Abstract
About a century ago, the French artillery commandant Charbonnier envisioned an intriguing result on the trajectory of a projectile that is moving under the forces of gravity and air resistance. In 2000, Groetsch discovered a significant gap in Charbonnier’s work and provided a valid argument for a certain special case. The goal of the present article is to establish a rigorous new approach to the full result. For this, we develop a theory of those functions which can be sandwiched, in a natural way, by a pair of quadratic polynomials. It turns out that the convexity or concavity of the derivative plays a decisive role in this context.
1. Introduction and Historical Background
Every differentiable real-valued function on an interval induces a canonical pair of polynomials of degree at most two, namely, the polynomials and that coincide with at the endpoints and , while their derivatives satisfy and Here we focus on the particular case when is sandwiched by these polynomials in the sense that either or holds on the entire interval The two possibilities are illustrated by the graphics in Figure 1, which are based on certain cubic polynomials .
We will see that, in fact, every cubic polynomial allows this kind of enveloping on any given interval , but, in general, the polynomial sandwich condition need not be satisfied even when is convex or concave on For instance, in each of the two cases for , the corresponding polynomials and coincide and hence miserably fail to sandwich
In the main results of this article, we connect the sandwich property to the convexity or concavity behavior of the derivative of Specifically, we prove that if is convex on and that if is concave on Moreover, we show that certain localized versions of our sandwich conditions actually characterize the convexity or concavity of From a broader point of view, while every calculus student knows what the conditions or on mean for a given function , here we offer an interpretation and visualization of the estimate on in terms of the parabolic sandwich condition on and its localized version.
All this has an interesting history and application. In Chapter V of his treatise on ballistics [1], published in 1907, the French artillery commandant Prosper Charbonnier studied the trajectory traced by a projectile under the forces of gravity and air resistance. Such a trajectory represents, of course, the solution of a certain system of differential equations, but an explicit solution formula is, in general, out of reach. Nevertheless, Charbonnier discovered that the trajectory is always sandwiched by the two parabolas associated with the data at the starting point and a chosen endpoint of the trajectory. These parabolas allow a natural interpretation in terms of projectile motion without air resistance in the classical sense of Galileo and serve to define a certain safety region for the complicated case of air resistance. Groetsch [2] noticed a serious gap in Charbonnier’s presentation and provided a rigorous proof but only for the special case of shooting from ground level to ground level.
Although our work is inspired by [2], our methods are rather different and remarkably elementary. The principal tool is a characterization of convex functions in terms of certain average values which is discussed in Section 2. Our sandwich results are then developed in Section 3, while Section 4 provides a simple approach to the full version of what Charbonnier proposed regarding projectile motion under air resistance.
2. Preliminaries on Convex Functions
We start by collecting a few tools from the theory of convex functions for which we refer, for instance, to Chapter III of [3] and Chapter I of [4]. A real-valued function on an interval is said to be convex (or concave upward) provided thatfor all and If the preceding inequality holds with “” instead of “” for all distinct , then is said to be strictly convex. Reversing the inequalities leads to the definitions of concave and strictly concave functions, while affine functions are those that are both convex and concave. It is well known that all such functions are automatically continuous on the interior of Moreover, a differentiable function is (strictly) convex precisely when its derivative is (strictly) increasing. In particular, if is twice differentiable on and satisfies for all , then is strictly convex. The following simple result will be useful.
Lemma 1. Let be a convex function on an interval If are points with such that the identityholds for at least one , then is actually affine on In particular, is strictly convex precisely when fails to be affine on each nondegenerate subinterval of
Proof. Assume that is not affine on By the convexity of , it then follows that there exists some point for which , where denotes the function representing the line through and On the other hand, again by convexity, the graph of on is bounded above by the line segments from to and from to Hence entails that on which means that for all , the desired contradiction. The final claim is now immediate.
The key to our main results will be the following characterization of convexity. For completeness, we include a short proof of this folklore result; see Problem 12.Q of [4] for the equivalence of and and also [5] for a long list of related results.
Proposition 2. For every continuous real-valued function on an interval , the following assertions are equivalent:(i) is convex on ;(ii)for all , there exists some for which(iii)for all with , we haveMoreover, if is convex, then equality holds in the estimate of assertion (iii) precisely when is affine on
Proof. If is convex, then, for arbitrary distinct , the substitution rule yieldswhich shows that implies . Next, if fails to hold, then there exist distinct points for which for all Arguing as in the preceding step, integration over the interval then reveals that is violated; hence implies . Finally, if is not convex, then there exist points with such that , where denotes the function representing the secant line from to Forwe conclude from the continuity of that , , , and on , which shows that condition cannot be satisfied. Thus implies , and consequently all the three assertions are indeed equivalent.
To establish the last claim, we have another look at the first step of our proof. Hence it is immediate that equality holds in if is affine on Conversely, suppose that is convex but not affine on By Lemma 1, we obtain for all Another glance at the first step of the proof then confirms that strict inequality is obtained in , as desired.
We conclude this section with the strict counterpart of the preceding result.
Corollary 3. For every continuous real-valued function on an interval , the following assertions are equivalent:(i) is strictly convex on ;(ii)for all distinct , there exists some for which(iii)for all with , we have
Proof. The implication is trivial, and follows from Proposition 2, since ensures that there is no nontrivial subinterval of on which is affine. Finally, is immediate from Lemma 1 and Proposition 2.
3. Main Sandwich Results
Throughout this section, let and be real numbers with , and let be a continuously differentiable function. We first associate with this function two polynomials that will serve as upper and lower bounds in our sandwich results.
Lemma 4. There exist unique polynomials and of degree at most two for whichnamely, the polynomials and with coefficients
Proof. The result follows from the solution of two obvious linear systems, each of them consisting of three linear equations for three unknowns. We leave the elementary details to the reader.
Hereafter, we refer to the polynomials and from Lemma 4 as the polynomials associated with
Lemma 5. Suppose that is either strictly convex or strictly concave on Then the polynomials and associated with satisfy and for all
Proof. First suppose that is strictly convex, and assume that for some Then Rolle’s theorem applied to the function on each of the intervals and ensures that there exist points and for which such that and Because , we conclude that vanishes at the three points , and On the other hand, is affine, since the degree of is at most two. Thus inherits strict convexity from We conclude that , the desired contradiction. Consequently, for all when is strictly convex. The remaining three cases may be handled mutatis mutandis.
If the derivative of is either strictly convex or strictly concave on , then a simple modification of the preceding argument shows that, for every quadratic polynomial , the equation has at most three solutions . This result may be viewed as a general version of Théorème III on page 192 in Charbonnier’s monograph [1]. As an immediate consequence, Charbonnier notes, without any further proof, that the trajectory traced by an arbitrary projectile is always sandwiched by certain canonical polynomials. We agree with Groetsch [2] that more work is needed for this conclusion. In fact, one of the main purposes of the theory of parabolic sandwiches is to bridge this gap. A complete proof of Charbonnier’s result will be provided in Corollary 13 below.
It turns out that the theory of parabolic sandwiches is dominated by the relationship between the quantities To simplify notation, we introduce The inequalities and admit simple geometric interpretations in terms of certain tangent lines, as shown next.
Lemma 6. Suppose that , and let denote the -coordinate of the point of intersection of the tangent lines for through the points and If , then precisely when , while if , then precisely when
Proof. The two tangent lines are given by and for all , and it is easily seen that the identity holds exactly when The verification of the stipulated equivalences is then straightforward.
The following two auxiliary results collect the information about the sign of which is needed in the proof of our main results.
Lemma 7. In the setting of Lemma 4, we have Moreover, the same equivalences hold when “” is replaced throughout by either “” or “”.
Proof. With the notation of Lemma 4, the estimate holds precisely when , which may be rewritten in the formand hence is equivalent to Similar arguments show that holds exactly when and that the equivalences remain valid for the case of “” instead of “”. The remaining assertion regarding “” is then immediate.
Lemma 8. The polynomials and associated with a function satisfy the following:(a)the identity holds precisely when ;(b) on implies that ;(c) on implies that ;(d)if is convex, then , and unless is affine on ;(e)if is concave, then , and unless is affine on
Proof. The equality version of Lemma 7 confirms that the identity holds exactly when By the uniqueness assertion of Lemma 4, this means precisely that .
From on and , we obtainfor all Taking the limit as then yields and hence Lemma 7 now completes the argument.
Since the polynomials associated with in the sense of Lemma 4 are and , the estimate on entails, by part , that and hence the assertion. Of course, one could also proceed as in the proof of .
This follows from an application of Proposition 2 to and in assertion .
This final claim is immediate from part applied to
We mention in passing that, for every differentiable function on , the convexity or concavity of ensures that is automatically continuous on the entire closed interval , not just on its interior. Indeed, it is a well-known consequence of the convexity or concavity condition that the one-sided limits of exist at the endpoints and , and it follows from the classical Darboux theorem that these one-sided limits coincide with and , respectively; see Lemma 4.48 of [6] and Theorem 5.12 of [7].
Theorem 9. A function is sandwiched by its associated polynomials and as follows:(a)if is strictly concave, then on ;(b)if is concave, then on ;(c)if is strictly convex, then on ;(d)if is convex, then on
Proof. By part of Lemma 8, the strict concavity of implies that Hence we conclude from Lemma 7 that and The first of these inequalities ensures thatfor all sufficiently close to and hence for all such , because On the other hand, we know from Lemma 5 that for all Consequently, by the intermediate value theorem, for arbitrary Similar reasoning shows that the inequality yields for all .
To reduce this claim to the preceding one, we consider an arbitrary and define for all Since for all , it follows that is strictly concave on as the sum of a concave and a strictly concave function. Consequently, if and denote the polynomials associated with , then assertion ensures that for all A glance at the formulas for these polynomials in Lemma 4 reveals that and as for each Thus for all , as desired.
((c) and (d)) These assertions follow from parts and , respectively, applied to , since, as already noted in the proof of part of Lemma 8, the polynomials associated with are and
Corollary 10. If and are the polynomials associated with a cubic polynomial , then on in the case , while on in the case
Proof. The result is immediate from Theorem 9, since the quadratic polynomial is strictly concave in the case and strictly convex otherwise.
It is fairly easy to construct examples of functions that are sandwiched by their associated polynomials in one way or the other but fail to be convex or concave. In fact, such examples may be found among small perturbations of strictly convex or strictly concave functions. To obtain a certain converse of Theorem 9, we need localized versions of our sandwich properties. For arbitrary with , let and denote the polynomials associated with the function when restricted to
Theorem 11. For every continuously differentiable function on , the following equivalences hold:
is concave on precisely when on for all with ;
is strictly concave on precisely when on for all with ;
is convex on precisely when on for all with ;
is strictly convex on precisely when on for all with
Proof. One implication of assertion is, of course, immediate from part of Theorem 9, while the other one follows from part of Lemma 8 together with Proposition 2 applied to This also settles assertion , since by now it should be obvious that and are equivalent. Similarly, assertions and follow from Corollary 3, Lemma 8, and Theorem 9.
4. Application to Projectile Motion
We first describe a general setting for the motion of a projectile in the -plane. Throughout we consider a projectile that is launched at time from the origin with muzzle speed and angle of inclination relative to the positive -axis. The position vector of the projectile at time is denoted by
We suppose that the motion of the projectile is governed by two forces. As usual, one of these forces is gravity in the direction of the negative -axis, which results in acceleration of a given magnitude The other force is air resistance whose direction is opposite to the velocity vector of the projectile and whose magnitude is proportional to the mass of the projectile but otherwise quite arbitrary. Specifically, the retarding force is supposed to be represented by a vector of the formwhere is a given nonnegative continuous function of five variables, defined on a suitable domain in In addition to the classical case of no air resistance represented by , this model covers the important case where the drag is proportional to some power of the speedof the projectile; see Chapter 3 of [8]. It also allows, for instance, for drag depending on the altitude of the projectile.
By Newton’s law, the motion of the projectile is then described by the initial value problemIn the following, we assume that the drag function is admissible in the sense that, for every choice of and , the initial value problem has a unique solution that is defined for all In practice, admissibility is guaranteed by existence and uniqueness results of Picard–Lindelöf type under mild Lipschitz conditions on the function with respect to the last four variables. In particular, is admissible if the partial derivatives of with respect to each of the last four variables exist and are continuous; see, for instance, [9].
At the present level of generality, explicit formulas for the solutions and of are clearly out of reach. In fact, even in the special case of drag quadratic in speed, no such formulas are known; see, for example, [8, 10]. Nevertheless, we will be able to obtain remarkable geometric insight into the shape of the trajectory traced by the projectile.
The clue is to focus directly on and the well-defined continuous function given byfor all Even though, in general, no explicit formula for is available, standard separation of variables shows that the solution of the differential equation is given byfor all , where is a real constant. Because , we conclude from thatand therefore for all It follows that the function is strictly increasing on and that its range is an interval of the form where For instance, it is well known that in the classical case of no drag, while in the case of drag that is either linear or quadratic in speed; see again [8, 10]. Moreover, the inverse of the function allows us to express the flight curve of our projectile in terms of a function on which will be investigated in the following result.
Theorem 12. Let denote an admissible drag function, let for denote the solution of the corresponding initial value problem , and consider the composition on the range of Then satisfiesfor all , and its third derivative exists and is continuous on Moreover, is strictly concave, while is always concave and even strictly concave provided that
Proof. We first observe that inherits differentiability from , since we know that for all From the definition of , it is then immediate that is differentiable and satisfies the equation for all Hence the chain rule implies that and thereforefor all Since confirms that both and are differentiable, so is Moreover, taking the derivative of the preceding identity and using , we obtain and consequentlyfor all In particular, this shows that is strictly concave on Repeating the procedure, we arrive atand thereforefor all , again on account of . We conclude that is continuous and that is always concave. Of course, is actually affine in the classical case of no air resistance , while the condition ensures that and hence that is strictly concave, as claimed.
As a simple application of the preceding results, we now obtain an extended version of Charbonnier’s Corollary 3; see page 193 of [1].
Corollary 13. In the setting of Theorem 12, consider an arbitrary time , and define the polynomialsThen the function induced by the solution curve of the initial value problem is sandwiched by and in the sense thaton the interval . Moreover, strict inequalities hold on provided that
Proof. The result is immediate from Theorems 9 and 12 together with Lemma 4 for the choices and
We conclude with a typical numerical example. Let m/sec2, m/sec, and radian, and consider the case of air resistance which is proportional to the square of the speed. This classical case is addressed by the initial value problemwhere we choose the quadratic air resistance coefficient to be ; see [8].
Although no explicit solution formula is known, one may use the NDSolve command of Mathematica to obtain a numerical solution based on some Runge–Kutta type method. The graphics in Figure 2 show the corresponding trajectory of the projectile over the time period of seconds together with two pairs of (dashed) sandwiching parabolas for the choices sec and sec.
We note that the case of projectile motion is rather special, since here both the function and its derivative happen to be concave. However, Theorem 11 holds in a more general setting and leads, as indicated earlier, to a geometric visualization of the conditions or on a given interval. Indeed, if the third derivative of a function exists on an interval , then the condition on is equivalent to the sandwich estimate on for arbitrary points with . We illustrate this characterization by the example of the function which satisfies for all . The graphics in Figure 3 display this function on the interval together with two typical parabolic sandwiches.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.