所有光滑的卵形曲线不是代数可积的

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《惠更斯与巴罗，牛顿与胡克》第87页

定理 任一代数可积的卵形曲线必有奇点：所有光滑的卵形曲线都是代数不可积的。

光滑的卵形曲线，例如：圆、椭圆，它们都是代数不可积的。

第89-90页
曲线$x=1-t^2, \quad y=t-t^3$的$t∈[-1,1]$这部分是一个代数可积的卵形线.


In[1]:=
        $\text{RegionMeasure}[\text{ParametricRegion}[k\{(1-t^2) ,(t-t^3) \},-1\leq t\leq1\&\&0\leq k\leq 1\},\{t,k\}]]$
Out[1]=
        $\dfrac8{15}$
In[2]:=
        $R=\left\{1-t^2,t-t^3,0\right\};\frac{1}{2} \int_{-1}^1 \left(R\times \partial _tR\right)[[3]] \, dt$
Out[2]=
        $\dfrac8{15}$

曲线$x=\left(t^{2}-1\right)^{2}, y=t-t^3$的$t∈[-1,1]$这部分是一个有连续变化曲率的局部代数可积卵形线.


In[1]:=
        $\text{RegionMeasure}\left[\text{ParametricRegion}\left[\left\{k
   \left\{\left(t^2-1\right)^2,t-t^3\right\},-1\leq t\leq 1\&\&0\leq k\leq
   1\right\},\{t,k\}\right]\right]$
Out[1]=
        $\dfrac{64}{105}$
In[2]:=
        $R=\left\{(t^2 - 1)^2, t - t^3, 0\right\};\frac{1}{2} \int_{-1}^1 \left(R\times \partial _tR\right)[[3]] \, dt$
Out[2]=
        $\dfrac{64}{105}$

hbghlyj

https://en.wikipedia.org/wiki/Newton%27s_theorem_about_ovals

In modern mathematical language, Newton essentially proved the following theorem:

There is no convex smooth (meaning infinitely differentiable) curve such that the area cut off by a line ax + by = c is an algebraic function of a, b, and c.

In other words, "oval" in Newton's statement should mean "convex smooth curve". The infinite differentiability at all points is necessary: For any positive integer n there are algebraic curves that are smooth at all but one point and differentiable n times at the remaining point for which the area cut off by a secant is algebraic.

Newton observed that a similar argument shows that the arclength of a (smooth convex) oval between two points is not given by an algebraic function of the points.