by KangSoo Lee and Uwe R. Fischer
I apologize for not finishing this paper. I just open this in the internet. But this document is flawed as it is originally written in LaTeX and not translated to the docuK format perfectly yet. This paper is published in Int. J. Mod. Phys. B 84, 1550021 (2014) and arXiv>condmat>arXiv:1301.2199.
##[.nosecN] Abstract
We introduce a scheme to describe the evolution of an interacting system of bosons, for which the field operator expansion is truncated after a finite number of modes, in a rigorously controlled manner. Using McLachlan's principle of least error , we find a set of equations for the manybody state. As a particular benefit, and in distinction to previously proposed approaches , our approach allows for the dynamical increase of the number of orbitals during the temporal evolution, because we can rigorously monitor the error made by increasing the truncation dimension. The additional orbitals, determined by the condition of least error of the truncated evolution relative to the exact one, are obtained from an initial trial state by a method we call steepest constrained descent.
## PH
 20220706: Update (images, bold, italic and so on.)
 20141217: To the docuK.
 20140623: First posting?

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 We illustrate this statement as follows. The reason why we can use $df = \frac{\partial f}{\partial z}dz + \frac{\partial f}{\partial z^*}dz^*$ is not because $dz$ and $dz^*$ are linearly independent differentials, but because $dz$ and $dz^*$ are differentials, that is infinitesimal quantities; $dz^*$ is in fact 100\% dependent on $dz$ since $dz^*$ is simply the complex conjugate of $dz$. Therefore we cannot simply argue that because $dz$ and $dz^*$ are independent variations, the partial derivatives $\frac{\partial f}{\partial z}$ and $\frac{\partial f}{\partial z^*}$ must be zero for the stationarity of $f$. There is a rather widespread misunderstanding of this mathematical fact found in the literature.
 When the variables and constraints are given in complex form, the method of Lagrange multipliers is expressed by $\frac{\partial f} {\partial z_k^*} = \sum_l \bigg[ \lambda_l \frac{\partial g_l} {\partial z_k^*} + \lambda_l^* \frac{\partial g_l^*} {\partial z_k^*} \bigg]$. Though $g_l (z_k, z_k^*)=c$ and $g_l^* (z_k, z_k^*)=c^*$ are the same constraint, we have to add the complex conjugate of it in the Lagrange equation. Be careful to note that $(\frac{\partial g_l}{\partial z_k^*})^*$ is not equal to $\frac{\partial g_l^*}{\partial z_k^*}$. The above equation gives the condition for the stationarity of the realvalued function $f$.
 $\vec{m}_k^l$ indicates the shorthanded notation of $\hat{a}_{l}^{\dagger} \hat{a}_{k}  \vec{m} \rangle = \sqrt{(m_{l}+1)m_{k}}  \vec{m}_k^l \rangle$ which means that, from the given configuration $\vec{m}$, one particle is removed from $k$orbital and one particle is added to $l$orbital and the state $ \vec{m}_k^l \rangle$ is properly normalized.
 A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev. A. 73, 063626 (2006). %General variational manybody theory with complete selfconsistency for trapped bosonic systems.
 J. J. McKeown, D. Meegan, and D. Sprevak, An Introduction to Unconstrained Optimization, IOP Publishing, 1990.
 The steepest descent direction in terms of real variables is given by the gradient: $d\vec{x} =  \vec{\nabla} f$. On the other hand, the steepest descent direction in terms of complex variables is given by the differential with respect to the complex conjugate of the variables: $dz_k =  \frac{\partial f}{\partial z_k^*}$. Then the steepest constrained descent is given by $\frac{d z_k} {d \tau} = \Delta(\tau) \bigg[ \frac{\partial f} {\partial z_k^*}  \sum_l \Big[\lambda_l \frac{\partial g_l} {\partial z_k^*} + \lambda_l^* \frac{\partial g_l^*} {\partial z_k^*} \Big] \bigg]$ where $\Delta(\tau)$ can be any arbitrary positive definite function.
 A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev. Lett. 99, 030402 (2007). %Role of Excited States in the Splitting of a Trapped Interacting BoseEinstein Condensate by a TimeDependent Barrier
 kipid's blog  Method of Lagrange multipliers
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