Regularity of generating families of functions

We describe the geometric structures involved in the variational formulation of physical theories. In presence of these structures, the constitutive set of a physical system can be generated by a family of functions. We discuss conditions, under which a family of functions generates an immersed Lagrangian submanifold. These conditions are given in terms of the Hessian of the family.


Introduction.
The constitutive set of a physical system is frequently a Lagrangian submanifold of a symplectic phase space. Such systems are considered reciprocal. It is convenient to be able to derive the constitutive set from a simpler generating object such as a Lagrangian in the case of dynamics and an internal energy function in the case of statics. The phase space is not usually the cotangent bundle of a manifold although it is normally isomorphic to a cotangent bundle. We refer to this isomorphism as a Liouville structure. For reasons of interpretation the Liouville structure can not be used to replace the phase space by the cotangent bundle. We stress the importance of Liouville structures for variational formulations of physical theories. It is the presence of a Liouville structure that permits the generation of a constitutive set from a generating object. We say that the system is potential if its constitutive set is derived from a generatig function or a function defined on a constraint manifold. Potentiality implies reciprocity. A more general generating object, such as a family of functions does not necesarily generate a Lagrangian submanifold. We discuss sufficient conditions for families of functions to generate Lagrangian submanifolds. We define the Hessian of a family of functions at its critical points. The sufficient conditions for families of functions to generate Lagrangian submanifolds are based on this definition.
Reciprocity is an important propery of the constitutive set. It can be established by examining directly this set. If the constitutive set is derived from a generating object, then it is more efficient to establish reciprocity by examining the generating object. A similar situation arises when conservatiom laws are examined. Conservation is a property of dynamics and can be established by direct examination of dynamics. Noether's theorems simplify the procedure by relating conservation properties to invariance properties of the generating object.
The paper is organized as follows. In Section 2 we describe some preliminary constructions. In Sections 3 and 4, we describe geometric structures involved in the variational formulation of physical theories, and the derivation of a set from a generating family of functions. The notion of a critical Research of the second author financed by the Polish Ministry of Science and Higher Education under the grant N N201 365636 point of a family is introduced. Section 5 contains examples of constitutive sets. In Section 6, we recall results concerning reductions of Lagrangian submanifolds. Then, we discuss the notion of the Hessian of a function (Section 7) and of a family of functions (Section 8), at a critical point. In Section 9 we introduce the notion of a regular family, less restrictive then the concept of a Morse family, and we show that the set generated by a regular family is an immersed Lagrangian submanifold.

Preliminary constructions.
Let (P, ω) be a symplectic manifold and let V be a vector subspace of the tangent space T p P . We denote by V ¶ the symplectic polar V ¶ = ṗ ∈ T p P ; ∀ δp∈V ω,ṗ ∧ δp = 0 . (1) If C ⊂ P is a submanifold, then T ¶ C will denote the set We recall that a submanifold C ⊂ P is said to be isotropic if T ¶ C ⊃ TC. A submanifold C ⊂ P is said to be coisotropic if T ¶ C ⊂ TC. A submanifold C ⊂ P is said to be Lagrangian if T ¶ C = TC.
A symplectic relation from a symplectic manifold (P 1 , ω 1 ) to a symplectic manifold (P 2 , ω 2 ) is a differential relation ρ from P 1 to P 2 . The graph of a symplectic relation is a Lagrangian sybmanifold of the symplectic manifold (P 2 × P 1 , ω 2 ⊖ ω 1 ). The form ω 2 ⊖ ω 1 is defined by where pr 1 : P 2 × P 1 → P 1 and pr 2 : P 2 × P 1 → P 2 are the canonical projections. If C is a coisotropic submanifold of a symplectic manifold (P, ω), then the set is called the characteristic distribution of the symplectic form ω restricted to C. At each p ∈ C the space D p = D ∩ T p P is the symplectic polar T ¶ p C of T p C. The characteristic distribution is Frobenius integrable. Its integral manifolds are isotropic submanifolds of (P, ω) called characteristics of ω|C. The set of characteristics may be a manifold P . In this case we introduce the reduction relation σ from P to P . Its graph is the set Let π : C → P be the canonical projection. The equality defines a symplectic form tω on P . The reduction relation σ is a symplectic relation from (P, ω) to ( P , tω). The graph of σ is the Lagrangian submanifold graph (σ) = (p, p) ∈ P × P ; p ∈ C, π(p) =p .
The projection π is the strict symplectic reduction from C onto the symplectic manifold ( P , tω) in the terminology of [1]. It is the essential part of the symplectic reduction relation σ.
Let F be a function on a differential manifold Q and let q ∈ Q be a point. The differential of F is a mapping dF : Q → T * Q. (8) and and where There are also projections and such that the mapping is the inverse of Φ : The space T f T * Q is a symplectic vector space with a symplectic form obtained as a restriction of the symplectic form ω Q to this vector space. Both subspaces H f and V f are Lagrangian subspaces. We choose a pair (δq, f ′ ) ∈ T q Q × T * q Q and use curves and such that δq = tγ(0), ϕ(0) = f ′ , and π Q • ϕ = γ. The mapping represents the pair in the sense that tχ(·, 0)(0) = i h (δq) and In the following calculation we use the facts that ω Q is the differential of the Liouville form ϑ Q , that the Liouville form is vertical and that for each s the curve χ(s, ·) : R → T * Q is vertical.
The formula shows that the mapping (17) is a linear symplectomorphism from the direct product T q Q ⊕ T * q Q with its canonical symplectic structure to the symplectic vector space (T f T * Q, ω f ). The formula is equivalent to (26).
3. Subsets of symplectic manifolds generated by families.
The geometric structures involved in the variational formulation of a physical theory are represented by the diagram for p ∈ P and each v ∈ TQ such that τ Q (v) = π(p). The canonical pairing is used. The relation (35) defines the pairing (34) in terms of the symplectomorphism α or the symplectomorphism in terms of the pairing. The set generated by the generating object (31) is expected to be a Lagrangian submanifold of the phase space (P, ω).
The formula (37) defines the set S directly in terms of the generating object. There is an alternate derivation of this set by the following sequence of operations.
(1) The function U is used to generate the Lagrangian submanifold S = im(dU ) ⊂ T * Q of the symplectic manifold (T * Q, ω Q ). (2) The phase lift symplectic relation of the fibration η is used to produce the set S = Ph η(S) ⊂ T * Q. The relation Ph η can be described in the following way. We denote by VQ the subbundle δq ∈ TQ; Tη(δq) = 0 (39) of the tangent bundle TQ composed of vertical vectors. The polar with δq ∈ T q Q and δq ∈ T q Q such that Tη(δq) = δq defines a differential fibrationη This fibration is the strict symplectic reduction (see [1]) from V • Q onto the symplectic manifold (T * Q, ω Q ). It is the essential part of the symplectic reduction relation (38) whose graph is the set The reduced set is not necessarily a Lagrangian submanifold.
of the injection ι is applied to the set S. The result is the set S = Ph ι( S) ⊂ T * Q. The relation Ph ι is, essentially, the strict symplectic reduction from a coisotropic submanifold π −1 Q (ι( Q)) of (T * Q, ω Q ) onto (T * Q, ω Q ). This reduction is the mappinĝ for each δq ∈ Tq Q, ι(q) = π Q (f ). If S is a Lagrangian submanifold of (T * Q, ω Q ), then is a Lagrangian submanifold of (T * Q, ω Q ), (4) The set S ⊂ T * Q is finally obtained as the inverse image α −1 ( S). This set is a Lagrangian submanifold of (P, ω) if S is a Lagrangian submanifold of (T * Q, ω Q ).
In the following example we have a nontrivial Liouville structure and a constrained generating family although the constraint is open. Example 1. Let M be the space time of general relativity with a Minkowski metric g : TM → T * M of signature (1,3). The Lagrangian of a free particle of mass m is the function defined on the open submanifold of time-like vectors in Q = TM . The dynamics of the particle is a differential equation in the energymomentum phase space T * M . It is therefore a subset D ⊂ TT * M . The space TT * M has a natural symplectic structure. The symplectic form is the total differential d T ω M of the canonical symplectic form ω M in T * M . d T is a derivation on the exterior algebra of forms on a manifold M with values in the exterior algebra of forms of the tangent bundle TM (for definition see, e.g., [2]). The dynamics is a Lagrangian submanifold of (TT * M, d T ω M ). The Liouville structure is used for generating dynamics from the Lagrangian (49). This Liouville structure was introduced in [3]. It is described rigorously in [4]. At each phase p ∈ T * M the pseudoriemannian structure of M defines subspaces H p ⊂ T p T * M and V p ⊂ T p T * M of horizontal and vertical vectors such that The dynamics is the set with Example 2. The Hamiltonian generating object for the dynamics of Example 1 is defined on P = P × R + , where P is the set is treated as a family of functions on fibres of the projection The Liouville structure is used. The Lagrangian generating family for the dynamics of a massless particle is the function defined on the space Q = Q × R + , where Q is the tangent bundle TM with the image of the zero section removed is treated as a family of functions on the fibres of the projection The dynamics is the set Example 4. The Hamiltonian generating object for the dynamics of Example 3 is the function defined on P = P × R + , where P is the cotangent bundle T * M with the image of the zero section removed is treated as a family of functions on fibres of the projection ζ : P → P : (p, µ) → p.
It is obvious that the set S is a Lagrangian submanifold if the first two operations listed above produce a Lagrangian submanifold. For this reason we will concentrate our attention on simpler generating objects with trivial Liouville structures and unconstrained families of functions. Such simple generating objects are encountered in the theory of partially controlled static systems. Variational formulations of dynamics require the use of nontrivial Liouville structures as is seen in the above example. We will derive conditions sufficient for obtaining Lagrangian submanifolds from the simple generating ojects.

Families of functions and sets generated by families.
The representing a simple generating object is relevant for our analysis. This simple object can be obtained from the diagram (28) by setting Q = Q and idenifying the symplectic space (P, ω) with (T * Q, ω Q ) or it can be considered an essential portion of the complete diagram (28). The set is the critical set of the family (U , η). Elements of the critical set are critical points of (U , η). There is a mapping κ(U , η) : for each δq ∈ T η(q) Q and each δq ∈ T q Q such that Tη(δq) = δq. The family of functions (U , η) generates a set S ∈ T * Q. This set is obtained by one of the two following constructions.
(1) Let S = im(dU ) ⊂ T * Q be the Lagrangian submanifold generated by the function U . The symplectic relation Ph η applied to S produces the set This is the set S generated by the family (U , η). (2) The set S is the image of κ(U , η). The formula gives an explicit description.

Examples.
We give examples of constitutive sets of static systems derived from variational principles applied to families of functions. Variational principles of statics are models for all variational principles of classical physics since at the basis of a variational principle there is a Liouville structure formally identifying the principle with that of a static system. Configuration spaces will be constructed using an affine space Q. The model space is a vector space V of dimension 3 with a Euclidean metric g : V → V * . Example 5. A material point with configuration q 2 in the affine space Q is connected to a fixed point q 0 with a rigid rod of length a. A second material point with configuration q 1 is tied elastically to q 2 with a spring of spring constant k. The internal configuration space Q is the product Q × D, with D = {q 2 ∈ Q; q 2 − q 0 = a} .
The set is the tangent bundle of D and the set is chosen to represent the dual of TD. We have the identifications and The internal energy of the system generates the internal constitutive set This set is the image of the differential dU . The configuration q 2 is not controlled. The control configuration space is the space Q. The projection η : is the control relation. The set is the vertical bundle and the set is its polar. The strict symplectic reduction is the mapping The set is the intersection S ∩ V • Q. The constitutive set of the partially controlled system is obtained from S by applying the symplectic reduction relation Ph η. It is the image of S ∩ V • Q by the mappingη. The internal energy is treated as a family of functions (U , η) defined on fibres of the projection η.
The constitutive set can be obtained directly from the variational definition We show that S is a submanifold of T * Q. With the exclusion of the set the set S is the union of images of the two smooth sections and The set is the set S with the exclusion of The set (90) is the image of the smooth section of the canonical projection of Q × V * onto V * . It follows that S is a submanifold of Q × V * of dimension 3.
Example 6. A material point with configuration q 1 in the affine space Q is tied elastically to a fixed point q 0 with a spring of spring constant k 1 . A second material point with configuration q 2 is tied elastically to q 1 with a spring of spring constant k 2 and rest length a. The internal configuration space Q is the product Q × Q and the internal energy is the function The internal energy generates the internal constitutive set The configuration q 2 is not controlled. The control configuration space is the space Q. The projection η : Q → Q : (q 1 , q 2 ) → q 1 (95) is the control relation. The set is the vertical bundle and the set is its polar. The strict symplectic reduction is the mapping The constitutive set of the partially controlled system is obtained from S by applying the symplectic reduction relation Ph η. It is the image of S ∩ V • Q by the mappingη. This constitutive set is the image of the mapping defined on the critical set The constitutive set can also be obtained from the variational construction 6. Regular reductions of Lagrangian submanifolds. Letη : N → T * Q (104) be a strict symplectic reduction from a coistropic submanifold N of a symplectic manifold (T * Q, ω Q ) onto a symplectic manifold (T * Q, ω Q ) and let S be a Lagrangian submanifold of the symplectic manifold (T * Q, ω Q ). We are extracting from [1] and [5] the following facts about the reduced set S =η(S). We assume that the intersection of S with N is not empty.
(1)) If the intersection of S with N is clean, then S is an immersed Lagrangian submanifold of (T * Q, ω Q ). (2)) If S is transverse to N , then S is an immersed Lagrangian submanifold of (T * Q, ω Q ) and η|(N ∩ S) is an immersion. Recall that submanifolds S and N have clean intersection if S ∩ N ⊂ T * Q is a submanifold and Example 7. We use the notation of Example 5. Let = (q 1 , f 1 , q 2 , f 2 ) ∈ S ∩ V • Q, i.e.
We have and For every δf = (δq 1 , δf 1 , δq 2 , δf 2 ) ∈ T f T * Q, we put and We conclude that S is transverse to N at f .
Example 8. We use the notation of Example 6. Let and Since δf 2 is proportional to g(q 2 − q 1 ), the algebraic sum T f V • Q + T f S is not, for dim V > 1, equal to T f T * Q and S is not transverse to V • Q. On the other hand, is a submanifold, and Comparing (117) with (114) and (115), we establish the equality It follows that S and V • Q have clean intersection.

The Hessian of a function at a critical point.
Let Q be a differential manifold and let q be a critical point of a function The image of the differential dU : is a Lagrangian submanifold S ⊂ T * Q of the symplectic space (T * Q, ω Q ) . It intersects the image of the zero section is a Lagrangian subspace of the symplectic vector space (T f T * Q, ω f ). We use the decomposition of the space T f T * Q introduced in Section 2. The function F = 0 is used. It follows that dF = O πQ . The decomposition makes it possible to define a quadratic generating function The Hessian of U at the critical point q is the bilinear symmetric function defined as the polarization of the quadratic function h. It follows from elementary linear symplectic algebra that the function h is quadratic and its polarization is a symmetric bilinear mapping. The space S f is generated by h in the sense that It follows from this expression for S f = TdU (T q Q) that H(U, q)(δ 1 q, δ 2 q) = p v (TdU (δ 1 q)), δ 2 q .
A useful expression is derived by using the formula (25).
is a Lagrangian subspace denoted by L f . This subspace is the graph of the linear mapping symmetric in the sense that λ f (δ 1 q), δ 2 q = λ f (δ 2 q), δ 1 q .
For the Hessian we have the expression In the following two propositions we are using a critical point q of a function U : Q → R, vectors δ 1 q and δ 2 q in T q Q, and a choice of a mapping χ : R 2 → Q such that χ(0, 0) = q, tχ(·, 0)(0) = δ 1 q, and tχ(0, ·)(0) = δ 2 q. Proposition 1. The derivative of a function U : Q → R depends on δ 1 q and δ 2 q but not on the choice of the mapping χ. Proof: and U − U (q)1 is in I 1 (Q, q) = (I 0 (Q, q)) 2 . I 0 (Q, q) is the maximal ideal of functions related to q. It is sufficient to examine the expression (133) for U = F G with F and G in I 0 (Q, q). The equality proves the proposition.
proves the proposition. The last proposition offers an alternate definition of the Hessian. This definition is closer to the usual definition of the Hessian in terms of local coordinates.
If q is not a critical point of the function U , then a Hessian of U at q can be defined in relation to a function F on Q such that dF (q) = dU (q). This relative Hessian is the Hessian H(U − F, q). 8. The Hessian of a family of functions at a critical point.
Hence, dF (q) = dU (q). We examine the bilinear mapping extracted from the relative Hessian The mapping χ : representing a pair (δ 1 q, δ 2 q) ∈ V q Q × T q Q can be chosen to be vertical in the sense that For the function F we have It follows that We had to choose a function F to be able to define the relative Hessian H(U − F , q). It turns out that the choice of this function has no effect on the construction of the mapping (144). We define the Hessian of the family (141) at the critical point q as the bilinear mapping Example 9. We consider the generating family of Example 5. Let q = (q 1 , q 2 ) ∈ Cr(U , η)), δ 2 q = (δ 2 q 1 , δ 2 q 2 ) ∈ T q Q, and δ 1 q = (0, δ 1 q 2 ) ∈ V q Q. A mapping χ : R 2 → Q can be choosen of the form where χ 1 represents the pair δ 1 q 1 , δ 2 q 1 ∈ T q1 Q, and χ 2 represents the vector δ 2 q 2 ∈ T q2 Q. We have from (149) and (150) Example 10. Here, we consider the generating family of Example 6. At we have H(U , η, q)(δ 1 q, δ 2 q) = k 2 1 a 2 g(q 2 − q 1 ), δ 2 q 2 − δ 2 q 1 gq(q 2 − q 1 ), δ 1 q 2 . η be a differential fibration, let q be a point in Q and let f be an element of V • q Q. We choose a function F : Q → R such that d η(q) F =η(f ) and use the function F = F •η to define a spliting T f T * Q = H f +V f at dF (q) = f . Note that dF (Q) ⊂ V • Q. Hence, The equality is a consequence of general properties of the injection i v . The two equalities (155) and (156) result in The This convenient expression for the symplectic polar is obviously independent of the choice of the function F . Let q ∈ Cr(U , η) be a critical point of a family (U , η), let S be the Lagrangian submanifold dU (Q) and let f = dU (q) ∈ S. Let F be one of the functions on Q used in Section 8 to define the Hessian H(U , η, q) at q. The function F = F • η is used to construct an isomorphism The space is the graph of a symmetric linear mapping We have H(U − F , q)(δ 1 q, δ 2 q) = λ f (δ 1 q), δ 2 q We introduce a rather obvious definition and a less obvious definition ker H(U , η, q) = ker λ f ∩ V q Q.
We have then ker and Consequently, Definition 1. A family (U , η) is called a Morse family if the rank of H(U , η, q) is maximal at each q ∈ Cr(U , η). The family (U , η) is said to be regular if the critical set Cr(U , η) is a submanifold of Q and the rank of H(U , η, q) at each q ∈ Cr(U , η) is equal to the codimension of Cr(U , η). We will show that a regular family generates a Lagrangian submanifold of T * Q and that a Morse family is regular. η) is a regular family, then the image of κ(U , η) is an immersed Lagrangian submanifold of T * Q. Proof: Let q ∈ Cr(U , η) and f = dU (q). The rank of κ(U , η) at q is equal to dim(T q Cr(U , η)) − dim(ker(T q κ(U , η))). (168) We have ker(T q κ(U , η)) = ker(T fη ) ∩ T f dU (Cr(U , η)) It follows from (169) and from (167) that Since the family (U , η) is regular, rank H(U , η, q) = dim V q Q + dim Q − dim Cr(U , η) and, consequently, It follows that dim(im(T q κ(U , η))) = dim Cr(U , η) − dim(ker(T q κ)) dim Q.
On the other hand, T q κ(U , η) is the composition of T q dU , restricted to T q Cr(U , η), and the strict symplectic reduction T fη , which is the essential part of the symplectic reduction relation The image T y dU (T q Cr(U , η)) is an isotropic subspace of T f T * Q and, consequently, im(T q κ(U , η)) is an isotropic subspace of Tη (f ) T * Q. This implies the inequality dim(im(T q κ) dim Q, and, consequently, dim(im(T q κ)) = dim(Q).
It follows from the constant rank theorem that S = κ(U , η)(Cr(U , ρ)) is an immersed submanifold of T * Q and dim(S) = dim(Q). Since S is isotropic it is Lagrangian.

Proposition 3. A Morse family is regular.
Proof: We have to show that the critical set of a Morse family (U , η) is a submanifold of dimension dim Q. Let q be a critical point of the family, f = dU (q) and the isomorphism constructed with a function on Q as in Section 8. The image L f = Ψ(T f S) of T f S is the graph of a symmetric mapping λ f : T q Q → T * q Q. The rank of the Hessian of (U , η) at q is the rank of λ f restricted to V q Q. Let λ (f ,v) : V q Q → T * q Q be this restriction. The dual mapping λ * (f ,v) : T q Q → V * q Q is of the same rank. Since λ f is symmetric, λ * (f ,v) = ρ q • λ f , where ρ q is the restriction of the canonical projection ρ : T * Q → V * Q to T * q Q. The injections i h , i v induce injections i (h,ρ) : T q Q → T ρ(f ) V * Q and i (v,ρ) : V * q Q → T ρ(f ) V * Q and an isomorphism Ψ ρ : With this isomorphism, the mapping T q (ρ • dU ) is represented by ρ q • λ f = λ * (f ,v) . The mapping ρ • dF is the zero section of V * Q. It follows that the image of i (h,ρ) is tangent to the zero section. We choose a local trivialization of V * Q in a neighbourhood O of q. We have Cr(U , η) ∩ O = (ζ • ρ • dU) −1 (0) and T q (ζ • ρ • dU): T q Q → V * q Q coincides with λ * (f ,v) . The rank of the Hessian of the family (U , η) at q is the rank of λ * (f ,v) and consequently, the rank of T q (ζ • ρ • dU ). It is maximal, hence equal dim V q Q = dim V * q Q. It follows that T q (ζ • dU ) is surjective and, by the implicit function theorem, Cr(U , η) is a submanifold of dimension dim Q − dim V * q Q = dim Q. Proposition 4.
The family (U , η) is regular if and only if S = dU (Q) and V • Q have clean intersection.
Proof: Let f ∈ S ∩ V • Q and π Q (f ) = q. As in the preceding proposition, we shall use the canonical projection (177) and the isomorphism (178). We have and Ψ ρ (T q (ρ • dU )) is the graph of ρ q • λ f = λ * (f ,v) : T q Q → V * q . The rank of λ (f ,v) is equal to the rank of the Hessian of the family (Q, η) at q. It follows that and the dimension of these spaces is dim Q + dim im(λ f ) = dim Q + rank H(U , η, q). Since the kernel of T f ρ is contained in T f V • Q, we have dim(T f V • Q + T f S) = dim(Tρ(T f V • Q + T f S)) + dim ker(T f ρ) = dim Q + rank H(U , η, q) + dim Q It follows that We conclude that T f Cr(U , η) = T f V • Q ∩ T f S if and only if dim(Cr(U , η)) = dim(Q) − rank H(U , η, q).
i.e., if and only if H(U , η, q) is of maximal rank.