This paper proposes a Fourier-Legendre spectral method to find the minimizers of a variational problem, called
σ
2
,
p
\sigma _{2,p}
-energy, in polar coordinates. Let
X
⊂
R
n
{\mathbb {X}}\subset \mathbb {R}^n
be a bounded Lipschitz domain and consider the energy functional
(
1.1
)
(1.1)
whose integrand is defined by
W
(
∇
u
(
x
)
)
≔
(
σ
2
(
u
)
)
p
2
+
Φ
(
det
∇
u
)
{\mathbf {W}}(\nabla u(x))≔(\sigma _2(u))^{\frac {p}{2}}+\Phi (\det \nabla u)
over an appropriate space of admissible maps,
A
p
(
X
)
\mathcal {A}_p({\mathbb {X}})
. Using Fourier and Legendre interpolation errors, we obtain an error estimate for the energy functional and prove a convergence theorem for the proposed method. Furthermore, we apply the gradient descent method to solve a nonlinear algebraic system which is obtained by discretizing the Euler-Lagrange equations. The numerical experiments are performed to demonstrate the accuracy and effectiveness of our method.