In [1]:

from sympy import *
init_printing()

In [2]:

X1 = Symbol('X_1')
X2 = Symbol('X_2')
a = Symbol('a')
phi = 1/(6*a)*(3*X1**2*X2 - X2**3)

In [3]:

psi_1 = integrate(-simplify(phi.diff(X2)),X1)

In [4]:

psi_1

Out[4]:

$$- \frac{X_{1}^{3}}{6 a} + \frac{X_{1} X_{2}^{2}}{2 a}$$

In [5]:

psi_2 = integrate(simplify(phi.diff(X1)),X2)

In [6]:

psi_2

Out[6]:

$$\frac{X_{1} X_{2}^{2}}{2 a}$$

In [7]:

K = Symbol('K')
psi = psi_2 - X1**3/(6*a) + K

In [8]:

chi = simplify(psi - 1/2*(X1**2 + X2**2)) #definition of chi ()

In [9]:

nsimplify(chi.subs(X1,a),rational=True)

Out[9]:

$$K - \frac{2 a^{2}}{3}$$

In [10]:

Ksol = solve(chi.subs(X1,a),K)[0]

In [11]:

chi = chi.subs(K,Ksol)

In [12]:

nsimplify(chi,rational=True)

Out[12]:

$$- \frac{X_{1}^{3}}{6 a} - \frac{X_{1}^{2}}{2} + \frac{X_{1} X_{2}^{2}}{2 a} - \frac{X_{2}^{2}}{2} + \frac{2 a^{2}}{3}$$

In [13]:

# Calculate integral of chi

$$2\mu \int\limits_{x_1 = -2a}^a\quad \int\limits_{x_2 = \frac{-2a-x_1}{\sqrt{3}}}^{\frac{2a+x_1}{\sqrt{3}}}\ \chi(x_1, x_2)\ dx_2\ dx_1 $$

In [14]:

chi_1 = integrate(chi,(X2,-2*a/sqrt(3) - X1/sqrt(3), 2*a/sqrt(3) + X1/sqrt(3)  ))

In [16]:

mu=Symbol('mu')
nsimplify(2*mu*integrate(chi_1,(X1,-2*a,a)),rational=True)

Out[16]:

$$\frac{9 \sqrt{3} a^{4} \mu}{5}$$

In [ ]: