Abstract This paper presents theoretical investigations of lattice Boltzmann method (LBM) to develop a completed LBM theory. Based on H-theorem with Lagrangian multiplier method, an amended theoretical equilibrium distribution function (EDF) is derived, which modifies the current Maxwell-Boltzmann distribution (MBD) to include the total internal energy as its parameter. This modification allows the three conservation laws derived directly from lattice Boltzmann equation (LBE) without additional small-parameter expansions adopted in references. From this amended theoretical EDF an improved LBM is developed, in which the total internal energy like the mass density and mean velocity is a new macroscopic variable to be updated for different times and cells during simulations. The developed method provides a means to consider external forces and energy generation sources as generalised forces in LBM simulations. The corresponding model and implementation process of the improved LBM are presented with its performance theoretically investigated. Analytically hand-workable examples are given to illustrate its applications and to confirm its validity. The paper will excite more researchers and scientists of this area to numerically practice the new theory and method dealing with complex physical problems, from which it is expected to further advance LBM benefiting science and engineering.