PhD thesis in English

Euler-Lagrange equations of motion for variational parameters q ∈ {ϕ σ , σ 0 , φ σ , u σ }have the formand for the above Lagrangian read:d ∂Ldt ∂ ˙q − ∂L∂q = 0 , ˙φ σ u σ − 22Mϕ σ − M ˙σ 0 + 2φ σ σ 0 = 0, (B.1) ˙ϕ σ + 2σ 0 ˙φσ + 22M ϕ σφ σ + 42M φ2 σσ 0 + Mω 2 σσ 0 = 0 (B.2)1u 3 σφ σ − M 2˙u σu σ= 0, (B.3)+ 22M φ2 σu σ + Mω2 σ2 u σ − g(t)2(2π) 3/2 1u x u y u z u σ= 0. (B.4)Eqs. (B.1) and (B.3) give the explicit relation between ϕ σ and σ 0 , and between φ σand u σ . By inserting Eqs. (B.1) and (B.3) into Eqs. (B.2) and (B.4), we obtainvariational equations which we will refer to as a Gaussian approximation:¨σ 0 (t) + λ 2 σ σ 0 = 0 , (B.5)ü σ (t) + λ 2 σ u σ(t) − 1u σ (t) − P(t)3 u σ (t)u x (t)u y (t)u z (t)= 0 . (B.6)We also note that solutions of Eqs. (B.5) and (B.6) for σ 0 (t) and u σ (t) can be afterwardsinserted into Eqs. (B.1) and (B.3) to obtain time-evolution of the phaseparameters ϕ σ and φ σ . The last two equations are given in the dimensionless form:we choose a convenient frequency scale ω (for example, the external trap frequencyin one of the spatial directions) and express all lengths in the units of the characteristicharmonic oscillator length l = √ /Mω, time in units of ω −1 , and externalfrequencies in units of ω: λ σ = ω σ /ω, σ ∈ {x, y, z}. The dimensionless interactionparameter P(t) is given byP(t) =√g(t) 2(2π) 3/2 ωl = 3 π N a(t) .l143