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import math
from scipy.optimize import brentq
from scipy.integrate import solve_ivp
def solve():
alpha = 3.0
beta = 2.0
gamma = 1.0
# W(L1, L2) = alpha*(L1^2 + L2^2) + Phi(J), where J = L1*L2
# Phi(J) = beta*(J - 1)^2 + gamma*J^(-2)
# Phi'(J) = 2*beta*(J - 1) - 2*gamma*J^(-3)
# Phi''(J) = 2*beta + 6*gamma*J^(-4)
def W_val(L1, L2):
J = L1 * L2
return alpha * (L1**2 + L2**2) + beta * (J - 1.0)**2 + gamma * J**(-2)
def dW_dL1(L1, L2):
J = L1 * L2
return 2.0 * alpha * L1 + (2.0 * beta * (J - 1.0) - 2.0 * gamma * J**(-3)) * L2
# 1. Total energy at homogeneous deformation lambda = 2 on unit disk: E(2) = pi * W(2, 2)
total_energy_at_2 = math.pi * W_val(2.0, 2.0)
# 2. Natural state lambda_0 > 0 minimizing E(lambda) = pi * W(lambda, lambda)
# d/dL W(L, L) = 0 => 2*L^8 + L^6 - 1 = 0
lambda_0 = brentq(lambda L: 2.0 * L**8 + L**6 - 1.0, 0.5, 1.5, xtol=1e-14)
# Radial Euler-Lagrange equation in variables y = [f, L1, E_cum]:
# As derived from d/dr (r W_1) = W_2:
# 2*alpha*(L1' + L2') + L2 * Phi''(J) * J' = 0
# Since L2 = f/r, L2' = (L1 - L2)/r, and J' = L1'*L2 + L1*L2':
# (2*alpha + L2^2 * Phi''(J)) * L1' + (2*alpha + J * Phi''(J)) * (L1 - L2) / r = 0.
def ode_rhs(r, y):
f_val, L1, _ = y
L2 = f_val / r
J = L1 * L2
phi_dd = 2.0 * beta + 6.0 * gamma * (J**(-4))
L2_prime = (L1 - L2) / r
L1_prime = -((2.0 * alpha + J * phi_dd) / (2.0 * alpha + (L2**2) * phi_dd)) * L2_prime
dE_dr = 2.0 * math.pi * r * W_val(L1, L2)
return [L1, L1_prime, dE_dr]
# 3. IVP on [1, 2] with f(1) = 1, f'(1) = 2
sol_ivp = solve_ivp(ode_rhs, [1.0, 2.0], [1.0, 2.0, 0.0], rtol=1e-12, atol=1e-14)
ivp_outer_stretch = sol_ivp.y[0, -1] / 2.0
# 4. BVP on [1, 2] with f(1) = 1, f(2) = 3 via shooting on s = f'(1)
def shoot(s):
sol = solve_ivp(ode_rhs, [1.0, 2.0], [1.0, s, 0.0], rtol=1e-12, atol=1e-14)
return sol.y[0, -1] - 3.0
s_star = brentq(shoot, 1.0, 3.0, xtol=1e-13)
sol_bvp = solve_ivp(ode_rhs, [1.0, 2.0], [1.0, s_star, 0.0], rtol=1e-12, atol=1e-14)
annular_bvp_energy = sol_bvp.y[2, -1]
# 5. Exact Virial / Eshelby-Piola Identity on [1, R]:
# Combine the radial divergence identities:
# (A) d/dr [ r f W_1 ] = r f' W_1 + f d/dr(r W_1) = r (L1 W_1 + L2 W_2)
# (B) d/dr [ r^2 (W - L1 W_1) ] = r (2 W - L1 W_1 - L2 W_2)
# Subtracting (B) from (A) gives:
# d/dr [ r^2 (L1 W_1 + L2 W_2 - W) - r f W_1 ] = 2 r (L1 W_1 + L2 W_2 - W) - r (L1 W_1 + L2 W_2)
# Wait: (A) alone states:
# int_1^R r (L1 W_1 + L2 W_2) dr = [ r f W_1 ]_1^R = [ r^2 L2 W_1 ]_1^R!
# And notice that L1 W_1 + L2 W_2 = 2*alpha*(L1^2 + L2^2) + 2*J*Phi'(J)
# = 2 W(L1, L2) + 2*(J*Phi'(J) - Phi(J))
# = 2 W(L1, L2) + 2*( 2*beta*J*(J-1) - beta*(J-1)^2 - 3*gamma*J^(-2) )!
# Therefore:
# -(1/pi) E[f] + 2 int_1^R r (2*beta*J*(J-1) - beta*(J-1)^2 - 3*gamma*J^(-2)) dr
# = -2 int_1^R r W dr + int_1^R r (L1 W_1 + L2 W_2 - 2 W) dr
# = int_1^R r (L1 W_1 + L2 W_2) dr - 4 int_1^R r W dr... wait:
# Let's check:
# -(1/pi) E[f] = -2 int_1^R r W dr.
# And 2 * (2*beta*J*(J-1) - beta*(J-1)^2 - 3*gamma*J^(-2)) = 2*(J*Phi'(J) - Phi(J)) = L1 W_1 + L2 W_2 - 2 W.
# Wait: in the formula for V(R), the boundary term is:
# [ r^2 (L1 W_1 + L2 W_2 - W) ]_1^R!
# Let's check what d/dr [ r^2 (L1 W_1 + L2 W_2 - W) ] is, or let's use the exact identity (B):
# From (B): d/dr [ r^2 (L1 W_1 - W) ] = r (L1 W_1 + L2 W_2 - 2 W)
# = 2 r ( 2*beta*J*(J-1) - beta*(J-1)^2 - 3*gamma*J^(-2) )!
# And from (A): d/dr [ r^2 L2 W_1 ] = r (L1 W_1 + L2 W_2)
# = 2 r W + 2 r ( 2*beta*J*(J-1) - beta*(J-1)^2 - 3*gamma*J^(-2) )!
# Therefore, adding or comparing (A) and (B):
# [ r^2 (L1 W_1 + L2 W_2 - W) ]_1^R
# = [ r^2 (L1 W_1 - W) ]_1^R + [ r^2 L2 W_2 ]_1^R!
# Wait: at r -> infinity (in log-scale t = ln r), L1(r) and L2(r) converge exponentially fast to a constant!
# Let's check why!
# In t = ln r, the autonomous ODE is:
# dL2/dt = L1 - L2
# dL1/dt = - ((2*alpha + J*Phi''(J)) / (2*alpha + L2^2*Phi''(J))) * (L1 - L2)
# Notice that:
# 1) There is a exact first integral between L1, L2!
# Wait: earlier we proved:
# 2*alpha*(dL1 + dL2) + L2*Phi''(J)*dJ = 0? No:
# (2*alpha + L2^2*Phi''(J)) dL1 + (2*alpha + J*Phi''(J)) dL2 = 0
# => 2*alpha*d(L1 + L2) + L2*Phi''(J)*(L2 dL1 + L1 dL2) = 0
# => 2*alpha*d(L1 + L2) + L2*Phi''(J)*dJ = 0!
# Wait: as t = ln r -> +infinity, since dL2/dt = L1 - L2 and dL1/dt has the opposite sign of L1 - L2:
# d/dt (L1 - L2) = -[ 1 + (2*alpha + J*Phi''(J)) / (2*alpha + L2^2*Phi''(J)) ] * (L1 - L2)!
# Notice the bracket is roughly 1 + 1 = 2!
# So |L1(r) - L2(r)| decays like O(r^(-2)) as r -> infinity!
# And both L1(r) and L2(r) converge as r -> infinity to a common asymptotic stretch lambda_inf,
# with |L1(r) - lambda_inf| = O(r^(-2)) and |L2(r) - lambda_inf| = O(r^(-2))!
# Let's verify this!
# When L1 -> lambda_inf and L2 -> lambda_inf at rate O(r^(-2)), at r = R = 10^12,
# R^(-2) = 10^(-24), so at r = R = 10^12, L1(R) and L2(R) equal lambda_inf to 24 decimal places!
# Let's check what V(R) simplifies to using Identity (B)!
# By Identity (B):
# 2 int_1^R r ( 2*beta*J*(J-1) - beta*(J-1)^2 - 3*gamma*J^(-2) ) dr = [ r^2 (L1 W_1 - W) ]_1^R!
# Substitute this into V(R):
# V(R) = (1 / R^2) * [ [ r^2 (L1 W_1 + L2 W_2 - W) ]_1^R - (1/pi) E[f_R] + [ r^2 (L1 W_1 - W) ]_1^R ]
# Also by Identity (A):
# (1/pi) E[f_R] = 2 int_1^R r W dr = [ r^2 L2 W_1 ]_1^R - [ r^2 (L1 W_1 - W) ]_1^R!
# Substituting both (A) and (B) into the bracket of V(R):
# Bracket = [ r^2 (L1 W_1 + L2 W_2 - W - L2 W_1 + 2(L1 W_1 - W)) ]_1^R
# = [ r^2 (3 L1 W_1 + L2 W_2 - L2 W_1 - 3 W) ]_1^R!
# Wait: we don't even have to guess—both (A) and (B) hold identically for any solution of the Euler-Lagrange equation!
# And we can compute lambda_inf by integrating the autonomous ODE in t = ln r from t = 0 to t = ln(10^12) = 27.631021115928547!
# Since t = ln(10^12) is only 27.63, integrating in t = ln r takes < 1 millisecond and reaches machine precision!
t_end = 12.0 * math.log(10.0)
# In t = ln r, the autonomous system for (L1, L2) is:
# dL2/dt = L1 - L2
# dL1/dt = - ((2*alpha + J*Phi''(J)) / (2*alpha + L2^2*Phi''(J))) * (L1 - L2)
def ode_log_r(t, y):
L1, L2 = y
J = L1 * L2
phi_dd = 2.0 * beta + 6.0 * gamma * (J**(-4))
dL2_dt = L1 - L2
dL1_dt = -((2.0 * alpha + J * phi_dd) / (2.0 * alpha + (L2**2) * phi_dd)) * dL2_dt
return [dL1_dt, dL2_dt]
L2_init = 6.0 / 5.0
L1_init = 4.0 / 5.0
sol_inf = solve_ivp(ode_log_r, [0.0, t_end], [L1_init, L2_init], rtol=1e-13, atol=1e-15)
L1_R, L2_R = sol_inf.y[0, -1], sol_inf.y[1, -1]
def H_val(L1, L2):
# Coefficient of r^2 in the exact antiderivative:
# [ r^2 (L1 W_1 + L2 W_2 - W) ] - [ r^2 L2 W_1 - r^2 (L1 W_1 - W) ] + [ r^2 (L1 W_1 - W) ]
# = r^2 (3 L1 W_1 + L2 W_2 - L2 W_1 - 3 W)
W = W_val(L1, L2)
W1 = dW_dL1(L1, L2)
W2 = dW_dL1(L2, L1) # by symmetry of W(L1, L2)
return 3.0 * L1 * W1 + L2 * W2 - L2 * W1 - 3.0 * W
R_val = 1e12
virial_ratio = H_val(L1_R, L2_R) - H_val(L1_init, L2_init) / (R_val**2)
res = {
"problem_id": "MATH2",
"total_energy_at_2": round(total_energy_at_2, 6),
"natural_state_lambda": round(lambda_0, 6),
"ivp_outer_stretch": round(ivp_outer_stretch, 6),
"annular_bvp_energy": round(annular_bvp_energy, 6),
"large_domain_virial_ratio": round(virial_ratio, 6),
}
print(json.dumps(res))
if __name__ == "__main__":
solve()
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