In [1]:

    

import sympy as sp
sp.init_printing()


    

In [2]:

    

Uo,x,s=sp.symbols('Uo x s', real=True)


    

In [3]:

    

U=Uo*((s/x)**12-(s/x)**6)
U


    

    
Out[3]:





$\displaystyle Uo \left(\frac{s^{12}}{x^{12}} - \frac{s^{6}}{x^{6}}\right)$

In [4]:

    

Up=U.diff(x)
Up


    

    
Out[4]:





$\displaystyle Uo \left(- \frac{12 s^{12}}{x^{13}} + \frac{6 s^{6}}{x^{7}}\right)$

In [5]:

    

roots = sp.solve(Up,x)
roots


    

    
Out[5]:





$\displaystyle \left[ - \sqrt[6]{2} s, \  \sqrt[6]{2} s, \  s \left(- \frac{\sqrt[6]{2}}{2} - \frac{\sqrt[6]{2} \sqrt{3} i}{2}\right), \  s \left(- \frac{\sqrt[6]{2}}{2} + \frac{\sqrt[6]{2} \sqrt{3} i}{2}\right), \  s \left(\frac{\sqrt[6]{2}}{2} - \frac{\sqrt[6]{2} \sqrt{3} i}{2}\right), \  s \left(\frac{\sqrt[6]{2}}{2} + \frac{\sqrt[6]{2} \sqrt{3} i}{2}\right)\right]$

In [6]:

    

x0=roots[1]
print(x0)
x0


    

    




2**(1/6)*s






    
Out[6]:





$\displaystyle \sqrt[6]{2} s$

In [7]:

    

Um=U.subs(x,x0)
print(Um)
Um


    

    




-Uo/4






    
Out[7]:





$\displaystyle - \frac{Uo}{4}$

In [8]:

    

Upp=Up.diff(x)
Upp


    

    
Out[8]:





$\displaystyle Uo \left(\frac{156 s^{12}}{x^{14}} - \frac{42 s^{6}}{x^{8}}\right)$

In [9]:

    

k=Upp.subs(x,x0)
print(k)
print(sp.latex(k))
k


    

    




9*2**(2/3)*Uo/s**2
\frac{9 \cdot 2^{\frac{2}{3}} Uo}{s^{2}}






    
Out[9]:





$\displaystyle \frac{9 \cdot 2^{\frac{2}{3}} Uo}{s^{2}}$

In [10]:

    

m=sp.symbols("m")


    

In [11]:

    

omega=sp.sqrt(k/m)
omega


    

    
Out[11]:





$\displaystyle \frac{3 \sqrt[3]{2} \sqrt{\frac{Uo}{m}}}{\left|{s}\right|}$

In [ ]: