Newton’s second law as limit of variational problems

We show that the solution of Cauchy problem for the classical ODE my″=f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m \mathbf {y}''=\mathbf {f}$\end{document} can be obtained as the limit of minimizers of exponentially weighted convex variational integrals. This complements the known results about weighted inertia-energy approach to Lagrangian mechanics and hyperbolic equations.


Introduction and statement of the result
governing the motion of a material point of mass m subject to the force field f .Our goal is to show that the solution to (1.1) is the limit as h → +∞ of the minimizers of the following functionals defined on trajectories y : R + → R N subject to the same initial conditions, as soon as (f h ) h∈N ⊂ L ∞ (R + ; R N ) is a sequence such that f h ⇀ f in w * − L ∞ (R + ; R N ) as h → +∞.More precisely, letting and f h ∈ L ∞ (R + ; R N ) for every h ∈ N, we may define the rescaled energy functional (see also Lemma 2.3 below) +∞ otherwise in W 2,1 loc (R + ; R N ), and we will prove the following result.
Theorem 1.1.For every h ∈ N, there exists a unique solution u h to the problem ∞ ((0, T ); R N ) for every T > 0, where y is the unique solution on R + of problem (1.1).
A variational approach based on the minimization of weighted inertia-energy (WIE) functionals can be used for approximating large classes of initial value problems of the second order.An example is the nonhomogeneous wave equation Indeed, it has been shown in [7] that given g ∈ L 2 loc ((0, +∞); , there exists a sequence (g h ) h∈N converging to g in L 2 ((0, T ); L 2 (R N )) for every T > 0 such that the following properties hold.First, the WIE functional Second, by setting w h (t, x) := u h (ht, x), the sequence (w h ) h∈N converges weakly in H 1 ((0, T )× R N ) for every T > 0 to a function w which solves in the sense of distributions in A similar result holds true for other classes of hyperbolic equations as shown in [5, ?, ?].
In particular it applies to the nonlinear wave equation w tt = ∆w − p 2 |w| p−2 w, p ≥ 2, as conjectured by De Giorgi [1] and first proven in [4], see also [6].Let us mention that (the scalar version of) Theorem 1.1 is not a direct consequence of the above result from [7], since one should apply the latter to constant-in-space forcing terms g and initial data α, β, and since the approximating sequence (g h ) h∈N in [7] is not arbitrary but obtained by means of a specific construction, not allowing for instance for the choice g h ≡ g for every h.
Concerning the WIE approach for odes, let us mention its application in [3] for providing a variational approach to Lagrangian mechanics, by considering an equation of the form for given potential energy U ∈ C 1 (R N ), bounded from below, and m > 0. The main theorem of [3] proves indeed that solutions to the initial value problem for (1.3) can be approximated by rescaled minimizers, subject to the same initial conditions, of the functionals It is worth noticing that, also in this case, Theorem 1.1 is not a consequence of the result from [3] since the latter requires that the force field is conservative and independent of t.
We have already observed that in the scalar case problem (1.1) is a particular case of problem (1.2), obtained by taking constant initial data and letting the forcing term depend only on time.Let us also mention another interpretation of (1.1) from a continuum mechanics point of view.Indeed, Newton's second law (1.1)governs the motion of the center of mass of a body occupying a reference configuration Ω ⊂ R N .More in detail, let ρ be the mass density of the body and let u(t, x) be the position of the material point x at time t.If T is the Cauchy stress tensor and b is the body force field acting on Ω, then the equation of motion, see for instance [2], takes the form (1.4) Therefore, by integrating in Ω both sides of (1.4), we formally get where f Ω = f Ω (t) is the total force acting on the body, accounting for surface and body forces, m Ω = Ω ρ(x) dx is the mass of the body and is the position at time t of the center of mass of the body during the motion.Therefore Newton's second law (1.1)can be viewed as the average in space of the equation of motion (1.4).In this perspective Theorem 1.1 can be seen as a result about the equation of motion in R N in the above average sense.
Let us finally stress that the methods that are described in this paper, here only devoted to the elementary problem (1.1), can be extended to nonlinear problems like y ′′ = ∇ y G(t, y) under suitable assumptions on G, but also to hyperbolic problems such as (1.2) allowing to get further results on these topics.In this perspective, we will develop our analysis in a forthcoming paper.

Existence of minimizers
In this section we provide some preliminary results that we are going to use for proving Theorem 1.1.First of all, it is worth noticing that if u ∈ A then u ∈ W 2,2 ((0, T ); R N ) for every T > 0 hence both u(0) and u ′ (0) are well defined.Moreover, if u ∈ A, by Cauchy-Schwarz inequality and the integral in the left hand side is finite (see Lemma 2.1 below), so that J h (u) is welldefined and finite.In fact, we have the following estimates Lemma 2.1.Let u ∈ A. Then e −t/2 u ∈ L 2 ((0, +∞); R N ), e −t/2 u ′ ∈ L 2 ((0, +∞); R N ) and Proof.We have u ∈ AC([0, T ]; R N ) and u ′ ∈ AC([0, T ]; R N ) for every T > 0. Therefore for a.e.t > 0.Moreover, given T > 0 we integrate by parts and obtain where we have used Young inequality.By letting T → +∞ we get (2.1).The same computation entails By letting T → +∞ and by taking advantage of (2.1) we obtain (2.2).
The next lemma proves the first statement of Theorem 1.1.
Lemma 2.2.For every h ∈ N there exists a unique solution to the problem Proof.We first observe that J h is strictly convex and that the minimization set is convex.Therefore if a minimizer exists it is necessarily unique, so we are left to prove existence.If Let (u k ) k∈N be a minimizing sequence for problem (2.3).Since u 0 + h −1 tv 0 is admissible for problem (2.3), we have for any large enough k whence by (2.4), by Young and Cauchy-Schwarz inequalities, and by denoting with C various constants only depending on f h ∞ , h, u 0 , v 0 , m, we get By taking into account of (2.4), (2.5), (2.6) we get that the sequence (e − t 2 u k ) k∈N is equibounded in W 2,2 (R + ; R N ), so there exists v ∈ W 2,2 (R + ; R N ) such that up to extracting a subsequence there holds e so eventually we find u ∈ A, and since We conclude that u is solution to (2.3).
for every y ∈ A h and equality holds if and only if y = y h , as claimed.

Proof of Theorem 1.1
Given y h minimizing F h over A h , here we prove suitable boundedness estimates for the sequence (y h ) h∈N , which is the main step towards the proof of Theorem 1.1.
Proof.Let h ∈ N, ϕ ∈ C c (R + ; R N ) and let ξ be the unique solution to By setting ψ h (t) := h −2 ξ(ht) we see that ψ h (0) = ψ ′ h (0) = 0 and that and the integral in the right hand side is finite since ϕ ∈ C c (R + ; R N ), thus we get y h +ψ h ∈ A h .The minimality of y h entails the validity of the first order relation Since ξ(0) = 0, using integration by parts we have for every ν > 0 and every τ > 0 Therefore, We recall from Lemma 2.3 that y h (t) = u h (ht), where u h is the unique solution to (2.3).Hence, by taking into account that and by using (3.1) and (3.2), we get By the arbitrariness of ϕ ∈ C c (R + ; R N ), and since Eventually, we have for every t ∈ [0, T ] The estimates (3.4), (3.5) and (3.6) prove the result, since the sequence Proof of Theorem 1.1.For every h ∈ N, let y h be as in Lemma 2. Since the Cauchy problem (1.1) has a unique solution y on R + and since T is arbitrary, we conclude that y h ⇀ y in w * − W 2,∞ ((0, T ); R N ) as h → +∞ for every T > 0 thus proving the theorem.