We demonstrate a model-independent method of estimating qualitative dynamics of an accelerating universe from observations of distant type Ia supernovae .
Our method is based on the luminosity-distance function , optimized to fit observed distances of supernovae , and the Hamiltonian representation of dynamics for the quintessential universe with a general form of equation of state p = w ( a ( z ) ) \rho .
Because of the Hamiltonian structure of FRW dynamics with the equation of state p = w ( a ( z ) ) \rho , the dynamics is uniquelly determined by the potential function V ( a ) of the system .
The effectiveness of this method in discrimination of model parameters of Cardassian evolution scenario is also given .
Our main result is the following , restricting to the flat model with the current value of \Omega _ { m, 0 } = 0.3 , the constraints at 2 \sigma confidence level to the presence of \rho ^ { n } modification of the FRW models are -0.50 \lesssim n \lesssim 0.36 .