Noether Symmetries of the Area-Minimizing Lagrangian

It is shown that the Lie algebra of Noether symmetries for the Lagrangian minimizing an (cid:2) n − 1 (cid:3) area enclosing a constant n -volume in a Euclidean space is so (cid:2) n (cid:3) ⊕ s R n and in a space of constant curvature the Lie algebra is so (cid:2) n (cid:3) . Furthermore, if the space has one section of constant curvature of dimension n 1 , another of n 2 , and so on to n k and one of zero curvature of dimension m , with n ≥ (cid:2) kj (cid:4) 1 n j (cid:5) m (cid:2) as some of the sections may have no symmetry (cid:3) , then the Lie algebra of Noether symmetries is ⊕ kj (cid:4) 1 so (cid:2) n j (cid:5) 1 (cid:3) ⊕ (cid:2) so (cid:2) m (cid:3) ⊕ s R m (cid:3) .


Introduction
Symmetries of the Lagrangian and the corresponding Euler-Lagrange EL equations play an important role in the study of the differential equations DEs . These symmetries can be used to reduce the order of the DEs or the number of variables in the case of partial differential equations PDEs and for the linearization of nonlinear DEs 1-3 . Symmetries that keep the action integral invariant called Noether symmetries are more important than the symmetries of the corresponding EL equation called Lie point symmetries as these symmetries give double reduction of the DEs and provide conserved quantities 4 .
As the differential equations "live" on manifolds, it is natural to search for the connection between symmetries of differential equations and geometry. The first such attempt looked for the connection through the system of geodesic equations 5, 6 . Some connections between Noether symmetries and isometries have been found in the context of general relativity 7-9 . There are some errors in 7, 8 and it is incomplete regarding all Noether symmetries under discussion, the corrected list was given in 10,11 . Recently, the relation of both the Lie and Noether symmetries of the geodesic for a general Riemannian 2 Journal of Applied Mathematics manifold has been given 12 . The geodesic equations are the EL equations for the arc-lengthminimizing action. Their symmetries and the corresponding geodesic equations are known for maximally and nonmaximally symmetric spaces. A connection was obtained between isometries the symmetries of the geometry and Lie symmetries of the geodesic equations of the underlying space, which leads to the geometric linearization for ordinary differential equations ODEs . An additional benefit of this approach is that one can obtain the solution of the linearized equations by the transformation to the metric tensor coordinates given by the geodesic equations from Cartesian coordinates 13-15 . In searching for an extension of the geometric methods to partial differential equations PDEs , we tried to obtain a relation between isometries and Noether symmetries for the area-minimizing Lagrangian. In this paper, it is shown that the Lie algebra of the symmetries for the area-minimizing Lagrangian in an n-dimensional Euclidean space is so n ⊕ s R n and in a space of constant curvature is so n . It is also shown that if the space has one section of constant curvature of dimension n 1 , another of n 2 , and so forth to n k , and one of zero curvature of dimension m and n ≥ k j 1 n j m, so that some of the sections have no symmetry, then the Lie algebra of Noether symmetries is A ⊕ k j 1 so n j 1 ⊕ so m ⊕ s R m . The plan of the paper is as follows. A brief review of the mathematical formalism is given in the next section. In the subsequent sections we present the symmetries of the areaminimizing Lagrangian for maximally and nonmaximally symmetric spaces. In the sixth section we present the symmetries for the less symmetric spaces. A summary and brief discussion are given in the last section.

Preliminaries
Let x x i and u u α be n-independent and m-dependent variables, respectively. The derivatives of u with respect to x are denoted by u α is the total derivative operator and we have used the Einstein summation convention. The set of all the first derivatives u α i is denoted by u 1 and the sets of all the higher-order derivatives are denoted by u 2 , u 3 , . . . , u i . Let L x, u, u 1 Isometries are the directions, k k a ∂/∂x a , along which the Lie derivative of the metric tensor g is zero, that is, This equation can be written in component form as where "," denotes derivative with respect to x a a 1, 2, . . . , n . This equation forms a set of n n 1 /2 linear first-order partial differential equations for n functions of n variables in general, called the Killing equations.
In the n-dimensional space we minimize the n−1 -area A S of a hypersurface S given by x n x n x α , α 1, 2, . . . , n − 1, keeping the n-volume V S fixed. We define y α ∂x n /∂x α . The area-minimizing action is 16 where g n−1 is the determinant of the n − 1 -metric of the hypersurface, g n is the determinant of the n-metric of the volume, and " , α " represents ∂/∂y α .

Two-Area Minimization
The flat space metric in spherical coordinates is

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Let the enclosing surface be r r θ, φ . The 2-area is then given by and the variational principle for the action 2.8 becomes Thus the Lagrangian is The Noether symmetry condition 2.5 results in the following system of linear PDEs:

3.6
This system gives us the following six symmetries:

3.7
Journal of Applied Mathematics 5 The corresponding Lie algebra of the Noether symmetries is so 3 ⊕ s R 3 , where ⊕ s is the semidirect sum, so 3 X 1 , X 2 , X 3 , R 3 X 4 , X 5 , X 6 , and are the vector gauge functions corresponding to the translations R 3 .

Three-Area Minimization
Following the same procedure for a 3-area enclosing a constant 4-volume in hyperspherical coordinates, the Lagrangian is

6
Journal of Applied Mathematics The Lie algebra of the Noether symmetries for this Lagrangian is so 5 ⊕ s R 5 and A 1 λr 4 20 sin 3 ψ cos ψ sin 3 χ sin 2 θ cos φ, cos χ cos φ sin 2 χ sin 2 ψ sin 2 θ, We can now prove the results generalized to m − 1 -area minimization for a constant m-volume in a flat space by using a method of reduction and induction as done earlier for the connection between geometry and Lie symmetries 6 . Proof. The Lie algebra of Noether symmetries of the Lagrangian that minimizes the 2area enclosing a 3-volume in Euclidean space, 3-area enclosing a 4-volume, and the 4area enclosing a 5-volume in Euclidean space are so 3 ⊕ s R 3 , so 4 ⊕ s R 4 , and so 5 ⊕ s R 5 , respectively. Now, suppose that the Lie algebra of Noether symmetries of the Lagrangian that minimizes an n−1 -area enclosing a constant n-volume in Euclidean space is so n ⊕ s R n . The Lagrangian for minimizing the n-area enclosing a constant n 1 -volume in Euclidean space contains a subset of Noether symmetries identical to the isometries of S n , that is, so n 1 . In the Euclidean space, S n minimizes the n-area enclosing a constant n 1 -volume. For the full set of Lie algebra, first reduce the n-area to an n − 1 -area and n 1 -volume to an n-volume. The Lagrangian minimizing the n-area enclosing a constant n 1 -volume reduces to the Lagrangian which minimizes the n − 1 -area enclosing a constant n-volume in the Euclidean space. The corresponding Lie algebra is so n ⊕ s R n the Lagrangian which minimizes the 4-area enclosing 5-volume can be transformed to the Lagrangian which minimizes 3-area enclosing 4-volume . Now, working in reverse, from the Lagrangian minimizing an n − 1area enclosing a constant n-volume to the Lagrangian minimizing the n-area enclosing a constant n 1 -volume, it takes n more generators of rotation and one generator of translation from the previous one, that is, n 1 more generators. Thus the Lie algebra of the Noether symmetries of the Lagrangian which minimizes m − 1 -area enclosing a constant m-volume in the Euclidean space is identical to the Lie algebra of isometries of the Euclidean space, that is, so m ⊕ s R m .

Two-Area Minimization
The metric for a three-dimensional curved space is ds 2 dχ 2 sinh 2 χdθ 2 sinh 2 χ sin 2 θdφ 2 . 4.1 Using the variational principle 2.8 for minimizing two area, we obtain the Lagrangian The Lie algebra of the Noether symmetries of the Lagrangian is so 3 . Notice that there is no translational symmetry arising here.

4.6
The Lie algebra of the Noether symmetries of the Lagrangian is so 4 , there is no translation in this case. For spaces of constant nonzero curvature we present the following theorem. Proof. The proof of this theorem can be provided by arguments similar to those in Theorem 3.1.
These two theorems provide the Noether symmetries of the area-minimizing Lagrangian for maximally symmetric spaces constant curvature and zero curvature . Notice that when we go to spaces of constant curvature from spaces of zero curvature we lose m symmetries of the area-minimizing Lagrangian. In the case of zero curvature m symmetries translational symmetries come out only with particular vector gauge functions, while the remaining symmetries rotational symmetries have a zero gauge function. In the case of nonzero curvature there is no translational symmetry and we have only rotational symmetries corresponding to a zero gauge function.

Three-Area Minimization in One-Dimensional Flat and Three-Dimensional Curved Space
The metric for a four-dimensional space having one-dimensional flat section is ds 2 dψ 2 dχ 2 sinh 2 χdθ 2 sinh 2 χ sin 2 θdφ 2 .

5.3
The Lie algebra of the Noether symmetries of this Lagrangian is so 4 ⊕ R 1 and R 1 corresponds to the vector gauge function, A λ 0, 0, φsinh 2 χ sin θ .

Four-Area Minimization in Two-Dimensional Flat and Three-Dimensional Curved Space
The metric for a five-dimensional flat space having two-dimensional flat section is ds 2 dr 2 r 2 dχ 2 dψ 2 sinh 2 ψdθ 2 sinh 2 ψ sin 2 θdφ 2 .

5.5
The Lie algebra of Noether symmetries for this Lagrangian is so 4 ⊕ so 2 ⊕ s R 2 , where R 2 corresponds to the vector gauge function 5.6

Symmetries for the Less Symmetric Spaces
The metric for a spheroid which is a less symmetric surface of positive curvature is ds 2 a 2 cos 2 θ b 2 sin 2 θ dθ 2 a 2 sin 2 θdφ 2 , 6.1 and the Lagrangian is

6.5
This Lagrangian admits no symmetry.
We can now generalize the results for spaces having sections of different constant curvatures using the geometric method as done in 13 . In the manifold M ⊥ , each section retains its Lie algebra of Noether symmetries. Thus, the full Lie algebra of Noether symmetries, in this case, is the direct sum of all these Lie algebras, that is, A ⊕ k j 1 so n j 1 ⊕ so m ⊕ s R m . If instead there is no reduction of dimension of the flat section, but one constant curvature section reduces by one-dimension, say n j → n j − 1 the Lie algebra of the other sections remains unchanged while that of the reduced section now becomes so n j . Now consider the case that there is only a one-dimensional flat section, that is, m 1, and n − 1 -dimensional section of constant curvature. We have an n-volume in a space having a one-dimensional flat section and an n − 1 -dimensional section of constant curvature. We minimize the n − 1 -area in a subspace of constant curvature keeping a constant n-volume. Then, the Lie algebra of Noether symmetries is so n ⊕ so 1 ⊕ s R 1 , that is, so n ⊕ R 1 as so 1 is the identity .
By increasing the dimension of the flat section by one, as in Theorem 3.1, the algebra for it becomes so m 1 ⊕ s R m 1 . Similarly, by increasing the dimension of one constant curvature section by 1, n j → n j 1 , the Lie algebra of that section becomes so n j 2 while the other sections retain their Lie algebras.
Thus, reduction and induction show that the formula continues to hold. This completes the proof.

Summary and Discussion
In this paper we have dealt with the Noether symmetries of the n − 1 -area-minimizing Lagrangian keeping a constant n-volume for the maximally and nonmaximally symmetric spaces. For spaces of maximal symmetry, the Lie algebra of the Noether symmetries is so n ⊕ s R n in an n-dimensional flat space and so n in an n-dimensional space of constant curvature. For an n-dimensional space of constant curvature, the area-minimizing Lagrangian has n n−1 /2 rotational symmetries with a zero gauge function and for the zero curvature there are n translational symmetries with specific vector gauge functions, along with n n − 1 /2 rotational symmetries with zero gauge function.
The second theorem provides the Noether symmetries for the area-minimizing Lagrangian in nonmaximally symmetric spaces. In this case, we have a space consisting of sections of different constant curvatures, one section of zero curvature, and possibly a section with no symmetry. The Lie algebra of Noether symmetries is then the direct sum of the Lie algebras of Noether symmetries of each section. If the space has a flat section of only one dimension the Lie algebra becomes ⊕ k j 1 so n j 1 ⊕ R 1 .