Temperature-Dependent Frequency Shifts in Collective Excitations of a   Bose-Einstein Condensate
  By including the contribution of the thermal cloud to the Lagrangian of the condensate of a Bose gas, we extend the time-dependent variational method at zero temperature to study temperature-dependent low collective excitation modes. A Gaussian trial wave function of the condensate and a static distribution density of the thermal cloud are used, and analytical expressions for temperature-dependent excitation frequencies obtained. Theoretical results are compared with measurements in the JILA and MIT experiments. 
Hualin Shi and Wei-Mou Zheng