Hamiltonian in Non-inertial Frame

Consider a particle inside an accelerating spaceship. Let \((X_0,Y_0,Z_0)\), \((X,Y,Z)\) and \((x,y,z)\) be the coordinates of the spaceship in the inertial frame, the coordinates of the particle in the inertial frame and in spaceship frame respectively. \begin{cases} X = X_0 + x\\ Y = Y_0 + y\\ Z = Z_0 + z. \end{cases} As discussed before, the Lagrangian should be written using inertial frame coordinates. The Lagrangian of the particle is \begin{align} L &\equiv T-V = \frac{1}{2}m(\dot{X}^2+\dot{Y}^2 + \dot{Z}^2)\\ &=\frac{1}{2}m\left[(\dot{X}_0+\dot{x})^2+(\dot{Y}_0+\dot{y})^2+(\dot{Z}_0+\dot{z})^2\right] \end{align} The generalized momentum of the particle in the spaceship frame is $$p_x\equiv \frac{\partial L}{\partial \dot{x}}=m(\dot{X}_0+\dot{x})$$ So, we have $$\dot{x}=\frac{p_x}{m}-\dot{X}_0.$$ Similarly for \(y\) and \(z\) component. The Hamiltonian of the particle in the spaceship frame is $$H_{\text{ship}} \equiv p_x\dot{x}+p_y\dot{y}+p_z\dot{z}-L.$$ Substitute the Lagrangian and the \(\dot{x}\), \(\dot{y}\), \(\dot{z}\) obtained before, the Hamiltonian becomes $$H_{\text{ship}} = \frac{1}{2m}(p_x^2 + p_y^2 + p_z^2)-(\dot{X}_0p_x+\dot{Y}_0p_y+\dot{Z}_0p_z).$$ Then, the Hamilton's equations are \begin{align} \dot{x} &= \frac{\partial H_{\text{ship}}}{\partial p_x} = \frac{p_x}{m} - \dot{X}_0\\ \dot{p}_x &= -\frac{\partial H_{\text{ship}}}{\partial x} = 0. \end{align} So, \(p_x\) is a constant. The equation of motion of the particle in the spaceship frame is $$\ddot{x} = -\ddot{X}_0,$$ which is the fictitious force that should be observed in an accelerating frame.