【英文版】【开普勒三定律的数学证明】Demostration of Kepler's Laws
证明开普勒猜想的原文(英文)并附译后汉语
Thus it can computed, the ratio of the volume of the internal tangent sphere of cube N to the volume of cube K is equal to √2π/6, i.e. π /√18. This ratio is just that ratio which the Kepler’s conjecture mentions.
Abstract
Heap together equivalent spheres into a cube up to most possible, then variant general volumes of equivalent spheres inside the cube depend on variant arrangements of equivalent spheres fundamentally. This π/√18 which the Kepler’s conjecture mentions is the ratio of the general volume of equivalent spheres under the maximum to the volume of the cube. We will do a closer arrangement of equivalent spheres inside a cube. Further let a general volume of equivalent spheres to getting greater and greater, up to tend upwards the super-limit, in pace with which each of equivalent spheres is getting smaller and smaller, and their amount is getting more and more. We will prove the Kepler’s conjecture by such a way in this article.
《开普勒三大定律》课件
开普勒三大定律推动了科学革命的发展，改变了人们对宇宙的认识 和理解。
开普勒三大定律的现代应用
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航天器轨道设计
开普勒三大定律在卫星轨 道设计、行星探测等方面 有重要应用。
导航定位
利用开普勒三大定律，可 以进行精确的导航定位， 如GPS系统。
科学研究
开普勒三大定律在天文学 、物理学、数学等领域的 研究中仍然具有重要意义 。
REPORTຫໍສະໝຸດ THANKS感谢观看CATALOG
DATE
ANALYSIS
SUMMAR Y
是太阳中心。
轨道的长轴在单位时间内扫过 的面积相等，即行星的轨道周
期与长轴的长度成正比。
定律的发现过程
开普勒通过对第谷·布拉赫观测数据的分析，发现行星轨道并非如哥白尼所设想的那 样是正圆，而是椭圆。
开普勒经过多次尝试和修正，最终确定了行星轨道定律的数学表达式，并成功解释 了行星运动规律。
开普勒的发现为后来的天文学和物理学发展奠定了基础，成为经典力学的重要基石 之一。
定律的意义与影响
总结词
开普勒第二定律对天文学和物理学的发展产生了深远 的影响，它不仅解释了行星轨道运动的规律，还为后 来的牛顿万有引力定律提供了启示。
详细描述
开普勒第二定律的提出，改变了人们对行星轨道运动 的认识。这一定律揭示了行星轨道运动中速度与轨道 半径之间的关系，为后来的天文学和物理学发展提供 了重要的启示。牛顿在研究万有引力定律时，也受到 了开普勒第二定律的启发，进一步揭示了天体运动的 规律。同时，开普勒第二定律也为后来的航天工程提 供了理论基础，帮助人们更好地理解和预测行星运动 。
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CATALOG
《开普勒三大定律》课件
证明方法：通过数学推导和计算，得出开普勒第三定律的公 式
添加项标题
证明意义：开普勒第三定律的发现，为牛顿的万有引力定律 提供了重要的依据，推动了天文学的发展
添加项标题
证明过程：通过观察和计算行星的轨道数据，发现行星的轨 道半长轴与周期之间存在一定的关系
添加项标题
证明结果：开普勒第三定律的公式为T^2=a^3，其中T为 行星绕太阳运动的周期，a为行星轨道半长轴
开普勒三大定律在天体物理学中的应用：可以用来计算行星的轨道参数，预测行星的位置和运动状态，以及研究太阳系和其他星系的结构和演化。
开普勒三大定律的拓展：在广义相对论中，开普勒三大定律被推广到任意引力场中，成为描述物体在引力场中运动的基本规律。
天文学：用于研 究行星运动和宇
宙结构
物理学：用于研 究天体物理和宇
开普勒第二定律还可以用来预测行星的位置和速度，这对于航天器的轨道设计和导航具有重 要意义。
开普勒第二定律在天文学研究中也具有重要意义，它可以帮助我们理解太阳系和其他星系的 形成和演化。
PART FIVE
开普勒第三定律是开普勒行星运动定律之一，描述了行星绕太阳运动的周期与它们到太阳的平均距离之间的关系。
宙学
数学：用于研究 几何学和代数
计算机科学：用 于研究人工智能
和机器学习
工程学：用于研 究航天器和卫星
的轨道控制
地理学：用于研 究地球自转和公
转规律
汇报人：PPT
开普勒第三定律描述了天体运动的周期与它们到太阳的距离之间的关系
在天体运动中，开普勒第三定律可以用来计算天体的轨道参数，如轨道半长轴、偏心率 等
开普勒第三定律在天体物理学、天文学等领域有着广泛的应用，如计算行星的轨道参数、 预测行星的位置等
开普勒第三定律的推导过程
开普勒第三定律的推导过程天文单位（英文：Astronomical Unit，简写AU）是一个长度的单位，约等于地球跟太阳的平均距离万有引力定律是用开普勒第三定律导出的，因此不能再用万有引力定律来推导开普勒第三定律，循环论证是不严谨的。
开普勒第三定律是开普勒根据第谷的观测数据来计算出来的，没有见过推导，推导过程只能是与万有引力定律的联系，不能叫推导。
把星球作的运动看成匀速圆周运动。
这时，万有引力提供向心力。
用质量、角速度、轨道半径表示出向心力，这样就可以写出一个方程.再将方程中的角速度用周期、圆周率表示。
再用绕同一中心天体运的星体列一个方程，两式相比就可证明开普勒第三定律：万有引力（1）向心力（2）（1）=（2），求出（3）又（4）将（3）代入（4）即可R为运行轨道半径，T=行星公转周期，常数这种方法只局限于匀速圆周运动的轨道模型，但现实中的星体运动的轨道都为椭圆，于是便有以下推导：利用微元，矢径R在很小的Δt时间内，扫过面积为ΔS，矢径R与椭圆该点的切线方向夹角为α，椭圆的弧长为ΔR。
在Δt→0时，扫过面积可以看作为三角形，R为半长轴面积速度为各行星绕太阳运行周期为T设椭圆半长轴为a、半短轴为b、太阳到椭圆中心的距离为c则行星绕太阳运动的周期选近日点A和远日点B来研究，由ΔS相等可得从近日点运动到远日点的过程中，根据机械能守恒定律得：得：，由几何关系得：，，所以整理得水星0.998 ，金星0.995 ，地球1，火星0.996，木星0.994，土星0.990 ，天王星1.00 ，海王星0.990。
开普勒三大定律的内容及意义
开普勒三大定律的内容及意义开普勒三大定律是什么，有什么重要的意义？想知道的小伙伴看过来，下面由小编为你精心准备了“开普勒三大定律的内容及意义”仅供参考，持续关注本站将可以持续获取更多的内容!开普勒三大定律的内容开普勒在1609年发表了关于行星运动的两条定律，一条是开普勒第一定律，也叫轨道定律，内容是所有的行星绕太阳运动的轨道都是椭圆的，太阳处在椭圆的一个焦点上。
开普勒第二定律，也叫面积定律，对于任何一个行星来说，它与太阳的连线在相等的时间扫过相等的面积。
用公式表示为：SAB=SCD=SEK到了1619年时，开普勒又发现了第三条定律，也就是开普勒第三定律，也称为周期定律，内容为所有的行星的轨道的半长轴的三次方跟公转周期的二次方的比值都相等。
开普勒不仅为哥白尼的日心说找到了数量关系，更找到了物理上的依存关系，使天文学假说更加的符合自然界本身的真实。
行星运动三大定律的发现为经典天文学奠定了基石，并导致数十年后万有引力定律的发现。
开普勒全名约翰尼斯开普勒，出生于1571年，死于1630年，开普勒是德国近代著名的天文学家，数学家，物理学家和哲学家。
开普勒以数学的和谐性探索宇宙，在天文学方面作出了巨大的贡献，开普勒是继哥白尼之后第一个站出来捍卫太阳中心说，并在天文学方面有突破性的成就的人物，被后世的科学家称为天上的立法者。
开普勒是哥白尼日心说的忠实信徒，为此开普勒做了不少天文测量，并在天文学方面作出了许多积极的贡献，1604年他观察到了银河系内的一颗超新星，历史上称它为开普勒新星，1607年，开普勒观测了一颗大慧星，就是后来的哈雷慧星，到了1609年，开普勒发表了多项有关行星运动的理论，当中包括了开普勒第一定律和开普勒第二定律，1618年，开普勒再次发表了有关行星运动的开普勒第三定律的论文。
开普勒三大定律的意义开普勒的三定律是天文学的又一次革命，它彻底摧毁了托勒密繁杂的本轮宇宙体系，完善和简化了哥白尼的日心宇宙体系。
开普勒第三定律的数学证明
开普勒第三定律的证明开普勒第三定律说轨道周期的平方 T 2 和轨道的半长轴的三次方 a 3 之比为常数。
我们从行星轨道所围成的面积来推导这个结论。
这里顺便推导椭圆的面积公式：如图，椭圆关于x 轴，y 轴均对称，故所求面积为第一象限部分面积的4倍，即S =4S 1=4∫ydx a利用椭圆的参数方程{x =a costy =b sint应用定积分的换元法dx =−a sint dt ,当x =0时 t =π2;当x =a 时，t =0 于是S =4∫bsint (−asint )dt =4ab ∫sin 2tdt =4ab ∫1−cos2t dt =π20π200π24ab (t −1sin2t )|π20=π∙a ∙b 几何公式：椭圆面积=π∙a ∙b (a 为半长轴，b 为半短轴) 定积分公式：椭圆面积=∫d =∫1| 0⃗⃗⃗⃗ || 0⃗⃗⃗⃗ |dt =1 | 0⃗⃗⃗⃗ || 0⃗⃗⃗⃗ | 0 0 两个面积相等可得=2πab | 0|| 0⃗⃗⃗⃗ |=2πa 2| 0|| 0⃗⃗⃗⃗ |∙√1−e 2 (b =a √1−e 2) (式1) 为求a ，利用|⃗ |=(1 e)∙| 0⃗⃗⃗⃗ |令 =π ，得椭圆极半径最大值|⃗|=1e|0⃗⃗⃗⃗ |而长轴2a=|0⃗⃗⃗⃗ ||⃗|=|0⃗⃗⃗⃗ |1e|0⃗⃗⃗⃗ |=2|0⃗⃗⃗⃗ |半长轴a=|0⃗⃗⃗⃗ | 1−e由（式1）平方后，可得2=4π2a4|0⃗⃗⃗⃗ |2|0⃗⃗⃗⃗ |2∙(1−e2)即2a3=4π2|0⃗⃗⃗⃗ |2|0⃗⃗⃗⃗ |2∙a∙(1−e2)=4π2|0⃗⃗⃗⃗ ||0⃗⃗⃗⃗ |2(1e)=4π2GM(a=|0⃗⃗⃗⃗ |1−e)对于特定的太阳系，等式a =4πGM右端是常数，这就是开普勒第三定律所要证明的结论。
开普勒三大定律公式及内容
开普勒三大定律公式及内容开普勒三大定律在天文学中可是超级重要的存在呀！这三大定律就像是解开宇宙奥秘的三把神奇钥匙。
咱们先来说说开普勒第一定律，也叫轨道定律。
它说的是所有行星绕太阳运动的轨道都是椭圆，太阳处在椭圆的一个焦点上。
想象一下，行星们就像一群调皮的孩子，绕着太阳这个“大家长”在椭圆轨道上欢快地奔跑。
我记得有一次在学校给学生们讲解这个定律的时候，有个小同学瞪着大眼睛问我：“老师，那为啥行星的轨道不是正圆呢？”我笑着回答他：“这就好像你跑步，不一定每次都沿着一个完美的圆形跑道跑，可能会有点偏差，行星们也是这样啦。
”这个小家伙似懂非懂地点点头，那模样可爱极了。
开普勒第二定律，又叫面积定律。
说的是行星和太阳的连线在相等的时间内扫过相等的面积。
这就好比行星在“赶路”的时候，离太阳近就跑得快，离太阳远就跑得慢，但是它们很努力地保证在相同时间里走过的“路程”是公平的。
说到这儿，我想起曾经在天文馆看到过一个演示模型，那模型清楚地展示了行星如何按照这个定律运动。
当时周围的小朋友们都看得入了神，嘴里还不停地念叨着：“太神奇啦！”最后是开普勒第三定律，也被称为周期定律。
它指出所有行星绕太阳运动的轨道半长轴的立方与公转周期的平方的比值都相等。
这有点复杂是不是？简单来说，就是不同的行星，它们的轨道大小和绕太阳一圈的时间之间有着固定的数学关系。
记得有一次我带着学生们到操场上，让他们模拟行星的运动，通过实际的体验来感受这些定律。
看着他们兴奋又认真的样子，我知道，他们对这些知识的理解更加深刻了。
在我们探索宇宙的过程中，开普勒三大定律为我们指明了方向。
它们让我们能够更好地理解行星的运动规律，预测天体的位置，甚至为我们探索更遥远的星系提供了基础。
所以呀，别小看这三个定律，它们可是天文学中的瑰宝，带领着我们不断去探索宇宙那无尽的奥秘！。
mathematical proof
Mathematical proofIn mathematics, a proof is a convincing demonstration (within the accepted standards of the field) that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproven proposition that is believed to be true is known as a conjecture.The statement that is proved is often called a theorem. Once a theorem is proved, it can be used as the basis to prove further statements. A theorem may also be referred to as a lemma, especially if it is intended for use as a stepping stone in the proof of another theorem.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice,quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.History and etymologyThe word Proof comes from the Latin probare meaning "to test". Related modern words are the English "probe", "proboscis”, "probation", and "probability", the Spanish "probar" (to smell or taste, or (lesser use) touch or test),[3] Italian "provare" (to try), and the German "probieren" (to try). The early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony.Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is probable that the idea of demonstrating a conclusion first arose in connection with geometry, which originally meant the same as "land measurement". The development of mathematical proof is primarily the product of ancient Greek mathematics, and one of its greatest achievements. Thales (624–546 BCE) proved some theorems in geometry. Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Aristotle (384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known. Mathematical proofs were revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today, starting with undefined terms and axioms (propositions regarding the undefined terms assumed to be self-evidently true from the Greek “axios” meaning“something worthy”), and used these to prove theorem s using deductive logic. His book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. In addition to the familiar theorems of geometry, such as the Pythagorean theorem, the Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.Further advances took place in medieval Islamic mathematics. While earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers (rather than geometric demonstrations) as he considered multiplication, division, etc. for ”lines.” He used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri (1000) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle. Alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate.Modern proof theory treats proofs as inductively defined data structures. There is no longer an assumption that axioms are "true" in any sense; this allows for parallel mathematicaltheories built on alternate sets of axioms (see Axiomatic set theory and Non-Euclidean geometry for examples).Nature and purposeThere are two different conceptions of mathematical proof. The first is an informal proof, a rigorous natural-language expression that is intended to convince the audience of the truth of a theorem. Because of their use of natural language, the standards of rigor for informal proofs will depend on the audience of the proof. In order to be considered a proof, however, the argument must be rigorous enough; a vague or incomplete argument is not a proof. Informal proofs are the type of proof typically encountered in published mathematics. They are sometimes called "formal proofs" because of their rigor, but logicians use the term "formal proof" to refer to a different type of proof entirely.In logic, a formal proof is not written in a natural language, but instead uses a formal language consisting of certain strings of symbols from a fixed alphabet. This allows the definition of a formal proof to be precisely specified without any ambiguity. The field of proof theory studies formal proofs and their properties. Although each informal proof can, in theory, be converted into a formal proof, this is rarely done in practice. The study of formal proofs is used to determine properties of provability in general, and to show that certain undecidable statements are not provable.A classic question in philosophy asks whether mathematical proofs are analytic or synthetic. Kant, who introduced the analytic-synthetic distinction, believed mathematical proofs are synthetic.Proofs may be viewed as aesthetic objects, admired for their mathematical beauty. The mathematician Paul Erdős was known for describing proofs he found particularly elegant as coming from "The Book", a hypothetical tome containing the most beautiful method(s) of proving each theorem. The book Proofs from THE BOOK, published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing.Methods of proofMain article: Direct proofIn direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to establish that the sum of two even integers is always even:Consider two even integers x and y. Since they are even, they can be written as x=2a and y=2b respectively for integers a andb. Then the sum x + y = 2a + 2b = 2(a + b). From this it is clearx+y has 2 as a factor and therefore is even, so the sum of any two even integers is even.This proof uses definition of even integers, as well as distribution law.Proof by mathematical inductionMain article: Mathematical inductionIn proof by mathematical induction, first a "base case" is proved, and then an "induction rule" is used to prove a (often infinite) series of other cases. Since the base case is true, the infinity of other cases must also be true, even if all of them cannot be proved directly because of their infinite number.A subset of induction is infinite descent. Infinite descent can be used to prove the irrationality of the square root of two.The principle of mathematical induction states that: Let N = { 1, 2, 3, 4, ... } be the set of natural numbers and P(n) be a mathematical statement involving the natural number n belonging to N such that∙(i)P(1) is true, i.e., P(n) is true for n = 1∙(ii)P(n + 1) is true whenever P(n) is true, i.e., P(n) is true implies that P(n + 1) is true.Then P(n) is true for all natural numbers n.Mathematicians often use the term "proof by induction" as shorthand for a proof by mathematical induction. However, the term "proof by induction" may also be used in logic to mean an argument that uses inductive reasoning.Proof by transpositionMain article: Transposition (logic)Proof by transposition or proof by contrapositive establishes the conclusion "if p then q" by proving the equivalent contrapositive statement "if not q then not p".Example:∙Proposition: If x² is even then x is even.∙Contrapositive proof:If x is odd (not even) then x = 2k + 1 for an integer k. Thus x² = (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, where (2k² + 2k) is integer. Therefore x² is odd (not ev en).To see the original proposition, suppose x² is even. If x were odd, then we just showed x² would be odd, even though it is supposed to be even; so this case is impossible. The only other possibility is that x is even.Proof by contradictionMain article: Proof by contradictionIn proof by contradiction (also known as reductio ad absurdum, Latin for "by reduction toward the absurd"), it is shown that if some statement were so, a logical contradiction occurs, hence the statement must be not so. This method is perhaps the most prevalent of mathematical proofs. A famousexample of proof by contradiction shows that is an irrational number:Suppose that were a rational number, so by definitionwhere a and b are non-zero integers with no common factor.Thus, . Squaring both sides yields 2b2 = a2. Since 2 divides the left hand side, 2 must also divide the right hand side (as they are equal and both integers). So a2 is even, which implies that a must also be even. So we can write a = 2c, where c is also an integer. Substitution into the original equation yields 2b2 = (2c)2 = 4c2. Dividing both sides by 2 yields b2 = 2c2. But then, by the same argument as before, 2 divides b2, so b must be even.However, if a and b are both even, they share a factor, namely 2.This contradicts our assumption, so we are forced to conclude that is an irrational number.Students can easily fall into erroneous proofs with this method. In searching for a direct proof, a mistake in reasoning will lead to false conclusions, which can often be detected as absurd, alerting the student to his or her error. But in constructing a proof by contradiction, a mistake in reasoning which implies absurd statements tends to be seen as the successful end of the proof.Proof by constructionMain article: Proof by constructionProof by construction, or proof by example, is the construction of a concrete example with a property to show thatsomething having that property exists. Joseph Liouville, for instance, proved the existence of transcendental numbers by constructing an explicit example.Proof by exhaustionMain article: Proof by exhaustionIn proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases.Probabilistic proofMain article: Probabilistic methodA probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory. This is not to be confused with an argument that a theorem is 'probably' true. The latter type of reasoning can be called a 'plausibility argument' and is not a proof; in the case of the Collatz conjecture it is clear how far that is from a genuine proof. Probabilistic proof, like proof by construction, is one of many ways to show existence theorems.Combinatorial proofMain article: Combinatorial proofA combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a bijection between two sets is used to show that the expressions for their two sizes are equal. Alternatively, a double counting argument provides two different expressions for the size of a single set, again showing that the two expressions are equal.Nonconstructive proofMain article: Nonconstructive proofA nonconstructive proof establishes that a certain mathematical object must exist (e.g. "Some X satisfies f(X)"), without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proved to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. A famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that a b is a rational number:Visual proofAlthough not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "proof without words". The left-hand picture below is an example of a historicvisual proof of the Pythagorean theorem in the case of the (3,4,5) triangle.∙Visual proof for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.∙Visual proof for the Pythagorean theorem by rearrangement. Elementary proofMain article: Elementary proofAn elementary proof is a proof which only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. For some time it was thought that certain theorems, like the prime number theorem, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.Two-column proofA two-column proof published in 1913A particular form of proof using two parallel columns is often used in elementary geometry classes in the United States. The proof is written as a series of lines in two columns. In each line, the left-hand column contains a proposition, while the right-hand column contains a brief explanation of how the corresponding proposition in the left-hand column is either an axiom, a hypothesis, or can be logically derived from previous propositions. The left-hand column is typicallyheaded "Statements" and the right-hand column is typically headed "Reasons".Statistical proofs in pure mathematicsMain article: Statistical proofThe expression "statistical proof" may be used technically or colloquially in areas of pure mathematics, such as involving cryptography, chaotic series, and probabilistic or analytic number theory. It is less commonly used to refer to a mathematical proof in the branch of mathematics known as mathematical statistics. See also "Statistical proof using data" section below.Computer-assisted proofsMain article: Computer-assisted proofUntil the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity. However, computers are now used both to prove theorems and to carry out calculations that are too long for any human or team of humans to check; the first proof of the four color theorem is an example of a computer-assisted proof. Some mathematicians are concerned that the possibility of an error in a computer program or a run-time error in its calculations calls the validity of such computer-assisted proofs into question. In practice, the chances of an error invalidating a computer-assisted proof can be reduced byincorporating redundancy and self-checks into calculations, and by developing multiple independent approaches and programs. Furthermore, although a computer might make a mistake when checking a proof, errors can never be completely ruled out in case of a human proof verifier as well, especially if the proof contains natural language and requires mathematical insight.Undecidable statementsA statement that is neither provable nor disprovable from a set of axioms is called undecidable (from those axioms). One example is the parallel postulate, which is neither provable nor refutable from the remaining axioms of Euclidean geometry.Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermelo-Fraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see list of statements undecidable in ZFC.Gödel's (first) incomplete ness theorem shows that many axiom systems of mathematical interest will have undecidable statements.Heuristic mathematics and experimental mathematicsMain article: Experimental mathematicsWhile early mathematicians such as Eudoxus of Cnidus did not use proofs, from Euclid to the foundational mathematics developments of the late 19th and 20th centuries, proofs werean essential part of mathematics. With the increase in computing power in the 1960s, significant work began to be done investigating mathematical objects outside of theproof-theorem framework, in experimental mathematics. Early pioneers of these methods intended the work ultimately to be embedded in a classical proof-theorem framework, e.g. the early development of fractal geometry, which was ultimately so embedded.Related conceptsColloquial use of "mathematical proof"The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing with mathematical objects, such as numbers, to demonstrate something about everyday life, or when data used in an argument are numbers. It is sometime also used to mean a "statistical proof" (below), especially when used to argue from data.Statistical proof using dataMain article: Statistical proof"Statistical proof" from data refers to the application of statistics, data analysis, or Bayesian analysis to infer propositions regarding the probability of data. While using mathematical proof to establish theorems in statistics, it is usually not a mathematical proof in that the assumptions from which probability statements are derived require empiricalevidence from outside mathematics to verify. In physics, in addition to statistical methods, "statistical proof" can refer to the specialized mathematical methods of physics applied to analyze data in a particle physics experiment or observational study in cosmology. "Statistical proof" may also refer to raw data or a convincing diagram involving data, such as scatter plots, when the data or diagram is adequately convincing without further anaylisis.Inductive logic proofs and Bayesian analysisMain articles: Inductive logic and Bayesian analysisProofs using inductive logic, while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in a similar manner to probability, and may be less than one certainty. Bayesian analysis establishes assertions as to the degree of a person's subjective belief. Inductive logic should not be confused with mathematical induction.Proofs as mental objectsMain articles: Psychologism and Language of thoughtPsychologism views mathematical proofs as psychological or mental objects. Mathematician philosophers, such as Leibniz, Frege, and Carnap, have attempted to develop a semantics for what they considered to be the language of thought, wherebystandards of mathematical proof might be applied to empirical science.Influence of mathematical proof methods outside mathematicsPhilosopher-mathematicians such as Schopenhauer have attempted to formulate philosophical arguments in an axiomatic manner, whereby mathematical proof standards could be applied to argumentation in general philosophy. Other mathematician-philosophers have tried to use standards of mathematical proof and reason, without empiricism, to arrive at statements outside of mathematics, but having the certainty of propositions deduced in a mathematical proof, such as Descarte’s cogito argument.Ending a proofMain article: Q.E.D.Sometimes, the abbreviation "Q.E.D."is written to indicate the end of a proof. This abbreviation stands for "Quod Erat Demonstrandum", which is Latin for "that which was to be demonstrated". A more common alternative is to use a square or a rectangle, such as □ or ∎, known as a "tombstone" or "halmos" after its eponym Paul Halmos. Often, "which was to be shown" is verbally stated when writing "QED", "□", or "∎" in an oral presentation on a board.。
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perpendicular to it. (Fig.2)
Then we prove Kepler’s second law before the other ones.
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demonstration to explain Newton’s work on demonstrating Kepler’s Laws.
1. Demonstration by Newton: Kepler’s Laws simply described the phenomenon of planets’ motion, which were definitely meaningful in the development of Physics and Astronomy, however lacked with some supports from theorem, which lately became the work of one of the great man in history of humanity, Newton, who mathematically find the essential of Kepler’s Law. In terms of Kepler’s second law, Newton demonstrated that it is equivalent between the second law and central motion, i.e., supposing a particle moving on a plane where the segment between it and a fixed point sweeps a fixed area, then the particle is moving under the central force, and vice versa. From the equivalence of the second law and central force, we know the acceleration vector of a planet always pointing the
Then discuss
Байду номын сангаас
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Report
Abstract:
Demonstration of Kepler’s Laws
HAN Fang-Zhou
The mathematical explanation about planet’s motion given by Newton and Kepler are two of
the most important achievement in science history. In this article we have a modern
2. Demonstrating Kepler’s Laws with modern method:
Suppose ������ to be radial vector emitting from sun pointing a planet with mass of ������.
Universal gravitational law told us that the attraction between planet and sun is ������������������
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Fig.5
or
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������0,������|������=0 = 0. ������0 = |������|������=0 = |������̇������̂ + ������������̇ ���̂���|������=0 = |������������̇ ���̂���|������=0 = |������������̇ ||���̂���|������=0 = (������������̇ )������=0