Accelerator physics needs advanced modeling and simulation concepts. In order to able to improve the design and performance of future colliders, models of the magnetic fields non-linearities needs deeper understanding. These non-linearities mainly come from magnet fringe fields and ends connections. In this context a more precise description of the particles' motion inside a quadrupole can be obtained using advanced matematical techniques to derive a symplectic integrator. This sophisticated tracking codes require as input a description of the vector potential in the whole space inside the quadrupole.
In this seminar the procedure to provide a realistic vector potential is presented. It must be both precise and expressed in a form that allows a fast tracking (crucial in long-term simulations), starting from magnetic field or harmonics' data (simulated or measured).