# Geometrical Applications Of Derivatives

Geometrical Applications Of Derivatives, Theories, and Graphs. Abstract A geometric solution is given to the Euler-Lagrange equation in the time-dependent Schrödinger picture, for $x \mapsto 1/x$ and $x \in D$. The solution can be found in the equilibrium state of the system, if one can find a suitable system of equations with the time-dependence of the potential $\phi$. The solution is then compared with the equilibrium state obtained by solving the differential-difference equations of the system. Introduction ============ The problem of finding the equilibrium state for the Euler equations has been the subject of a great deal of research for the past thirty years. Some of the major results of this era were obtained in the works of John von Neumann, C. S. Lewis, and W. Zoller [@Zoller] and the pioneering work of J. P. Cirac [@Cirac]. Much of the work on the equilibrium state was done by W. J. Bardeen and A. J. Tersoff. The equilibrium state was obtained by solving differential-differences equations of the form, \begin{aligned} \label{eq:hamiltonian} \partial_t v_j &=& -\frac{1}{\Omega_j} \frac{\partial u_j}{\partial x_j} + \frac{1-\Omega_{j-1}}{1+\Omega} \frac{u_j}{1-\frac{\Omega_{ji}}{1-x^i}},\\ \label{eq.hamiltonian2} v_j & = & \frac{-\Om_j}{(\Omega_1+\omega_j\sqrt{\Omega_k})^2},\end{aligned} where $\Omega$ is a constant, and $\Omega_i$ are the couplings of the $i$th and $i$-th particle. The solution given by the Hamiltonian is then given by the right-hand side of the Hamiltonian, $$\label{hamiltonian1} H = \Omega_0 + \Omega_{\rm c} + \Om_0,$$ where $\omega_0$ is the frequency of the nonlinear mode, $\Omega_{0}$ is the value of $\Omega$, and $\Om_i$ is the coupling of the $iv$-th and $j$-th particles. The Hamiltonian has the form, $eq:hamil$$$\begin{gathered} \Omega = \left( \begin{array}{c} \omega/2 \\ -\omega \\ -2\sqrt{1+x^i+x^j} \end{array} \right) \left( \begin{matrix} \sqrt {1-x} & -\sqrt \Omega \\ -x & \sqrt {x} \ \end{matrix}\right) \end{gathered}\label{eq1}$$ where $\Omega = 2\sqrt {\Omega_2}$.