Variational Principle and Approximate Solution for the Fractal Vibration Equation in a Microgravity Space

Under the microgravity space, many theories derived from the earth's surface become untenable, so a modified vibration equation with fractal derivative is presented in this work. With the help of the semi-inverse method, we successfully develop the fractal variational principle, which not only provides conservation laws in an energy form but also provides physical insight into the nature structures of the solutions. Finally, the variational iteration method, together with the two-scale transform, is applied to find the solution of the fractal vibration equation. The obtained results show that the method is powerful and accurate.