Throughout this paperM denotes an n-dimensional smooth manifold, C∞(M) de-notes the ring of real-valued smooth functions, X(M) is the C∞(M)-module of vector fields onM,π :T M →M is the tangent bundle ofM,TM =T M\{0}is the slit tangent space. V T M = Kerπ∗ is the vertical sub-bundle of T T M. We denote by Λk(M),Sk(M) and Ψk(M) the C∞(M)-modules of skew-symmetric, symmetric and vector valued k-forms respectively, and by Λkv(T M), Svk(T M) and Ψkv(T M) the corresponding semi-basic C∞(T M)-modules. We consider Λ(M) = L
k∈NΛk(M) the graded algebra of differential forms on M and Ψ(M) = L
k∈NΨk(M) for the graded algebra of vector-valued differential forms onM.
The Fr¨olicher–Nijenhuis formalism
The Fr¨olicher-Nijenhuis theory provides a complete description of the derivations of Λ(M) with the help of vector-valued differential forms, for details we refer to [39].
The i∗ and the d∗ type derivations associated to a vector valued l-form L will be denoted byiL and dL. They can be introduced in the following way: if L∈Ψl(M), then
iLω(X1, . . . , Xl) = ω(L(X1, . . . , Xl)),
where X1, . . . , Xl ∈ X(M), ω ∈Λ1(M). Furthermore, dL is the commutator of the derivations iL and d, that is
dL:= [iL, d] =iLd−(−1)l−1diL.
We remark that for X ∈ X(M) we have dX = LX the Lie derivative, and iX is the substitution operator. The Fr¨olicher–Nijenhuis bracket of K ∈ Ψk(M) and L∈Ψl(M) is the unique [K, L]∈Ψk+l form, such that
[dK, dL] =d[K,L].
In the special case, when K ∈ Ψ1(M), X, Y ∈ X(M) we have [K, X] ∈ Ψ1(M) defined as
[K, X](Y) = [KY, X]−K[Y, X].
Spray and associated geometric quantities
LetJ:T T M −→T T M be thevertical endomorphismandC ∈X(T M) theLiouville vector field. In an induced local coordinate system (xi, yi) on T M we have
J =dxi⊗ ∂
∂yi, C =yi ∂
∂yi.
Euler’s theorem for homogeneous functions implies that L ∈ C∞(T M) is a k-homogeneous function in they = (y1, . . . , yn) variables if and only if
yi∂L
∂yi −Lk = 0. (1.1)
The vertical endomorphism satisfies the following properties:
J2 = 0, KerJ = ImJ =V T M, [J, C] =J. (1.2) Asemispray is a vector fieldS onTM satisfying the relation JS =C. A semispray is called spray if [C,S] = S. The coordinate representation of a spray S takes the form
S =yi ∂
∂xi −2Gi(x, y) ∂
∂yi, (1.3)
where the functionsGi(x, y) are homogeneous of degree 2 in y.
The geodesics of a spray are curves γ : I → M such that S ◦γ˙ = ¨γ. Locally, they are the solutions of the second order ordinary differential equation (SODE)
¨
xi =−2Gi(x,x)˙ , i= 1, . . . , n. (1.4) Sprays describe a global and coordinate free way the systems of second order differ-ential equations.
To every spray S a connection Γ := [J,S] can be associated [46]. Γ is called the natural connection associated to S. One has Γ2 = Id. The eigenspace of Γ corresponding to the eigenvalue −1 is the vertical space V T M, and the eigenspace corresponding to +1 is called the horizontal space. For any z ∈ T M, we have TzT M =HzT M ⊕VzT M. The horizontal and vertical projectors are denoted byh and v. One has
h = 1
2(Id + Γ), v = 1
2(Id−Γ). (1.5)
Locally, the above two projectors can be expressed ash= δxδi⊗dxi,andv = ∂y∂i⊗δyi, where
δ
δxi = ∂
∂xi −Gji(x, y) ∂
∂yj, δyi =dyi+Gij(x, y)dxj, (1.6)
Gji = ∂G∂yij. In the sequel, F ∈ Ψ1(T M) denotes the almost complex structure associated to the connection Γ.
The parallel translation of a vector along curves is defined through horizontal lifts. Let γ: [0,1]→ M be a curve such that γ(0) = p and γ(1) = q. The parallel translationτ: TpM →TqM along γ is defined as follows: if γh is the horizontal lift of γ (ie. ˙γh(t) ∈ HT M and π◦γh = γ) with γh(0) = v, then τ(v) = w, where γh(1) =w. More details about the parallel translation can be found in Section 3.2.
The Berwald connection D:X(TM)×X(TM)→X(TM) is a linear connection onTM defined as follows:
DXY =v[hX, vY]+h[vX, hY]+J[vX,(F+J)Y]+(F+J)[hX, J Y]. (1.7) Using formula (1.7), it follows that Dh = 0 and Dv = 0, which means that the Berwald connection preserves both the horizontal and vertical distribution. More-over, we have DJ = 0, which implies that the Berwald connection has the same action on horizontal and vertical vector fields. Considering the (h, v, v) components of the classical curvature of the Berwald connection we obtain a tensor-field
B(X, Y, Z) =DhXDJ YJ Z − DJ YDhXJ Z − D[hX,J Y]J Z (1.8) called theBerwald curvature. Locally B(x,y)=Bijkl(x, y)dxj⊗dxk⊗dxl⊗∂x∂i where
Bjkli (x, y) = ∂Gijk
∂yl = ∂3Gi
∂yj∂yk∂yl. (1.9)
It is identically zero if and only if the connection Γ is linear. The mean Berwald curvature tensor field B(x,y)=Bjk(x, y)dxj ⊗dxk is the trace
Bjk(x, y) = Bjkll (x, y) = ∂3Gl
∂yj∂yk∂yl. (1.10) Thecurvature of the nonlinear connection Γ is R= 12[h, h], the Nijenhuis torsion of the horizontal projection h. The curvature tensor
R(X, Y) =v
hX, hY
, (1.11)
for X, Y ∈ X(T M), characterizes the integrability of the horizontal distribution HT M: it is integrable, if and only if the curvature is identically zero.
The Jacobi endomorphism (or Riemann curvature in [82]) is defined as
Φ = iSR. (1.12)
The Jacobi endomorphism determines the curvature by the formula R = 13[J,Φ].
The spray S is called flat if its Jacobi endomorphism has the form Φ = ρJ and isotropic if
Φ =ρJ−α⊗C (1.13)
with some λ ∈ C∞(TM), α ∈ Λ1v(TM). We consider also the Ricci curvature Ric, and theRicci scalar ρ, [10], [82, Def. 8.1.7], which are given by
Ric = (n−1)ρ=Rii = Tr(Φ). (1.14)
The Euler–Lagrange partial differential equation
ALagrangian is a functionE: T M →Rsmooth onTM and C1 on the zero section.
E is called regular, if the Euler–Poincar´e 2-form ΩE := ddJE has maximal rank.
If LCE = 2E, and E is C2 on the 0-section, then E is quadratic and it defines a (pseudo)-Riemannian metric onM byg(v, v) = 2E(v), v ∈T M. If LCE = 2E and E isC1 on the null-section, then E defines aFinsler structure. Note that from (1.1) we find that iJddJ =d2J =d[J,J] = 0, so for every Lagrangian E we have
iJΩE = 0. (1.15)
A regular Lagrangian E allows us to define a pseudo-Riemannian metric on the vertical bundle, by putting gE(J X, J Y) = ΩE(J X, Y). The local expression of the and the Lagrangian E is regular if and only if
det ∂2E
∂yα∂yβ
6= 0.
LetE :T M →R be a regular Lagrangian. From [41] we know that the vector field S onT M defined by
iSΩE =d(E− LCE) (1.17)
is a spray and the paths ofS are the solutions to the Euler-Lagrange equations:
d
Definition 1.2.1. A spray S is called variational if there exists a smooth regular LagrangianE which satisfies (1.17), the Euler-Lagrange equation.
When a regular Lagrangian E is given, (1.17), resp. (1.18), is a second order ordinary differential system. On the other hand, when the spray S is given, then (1.17), resp. (1.18), is a second-order partial differential system on the Lagrange functionE. We introduce the following
Definition 1.2.2. LetE be a Lagrangian and S a spray on the manifoldM, then the Euler-Lagrange form associated withE and S is
ωE :=iSΩE +dLCE−dE. (1.19) It is easy to see that ωE is semi-basic, and the local expression in the standard coordinate system onT M isωE =Pn
dxi.Therefore, along a curve γ = (x(t)) associated with S we have
where d/dt denotes the derivation along γ. One can recognize that the coefficients of the form (1.20) are the left hand sides of the Euler-Lagrange system (1.18).
Consequently, in order to find the solution to the inverse problem for a given second order ordinary differential system, we have to look for a regular Lagrangian such that
ωE = 0, (1.21)
or, using local coordinate system:
yj ∂2E
∂xj∂yi −2Gj(x, y) ∂2E
∂yj∂yi − ∂E
∂xi = 0, i= 1, . . . , n. (1.22) Conclusion 1.2.3. To solve the inverse problem of the calculus of variations for a given system of second-order ordinary differential equations or spray, one has to find a regular Lagrangian E such that it solves the Euler–Lagrange partial differential equation (1.21).