We propose a new geodesic based algorithm for fiber tracking in diffusion tensor imaging data. Our algorithm computes the multi-valued solutions from the Euler-Lagrange form of the geodesic equations. Compared to other geodesic based approaches, multi-valued solutions at each grid point are computed rather than just computing the viscosity solution. This allows us to compute fibers in a region with sharp orientation, or when the correct physical solution is not the fiber computed from the first arrival time. Compared to the classical stream-line approach, our method is less sensitive to noise, since the complete tensor is used. We also compare our algorithm with the Hamilton-Jacobi equation (HJ) based approach. We show that in the cases where U-shaped bundles appear, our algorithm can capture the underlying fiber structure while other approaches may fail. The results for synthetic and real data are shown for both methods.