Integrability of Newton ovals, computation of air damper inlets

Following Arnol'd ( [1]), a curve in the real plane is said to be algebraic if its points (x, y) satisfy an equation P (x, y) = 0 where P is a non-zero polynomial. Following Newton in its Principia ( [2]), an oval is a closed convex algebraic curve; following Newton (and Chandrasekhar) again, an oval is said to be smooth if at every point there is the tangent line and its position varies with continuity when the point varies with continuity along the curve. An oval is said to be algebraically squarable if exists a polynomial Q(p, q, r, s) such that the area S of an arbitrary segment formed by a cartesian line ax + by + c = 0 satisfies the equation Q(S, a, b, c) = 0 ( [1], [4]). Newton, during its studies on planets trajectories, has proven in its Lemma 28 of the First Book of Principia the following theorem (see ( [1]) for the original Newton proof, which is based on a geometric and algebric, not analytic, argument):

If an oval is smooth, it is not algebraically squarable.

For example, Kepler elliptical ovals are not algebraically squarable, and so the convex Newton apple (Fig. 2; see forward for its equation). The unit square of equation x 2 y 2 -x 2 y -xy 2 + xy = 0 is algebraically squarable and a polynomial satisfying the definition is In their works, Arnol’d and Pourciau describe how the Newton result about ovals integrability can be interpreted in a weaker sense too, using the same logical argumentation used by Newton in his proof. They introduce the concept of locally algebraically integrability, without giving a clear formal definition. But from the examples and from the proofs which they offer, one can deduce the following definition:

An oval is said to be locally algebraically squarable if for each point P 0 exist a line L 0 of equation a 0 x + b 0 y + c 0 = 0 and a polynomial Q 0 (p, q, r, s) such that the areas S of the segments formed by all the lines ax + by + c = 0, in a suitable geometrical neighbourhood of the line L 0 , satisfy the equation

In practice, in (global) algebraic integrability for the whole oval there is a polynomial Q such that Q(S, a, b, c) = 0, while in local algebraic integrability for each point P 0 of the oval there is a polynomial

As Newton understood (see [1]), global and local algebraic integrability seem to depend on the degree of smoothness of the curve. A curve F (x, y) is said to be of class C m at its point P 0 = (x 0 , y 0 ) if its partial derivatives until the m-th order exist at P 0 , they are continuous, and the gradient (∂ x F (x 0 , y 0 ), ∂ y F (x 0 , y 0 )) is not the null vector (that is P 0 is regular). A curve is said to be of class C m if all its points are regular and its partial derivatives until the m-th order exist and are continuous. If also the curve is infinitely derivable with continuity, it is of class C ∞ . The curve is said to be analytic at P 0 if it is expandible in a convergent power series of integer exponents in a neighbourhood of P 0 .

Theorem An algebraic curve of class C 1 at its point P 0 , is analytic at P 0 .

Proof. Suppose ∂ y F (x 0 , y 0 ) = 0. From the theorem of implicit function, there is a function y = φ(x) such that F (x, φ(x)) = 0 for all the x in a neighbourhood of P 0 . Also, φ is derivable and its first derivative is

Note that, using permanence of sign, the neighbourhood of P 0 can be chosen so that ∂ y F (x, φ(x)) > 0 on it. Therefore, being F (x, y) partially derivable indefinitely, from (1) the successive derivatives φ ′′ , φ ′′′ , … have only integer powers of ∂ y F (x, φ(x)) at denominator, and we deduce that φ is indefinitely derivable. So locally the curve F (x, y) is the graph of an infinitely differentiable function y = φ(x). From a theorem of Newton ( [1]), it follows that F is analytic at P 0 .

A curve is analytic if it is analytic at every point. Newton has proven ( [1]) that a C ∞ oval is analytic and it is not algebraically squarable, even locally. From previous theorem it can be deduced that A C 1 oval is not algebraically squarable, even locally.

Hence, if you want an oval locally algebraically squarable, you must search among ovals with at least a singular point, that is a point P 0 = (x 0 , y 0 ) such that ∂ x F (x 0 , y 0 ) = ∂ y F (x 0 , y 0 ) = 0. Note that an oval can be smooth in the sense of Newton, that is to have continuous tangent line at every point, but it could not be of class C 1 (in the paper of Pourciau, smooth is equivalent to C 1 , but this seems not to agree with Newton assumptions). Arnol’d ( [1]) offers the example of the parametric curve x = (t 2 -1) 2 , y = t 3 -t for -1 ≤ t ≤ 1, which have continuous and never null tangent vector (4t(t 2 -1), 3t 2 -1), that is the curve is smooth. But if we consider its algebraic equation, that is y 4 -2xy 2 -x 3 + x 2 = 0, we can see that the origin (0, 0) is the unique singular point, that is the curve is not of class C 1 . Therefore, from Newton this oval is not algebraic squarable. It is locally algebraic squarable (see ( [1]), b

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