Newton's Second Law in Field Theory

In this article we present a natural generalization of Newton's Second Law valid in field theory, i.e., when the parameterized curves are replaced by parameterized submanifolds of higher dimension. For it we introduce what we have called the geodesic $k$-vector field, analogous to the ordinary geodesic field and which describes the inertial motions (i.e., evolution in the absence of forces). From this generalized Newton's law, the corresponding Hamilton's canonical equations of field theory (Hamilton-De Donder-Weyl equations) are obtained by a simple procedure. It is shown that solutions of generalized Newton's equation also hold the canonical equations. However, unlike the ordinary case, Newton equations determined by different forces can define equal Hamilton's equations.


Introduction
At the core of Classical Mechanics is Newton's second law: force equals mass times acceleration. Later elaborations led to the equivalent and successful Lagrangian and Hamiltonian formulations of mechanics. The passage from ordinary mechanics to field theory is done precisely in Lagrangian and Hamiltonian terms. Therefore, a natural question arises: is there a version of Newton's second law in field theory? We give an affirmative answer in this article.
Let us recall the basic points of the ordinary version. Newton's second law says, essentially, that the motions are governed by second order differential equations (a specific type of vector fields defined on the tangent bundle of the configuration manifold Q). When Q is endowed with a pseudoriemannian metric, we can set which are the inertial motions (integral curves of the geodesic field). In addition, the metric gives us an isomorphism of the tangent bundle with the cotangent bundle, where the Liouville form is defined. With the above ingredients (a tangent field describing inertial motion and a symplectic structure on T Q) we can associate a second order differential equation to each given force (see [11,3]).
When dealing with field theory, the curves describing the motions are replaced by submanifolds of dimension k greater than one. Hence, in order to get the generalized Newton's law, we have to replace the tangent bundle T Q with an analogous space for k > 1: the bundle of (k, 1)-velocities, Q 1 k , a particular case of Weil bundle (see [22,10,16]). It is also necessary to consider second order partial differential equations and, in addition, to obtain a generalized geodesic field to be able to define the inertial motion: we will see that this is possible by using the metric. Also by means of the metric, we obtain an isomorphism of Q 1 k to its dual (see 1 below), where the generalized Liouville form is defined (polysimplectic structure [9]). Once this is done, the rest goes parallel to the ordinary case.
This generalized Newton's second law is an equation with values in certain linear endomorphisms space. When we take the trace, we recover the Euler-Lagrange and the canonical Hamiltonian formalisms (Hamilton-De Donder-Weyl equations). As a consequence, all the solutions of the Newton's equation also hold the associated Euler-Lagrange and canonical equations. However, in general, the converse is not true. Hence, there is an essential difference with the classical case: now (k > 1), Newton's law is no longer equivalent to the Euler-Lagrange equations or the canonical equations. In that sense, we can say that Newton's equation distinguishes between systems that, from the Lagrangian and Hamiltonian points of view, look identical.
To develop this program, we will follow [11] (see also [3]) where has been given a presentation of Mechanics allowing several conceptual advances (v.g., a clear understanding of what absolute time [14] and relativistic forces [12,3] are).
The framework used in this paper is close to that of Günther [9], who developed the so-called polysymplectic formalism, which has been extended in several directions since then (see for instance [5,18,19,6] and references therein; it should be clarified that there are other different approaches to field theory in their Lagrangian and Hamiltonian forms; see, v.g., [20,8,21])).
The structure of this paper is the following: in Section 3 they are introduced the bundles of (k, 1) and (k, 2)-velocities; Section 4 presents the definition of second order PDE and, a particular case, the geodesic k-field, is constructed; in Section 5 the dual spaces are considered: bundles of covelocities joint with the generalized Liouville form; with all the previous work done, in Section 6 the Newton's second law for fields is obtained; by taking the trace we get the Euler-Lagrange and Hamilton canonical equations in Section 7 and, also, we obtain their variational properties.
Notation and conventions. Along the paper Q will be an n-dimensional C ∞ manifold. Einstein convection on repeated indexes will be assumed. For an smooth map ϕ between manifolds, ϕ * will denote its tangent map (at a given point), and ϕ * will be the associated pull-back of differential forms. The inner contraction of a vector field D with a differential form ω is denoted by ι D ω.

Newton's second law
In order to present our goal and also to fix notation, it will be useful to recall the model of the Classical Mechanics. We will follow the presentation in [11,3].
Let Q be an n-dimensional smooth manifold, and π : T Q → Q be its tangent bundle. There is a canonical derivatioṅ whereḟ is df considered as a function on T Q; that is to say, When {q 1 , . . . , q n } is a coordinate system on U ⊂ Q, the set {q 1 , . . . , q n ,q 1 , . . . ,q n } defines a local chart on π −1 (U) ⊆ T Q. In these coordinates, . A vector field D on T Q is a second order differential equation when its restriction (as derivation) to the subring Locally, a second order differential equation is for suitable functions f i on T Q. A curve q i = q i (γ(t)),q i =q i (γ(t)) in T Q is integral for such a vector field D if and only if dq i (γ(t))/dt =q i (γ(t)) and dq i (γ(t))/dt = f i (q j (γ(t)), dq j (γ(t))/dt). That is to say, if it holds the following system of second order ordinary differential equations Let T * Q be the cotangent bundle of Q and π : T * Q → Q the canonical projection. Recall that the Liouville form θ on T * Q is defined by θ σq = π * (σ q ) for each σ q ∈ T * q Q. The 2-form dθ is the natural symplectic form associated to T * Q. In local coordinates {q 1 , . . . , q n } in Q, and the corresponding {q 1 , . . . , q n , p 1 , . . . , p n } for T * Q (defined by p i (σ q ) := σ q (∂/∂q i )), we have Let g be a pseudoriemannian metric in Q. Then we have an isomorphism of vector fiber bundles T Q ≃ T * Q, v q → ι vq g (ι vq g is the inner contraction of v q with g). Using the above isomorphism we can transport all the structures of T * Q to T Q. In particular, we will work with the Liouville form θ and the symplectic form dθ transported to T Q with the same notation.
If the expression in local coordinates of the metric is g = g ji dq j dq i , then the local equations for the isomorphism T Q ≈ T * Q are and the Liouville form on T Q is given by Definition 2.2. (Kinetic Energy). The function T on T Q defined by T (v q ) := 1 2 g(v q , v q ), v q ∈ T Q, is the kinetic energy associated to the metric g. It holds, equivalently, T = 1 2 g(ḋ,ḋ). Given a tangent vector v q ∈ T q Q, there is a geodesic curve on Q such that its initial velocity at q is exactly v q . As a consequence, trough each point v q ∈ T Q passes the prolongation of a unique geodesic curve. In this way, a vector field D G on T Q is defined. In coordinates, In fact, D G can be characterized in the following terms: Lemma 2.1. The geodesic field D G of the metric g is the second order differential equation determined by The geodesic field D G is chosen as the origin in the affine bundle of second order differential equations. With this choice we establish a one-to-one correspondence between second order differential equations D and vertical tangent fields V.
On the other hand, a differential 1-form on T Q is said to be horizontal if vanishes when is applied to a vertical vector (i.e., a vector in the kernel of π * , where π : T Q → Q is the canonical projection). In particular, Liouville form θ is horizontal.
Then, it can be proved the following main result Theorem 2.1 ( [11]). The metric g establishes a one-to-one correspondence between second order differential equations and horizontal 1-forms in T Q. The second order differential equation D and the horizontal 1-form F that correspond to each other are related by The correspondence (2.3) will be called the Newton's Second Law : under the influence of a force F the trajectories of the mechanical system satisfy the second order differential equation D associated with F . Indeed, F is a force, V = D − D G will be the acceleration (relative to the inertial motion D G ) and ι V dθ = F relates both of them.

Bundle of contact elements. k-Velocities
The points of the manifold Q correspond to morphisms of algebras C ∞ (Q) → R. If we substitute R by the dual numbers R[ǫ]/ǫ 2 we get the tangent vectors as a kind of generalized "points" of Q. That is the idea behind the theory developed by Weil in his paper on 'points proches' or near elements [22] (see also [16,10]). In that theory they are considered morphisms of the ring of smooth functions of a given manifold to a Weil algebra, i.e., a finite dimensional local and rational R-algebra. The main examples of Weil algebras are the quotients are the polynomials in k undetermined ǫ i , and (ǫ 1 , . . . , ǫ k ) is the maximal ideal generated by them (so that (3.1) is the ring of polynomials truncated at the degree ℓ).
The tangent map d t γ (or, equivalently, its dual (d t γ) * , the cotangent map) associated with an smooth map γ : U → Q, where U is a neighborhood of t ∈ R k and γ(t) = q.
The set of (k, 1)-contact elements of Q is denoted by Q 1 k . As a particular case, Q 1 1 = T Q. The equivalence of the above items 1, 2 and 3 is obvious because T t U = R k . Item 4 follows from 3 by taking γ * : C ∞ (Q) → C ∞ (U) followed be the quotient map ..,n are local coordinates around q ∈ Q, we have for suitable scalarsq j α =q j α (X q ). Equivalently, X q considered as a morphism C ∞ (Q) → R 1 k is given by q j → q j (q) +q j α (X q )ǫ α . The set {q j ,q j α } defines a local chart on Q 1 k ; so Q 1 k is endowed with a smooth manifold structure. With this structure, the natural map π : Q 1 k → Q is a C ∞ regular projection.
For each α = 1, . . . , k, we define the derivative: for all f ∈ C ∞ (Q) and X q = (X 1 , . . . , X k ) ∈ Q 1 k . In local coordinates, 3.1. First order prolongation of a parameterized submanifold. Let U be an open subset of R k with coordinates t 1 , . . . , t k (associated with the canonical basis {e α }). Each smooth map γ : U → Q can be prolonged to the manifold of (k, 1) contact elements in the natural way: 3.2. First order differential equation defined by a field of k-velocities. Let X : Q → Q 1 k a section of the natural projection Q 1 k → Q. This is equivalent to a collection of k vector fields (X 1 , . . . , X k ) on Q. In local coordinates . From X is easy to define a system of first order partial differential equations: By using (3.4), in local coordinates, γ is a solution of X if , . . . , q k (γ(t))), α = 1, . . . , k.
When k = 1, Definition 3.2 matches the well-known differential equation determined by a tangent vector field.
3.3. Second order contact elements. Item 4 in Definition 3.1 is the most easily generalizable: it is sufficient to substitute R 1 k by other suitable Weil algebra. For example, we can define second order contact elements: In local coordinates such a map is described by The functions q i ,q i α ,q i αβ so defined, serve as a local chart. In this way, Q 2 k is endowed with a smooth manifold structure. There is a natural projection Q 2 k → Q 1 k , by composing with R 2 k → R 1 k and is given in coordinates by (q i ,q i α ,q i αβ ) → (q i ,q i α ). In fact, given an arbitrary Weil algebra A is possible to define the manifold of A-near points Q A [22] (also known as Weil bundle and denoted by T A Q [10], or bundle of A-contact elements in [16] in a terminology closer to that employed by S. Lie). For that, it is sufficient to consider morphisms C ∞ (Q) → A, etc.
It is possible, anyway, to relate this algebraic definition with infinitesimal calculus as follows: consider an smooth map γ : U → Q with γ(t) = q (as in Definition 3.1). By transposition we get the associated map between rings of functions γ * : C ∞ (Q) → C ∞ (U). Next, we compose with the quotient by m 3 t (or, that is the same, we take the second order Taylor expansion): is, by definition, the (k, 2)-contact element with: From a γ as above, we can define its second order prolongation We can consider a different but related concept: iterated bundles of velocities: (Q 1 k ) 1 k . The relation between Q 2 k and (Q 1 k ) 1 k generalizes that existent between the second order tangent bundle T 2 Q and the iterated tangent bundle T (T Q) (case k = 1).
The proof, valid for a more general situation, can be found in [16] according the following steps: it is a local question and, when Q = R n , it is easy to see that Q A = Q ⊗ A for any Weil algebra A; as a consequence, if B is another Weil algebra, we have (Q A ) B = Q A⊗B for Q = R n and, so, also for an arbitrary manifold Q (this is a result in [22]); now, for k as follows: given S q ∈ Q 2 k we consider the composition which defines a point of (Q 1 k ) 1 k . Remark 3.1. The theory of Weil bundles is strongly related to that of jet bundles: the points of a Weil bundle Q r k are, by definition, homomorphisms C ∞ (Q) → R ℓ k . Those that are epimorphisms constitute an open subsetQ ℓ k ⊂ Q ℓ k . Taking the kernels of the homomorphisms iň Q ℓ k , we obtain the bundle of jets of order r of k-dimensional submanifolds of Q, J r k (Q), as a set of ideals of C ∞ (Q) (jets of sections of a fiber bundle Q → M are recovered as certain open subset of J ℓ k (Q)); this point of view was developed in [16] and entails a number of advantages; as an example of application of this approach, see [17]). The theory can be generalized to Weil bundles for any Weil algebra A as demonstrated in [1], where it was showed thatQ A → J A (Q) is a principal fiber bundle; even the notion of contact system was extended to these spaces [2]. Finally, a general version of jets understood as a part of the spectrum of the ring C ∞ (Q) was presented in [4]. 4. Second order partial differential equations. The geodesic k-field In this section we will generalize the concept of second order differential equation (Definition 2.1) to the k-parametric case.
is called a second order partial differential equation. In local coordinates, we have for suitable functions A i αβ ∈ C ∞ (Q 1 k ). Moreover, if we identify Q 2 k as submanifold of the iterated (Q 1 k ) 1 k (Proposition 3.1), then D is a k-tuple of vectors fields (D 1 , . . . , D k ) on Q 1 k ; in local coordinates, In other words, D = (D 1 , . . . , D k ) can be characterized by the following property, When D is considered as a field of k-velocities on Q 1 k , defines a system of first order partial differential equations on Q 1 k (see Definition 3.2). In local coordinates (as in Equation (3.5)) we get the system: (with t = (t 1 , . . . , t k )) which turns to be the following system of second order partial differential equations: (where, in order to lighten the notation, we have put q i (t) instead of q i (γ(t))).

4.1.
Geodesic k-field. It will be essential in the sequel the concept of differential equation describing 'inertial motion' or 'evolution in the absence of forces' (Newton's first law): Theorem 4.1 (Geodesic k-field). Each given metric g on Q determines canonically a second order partial differential equation D G which will be called geodesic k-field.
Proof. Let us denote by exp the exponential map associated with g. Given a X q = (X 1 , . . . , X k ) ∈ Q 1 k , we consider the smooth map γ : R k → Q, (t 1 , . . . , t k ) → exp(t 1 X 1 + · · · + t k X k ), obtained by composing X q : R k → T q Q with the exponential map exp : T q Q → Q (just defined in a neighborhood of 0 ∈ T q Q). Then we can define D G : where γ 2 is the second prolongation (3.6) of γ.
A straightforward computation gives D G = (D 1 , . . . , D k ) with β and the corresponding system of second order partial differential equations:

5.
The dual: k-symplectic canonical structure. The Liouville form The bundle Q 1 k → Q possesses a dual (Q 1 k ) * , defined as the set of linear maps T q Q → R k , q ∈ Q. These elements, that will be called (k, 1)-covelocities, correspond to tangent maps of smooth maps from (a neighborhood of q in) Q to R k . Equivalently, the elements of (Q 1 k ) * are tuples of k differential 1-forms σ q = (σ 1 , . . . , σ k ), σ α ∈ T * q Q or also σ = σ α ⊗ e α where {e α } is the canonical basis for R k . We can endow (Q 1 k ) * with a differentiable structure in the following way: if {q i } is a local chart on Q, each covelocity σ q ∈ (Q 1 k ) * can be expressed as for appropriate scalars p α i ∈ R. The sets of functions {q i , p α i } so defined, provide us with local charts on (Q 1 k ) * . In local coordinates the duality or coupling of elements X q =q i α (∂/∂q i ) q ⊗ e α ∈ Q 1 k and σ q = p α i d q q i ⊗ e α ∈ (Q 1 k ) * is given by On the other hand, it is also well defined the coupling with values in the endomorphisms of R k , that we denote as an interior product: In local coordinates, i Xq σ q = p α iq i β e β ⊗ e α If we denote by tr the trace map End(R k ) → R, then σ q , X q = tr i Xq σ q .

Liouville form.
As in the case of the cotangent bundle, (Q 1 k ) * is endowed of a tautological structure: at each σ q ∈ (Q 1 k ) * we can define a k-tuple of differential 1-forms θ σq = (θ 1 , . . . , θ k ) (a (k, 1) covelocity on (Q 1 k ) * ) by the rule where π : (Q 1 k ) * → Q is the canonical projection. We call θ the (generalized) Liouville form and, in local coordinates, Remark 5.1. When we work with differential forms, vector fields or, more in general, tensor fields with values in R k , (R k ) * or End R k , we can perform 'differential operations' (exterior differential, Lie derivatives, etc.) componentwise: as an example, given a basis {e α } of R k , if ν = ν α ⊗ e α , where ν α are ordinary differential forms, then it is well defined its exterior differential dν := (dν α ) ⊗ e α .
The exterior differential of the Liouville form θ defines the so called polysymplectic structure of (Q 1 k ) * : or, in local coordinates, dθ = dp α i ∧ dq i ⊗ e α .

5.2.
Isomorphism induced by a metric. When Q is endowed with a pseudoriemannian metric g (non singular symmetric 2-covariant tensor field of arbitrary signature), we can translate the structures of the velocity bundle to the covelocities one and conversely. Specifically, given X q ∈ Q 1 k we can define σ q ∈ (Q 1 k ) * to be the transposed of the composition: ≃ T * q Q followed with the identification (R k ) * ≃ R k , e α → e α (via the canonical basis or the canonical euclidean metric). Hence, we obtain an isomorphism of bundles on Q, Remark 5.2. The raising and lowering of indices α is made by means of the canonical euclidean metric on R k ; as an example,q i α =q iβ δ αβ , where δ αβ is the Kronecker delta.
From now on, we suppose g is given and we work indistinctly on Q 1 k or (Q 1 k ) * under the isomorphism (5.4). For instance, we have a Liouville form on Q 1 k and second order partial differential equations on (Q 1 k ) * . Remark 5.3. The isomorphism induced by g, plays the role of the Legendre transformation in [9]. In fact, in a sense, it is a particular case.

The generalized Newton's second law
With all necessary tools previously introduced, in this section we will get a k-dimensional version of the Newton's second law.
Let g a (non singular) metric on Q of arbitrary signature. By means of g we translate the (generalized) Liouville form θ to Q 1 k and define T : the differential 1-form with values in the matrixes End R k determined by In a local chart, a straightforward by lengthy computation shows the following coordinate expression: T = T β α ⊗ e β ⊗ e α with (6.2) Lemma 6.1. Let D, D ′ : Q 1 k → Q 2 k be two second order partial differential equations. Then, their difference V : k is vertical with respect to the tangent map of the projection Q 1 k → Q. Conversely, the sum of a second order differential equation D and a vertical V gives a new second order differential equation.
Proof. It follows from (4.1) that A differential 1-form on Q 1 k is said to be horizontal if vanishes when is applied to a vertical vector (as before, we mean vertical with respect to the projection Q 1 k → Q). In local coordinates a form is horizontal if locally is a linear combination of dq i , i = 1, . . . , n, with coefficients in C ∞ (Q 1 k ). Lemma 6.2. For each vertical section V : Proof. It is enough to take local coordinates: by (5.5), V α p β i = g ij V iβ α . Hence, if V is vertical, then we get ι Vα dθ β = g ij V iβ α dq j which is a horizontal form. Conversely, if ι Vα dθ β are horizontal, then V α q i = 0, so that V is vertical.
From the above lemmas we get the Theorem 6.1 (Newton's second law for fields). With the same notation. Each metric g on Q determines a univocal correspondence between horizontal 1-forms F valued in End R k and second order differential equations D : Q 1 k → Q 2 k given by the rule Proof. I follows from Lemmas 6.1 and 6.2 by taking into account that D − D G is vertical and By combining Equations (6.4), (4.4) we get the coordinate expression of this generalized Newton's law (once the 'form of work' or force F is given): where t = (t 1 , . . . , t k ), q i = q i (t), g ij is the (i, j) entry of the inverse matrix of (g ij ), etc., as usual.

Canonical equations and the Hamilton-Noether principle
In order to connect the Newton's second law with the Hamiltonian or Lagrangian theory we just have to take trace: tr : End R k → R, a β α → a α α . In particular, from (6.2) we see that and taking the trace on (6.1) we get: The function k-kinetic energy T can be globally defined on Q 1 k as follows: For the same reason, if we take trace on the relation (6.3) we arrive to In the 'conservative case', say F α α = −dU for certain function U ∈ C ∞ (Q), we get i Dα dθ α + dH = 0, where H := T + U. So, for such an H, we obtain the classical Hamilton's canonical equations in field theory (or Hamilton-De Donder-Weyl equations; see for instance [20], Ch. IV, formulae (2.26)+(2.35)): By construction, a solution of D in (6.3) is always a solution of (7.3). However, the converse is not true. In order to illustrate this fact, we give the following elementary Example. Let us consider Q = R n endowed with the euclidean metric g ij = δ ij , so that Christtoffel symbols Γ ℓ ij vanish. Consider also the null force F = 0. Assume that D holds (6.3), in such a way that D is the geodesic k-field for the flat metric: D α =q i α ∂/∂ q i . Therefore, D defines the following differential equations system (Newton's second law) ∂t α ∂t β = 0 whose solutions are If, on the other hand, we take the trace, we get: tr F = 0, so that, H = i,α (q i α /2) 2 and then, a little calculation gives (for (7.3)) the Laplace equation: q i (t) = harmonic functions of t 1 , . . . , t k Obviously, the set (7.5) is only a part of (7.7).
Equations (7.3) show, by mere inspection, that taking the trace on the Newton's equation, we get the canonical equations. However, in the sequel we will prove this fact intrinsically by obtaining the needed variational properties, including Hamilton's principle and Noether's theorem.
Infinitesimal transformations. Let us consider a tangent vector field v = v i ∂/∂q i on Q and denote by {τ t } t∈R its local uniparametric group (or flow). Then it define by means of the tangent map a new uniparametric group on Q 1 k , say {τ t } t . The vector field that generates τ is, in local coordinates, (7.8) δ More intrinsically, δ is determined by the following feature: if f ∈ C ∞ (Q), then δ v f = vf and (7.9) δ v∂α f =∂ α δ v f.
Now, the consequences of the above theorem are obtained as in the classical case: Hamilton's principle. Let γ : U → Q be a solution of the second order equation D: that is to say, (γ * ) t (∂/∂t α ) t = D α,γ(t) . In that case, for an arbitrary infinitesimal variation δ = δ v we have where dt := dt 1 ∧· · ·∧dt k . By Stokes' theorem, the last integral equals to ∂ U (γ 1 ) * θ α (v) i ∂/∂t α dt.
Hence, when the variation v vanishes along the boundary ∂ U we get the Hamilton principle U (γ 1 ) * δ(L) dt = 0.
Noether conservation laws. Let us suppose that v is a symmetry of the Lagrangian function L; that is to say, δL = 0. Let us take an arbitrary solution γ : U → Q of D. Then, substitution in (7.11) and pull-back by γ 1 implies ∂ ∂t α ((γ 1 ) * θ α (v)) = 0, which defines the conservation law associated to the symmetry v.