and can be used to express $\psi[t]$ in terms of $\phi[t]$. 4. To solve numerically, one would typically use a numerical differential equation solver such as NDSolve in Mathematica, specifying the interval $t \in [-\frac{1}{2}, \frac{1}{2}]$. 5. The problem is complex and requires numerical methods; the exact closed-form solution is not feasible. 6. The key steps are: - Define $g[t]$ as given. - Use the algebraic equation to solve for $\psi[t]$ in terms of $\phi[t]$. - Substitute $\psi[t]$ into the differential equation for $\phi[t]$. - Solve the resulting ODE numerically with initial conditions. Final answer: The solution is the numerical solution of the system above on $t \in [-\frac{1}{2}, \frac{1}{2}]$ with given parameters and initial conditions.