This is a homework of computational nano mechanics. The basic requirement is to use Lagrangian function to describe the motion of particles. In homework, it requires 5 particles. As an enhancement, I rewrote the code with Qt and use GNU Scientific Library(GSL) to finish the task.

**Concept**

The Lagrangian equation of motion describe the particle motion with energy method, the equation is:

\[ \displaystyle \frac{d}{dt}\frac{\partial L}{\partial \dot{x}_k}-\frac{\partial L}{\partial x_k}=0\]

\( L\) is the Lagrangian function of the system, \( k\) denotes the degree of freedom.

\[ \displaystyle L = T – U\]

\( T\) is the kinetic energy and \( U\) is the potential energy.

Since the derivative of \( U\) with respect to \( x\) is the force

\[ \displaystyle f = -\frac{dU(r)}{dr}\]

the Lagrangian equation of motion uses another way to illustrate the Newton’s second law.

**Single particle example**

For a single particle in a chamber, two degree of freedom $latex x_1 = x, x_2 = y$, the kinetic energy is:

\[ \displaystyle T =\sum \frac{1}{2} m \left({\dot{x}}^2 + {\dot{y}}^2\right)\]

\( k\) is the degree of freedom. $latex m$ denotes the mass of the particle.

The potential energy of the particle in a chamber is expressed as:

\[ \displaystyle U = \alpha e^{- \beta(R-r)}\]

Here the exponent function is used to describe the repulsion from the chamber wall. when the distance $latex R-r$ close to zero, the repulsion potential will be much large and prevent the particle penetrating the chamber wall. The Lagrangian function and equations are:

\[ \displaystyle L = \frac{m}{2} (\dot{x}^2+\dot{y}^2)-\alpha e^{-\beta (R-\sqrt{x^2+y^2})}\]

and for each degree of freedom:

\[

\displaystyle \left\{\begin{matrix}

m \ddot{x}(t) = -\alpha \beta e^{-\beta (R-\sqrt{x^2+y^2})} x(t) / \sqrt{x^2+y^2} \\

m \ddot{y}(t) = -\alpha \beta e^{-\beta (R-\sqrt{x^2+y^2})} y(t) / \sqrt{x^2+y^2}

\end{matrix}\right.

\]

For this example, we take the following parameters:

\[ \displaystyle \alpha = 10^{-27}, \beta = 4nm^-1, R = 10nm, m = 10^{-9}kg\]

The initial conditions are:

\[ \displaystyle x(0) = -3nm, \dot{x}(0) = 10nm/s, y(0) = 5nm, \dot{y}(0) = 20nm/s\]

Using Mathematica (FF01Y5.nb) to plot the trajectory or use this script (YR29Q8.nb) to plot the animation:

**Lennard-Jones potential function**

Lennard-Jones potential function is almost the world-famous potential function in nano mechanics. It describes the interaction between atoms. The function was proposed through quantum perturbation theory in which the attraction should follow a $latex r^{6}$ relation. It combines the attraction and repulsion together.

\[ \displaystyle U(r_{ij})=4 \varepsilon \left[ \left(\frac{\sigma}{r_{ij}} \right)^{12} – \left( \frac{\sigma}{r_{ij}}\right)^{6} \right]\]

where: \( \varepsilon\) denotes the depth of the energy well, and \( \sigma\) denotes the equilibrium spacing of atoms.

Take \( \sigma = 3, \varepsilon = 0.2\), the result is shown below:

As mentioned before, the derivative of \( U\) with respect to $latex r$ is the force. The force from Lennard-Jones interaction is simply get the derivative of \( U\):

\[ \displaystyle F=-U^{\prime}(r_{ij}) = -4 \varepsilon \left(\frac{6 \sigma ^6}{r^7}-\frac{12 \sigma ^{12}}{r^{13}}\right) \]

The equilibrium distance between two atoms is the position \( F=0\).

\[ \displaystyle \sigma \sqrt[6]{2} = r_0\]

\[ \displaystyle r_0 = 1.225 \sigma\]

**Two particles example**

If two particles are put into the chamber, Lennard-Jones potential should be considered in calculation. Here, I try to use GNU Scientific Library(GSL) to solve the problem in three dimension.

The *Explicit 4th order (classical) **Runge-Kutta**(RK4) *method is chosen for this problem. Two particles are denoted as \( i\) and \( j\) respectively. The Lagrangian function is (\( k\) denotes the degree of freedom):

\[ \displaystyle L=T-U \]

\[ \displaystyle T = \sum \frac{1}{2}m_i (\dot{x}_{ik}^2)+\sum \frac{1}{2}m_j (\dot{x}_{jk}^2) \]

\[ \displaystyle U = E_{pi} + E_{pj} + W_{LJ}\]

\( E_{pi}\) and \( E_{pj}\) denotes the repulsive potential from the chamber wall, the equations of them are similar to **Single particle example**. Here, I just analyze the Lanner-Jones potential:

\[ \displaystyle W_{LJ-ijk} = 4 \varepsilon \left[ \frac{\sigma ^ {12}}{\left(\sum_{k=1}^3 \left( x_{ik} – x_{jk}\right)^2 \right)^6} – \frac{\sigma ^ 6}{\left(\sum_{k=1}^3 \left( x_{ik} – x_{jk}\right)^2 \right)^3} \right]\]

For the derivative \( W_{LJ-ijk}\) with respect to \( x_{ik}\) and \( x_{jk}\):

\[ \displaystyle \frac{\partial W_{LJ-ijk}}{\partial x_{ik}} = 4 \varepsilon \left[ \frac{6 (x_{ik}-x_{jk})\sigma ^ {6}}{\left(\sum_{k=1}^3 \left( x_{ik} – x_{jk}\right)^2 \right)^4} – \frac{12 (x_{ik}-x_{jk}) \sigma ^ {12}}{\left(\sum_{k=1}^3 \left( x_{ik} – x_{jk}\right)^2 \right)^7} \right]\]

\[ \displaystyle \frac{\partial W_{LJ-ijk}}{\partial x_{jk}} = 4 \varepsilon \left[ -\frac{6 (x_{ik}-x_{jk})\sigma ^ {6}}{\left(\sum_{k=1}^3 \left( x_{ik} – x_{jk}\right)^2 \right)^4} + \frac{12 (x_{ik}-x_{jk}) \sigma ^ {12}}{\left(\sum_{k=1}^3 \left( x_{ik} – x_{jk}\right)^2 \right)^7} \right]\]

With these two equations, we can simply calculate the motion of these two particles:

大神，原来的内容有没有备份？最近发现个人主页好像改版了。。

有备份，最近比较忙，暂时还没有时间整理以前的东西。