Shulou(Shulou.com)11/24 Report--

There is no doubt that William Ron Hamilton is the greatest scientist in Irish history. Hamilton was born in Dublin, Ireland, on August 3, 1805. His uncle was an accomplished linguist-he could blurt out Greek, Latin, Hebrew, Sanskrit, Semitic, Pali, and various dialects. Under the influence of his uncle, Hamilton was already very good at English at the age of 3; a good geographer at the age of 4; able to read and translate Latin, Greek and Hebrew at the age of 5; and mastered Italian and French at the age of 8; he can also improvise in Latin. Finally, he began to learn Arabic and Sanskrit when he was less than 10 years old, laying a solid foundation for his extraordinary academic achievements in oriental languages. At the age of 13, Hamilton became one of the most shocking linguistic monsters in history.

When Hamilton was 12 years old, he met Colburn, an American math kid, at Westminster School in London. Colburn gave Hamilton some scientific enlightenment. At the age of 17, Hamilton mastered mathematics through integral calculus and acquired a sufficient knowledge of mathematical astronomy to enable him to calculate solar and lunar eclipses. He studied the works of Newton and Lagrange. All of this is his amateur pastime (but he has made "some strange discoveries").

The discovery Hamilton mentioned may be the beginning of his first great study, the ray system in optics. Before that, he had found a mistake in Laplace's parallelogram law of force.

Hamilton never went to school before he went to college, and all his elementary training came from his uncle and self-study. While preparing for the entrance examination for Trinity College, University of Dublin, he was forced to concentrate on classical literature, which did not take up all his time, because he wrote to his cousin Arthur on May 31, 1823.

In optics, I made a very strange discovery-at least that's what I thought.

This refers to the "eigenfunction", a discovery that indicates that Hamilton can compete with any real young mathematician in history. On July 7, 1823, young Hamilton easily won the first place among 100 applicants and entered Trinity College. He soon became famous, and when he was a college student, his outstanding talents in classical literature and mathematics had aroused curiosity in academic circles in England, Scotland and Ireland. Some people even claim that the second Newton has appeared. Most importantly, he completed the first draft of the first part of his epoch-making paper on X-ray systems. When Hamilton submitted his paper to the Royal Irish Academy of Sciences, Dr. Brinkley commented

This young man, I do not say that he is going to be the first mathematician of his generation, but that he is the first mathematician of his generation.

The end of Hamilton's college career at Trinity College was more surprising than its beginning. Dr. Brinkley resigned as professor of astronomy. According to the usual British custom, several famous astronomers, including the later British astronomer Royal George Bedell Eli, competed to "advertise" the vacant professorship. After discussion, the council abandoned all applicants and unanimously elected Hamilton, a 22-year-old college student at the time, as a professor. Hamilton did not apply.

He began to do a good job. Hamilton wisely devoted his main energy to mathematics. At the age of 23, he published the completion form of those "strange discoveries" he made when he was a 17-year-old, that is, the first part of the theory of X-ray systems, which is a great masterpiece for optics. like Lagrange's Analytical Mechanics to Mechanics. It was extended to dynamics in Hamilton's own hands, expressing the basic discipline in its perhaps final and perfect form.

Hamilton introduced some methods of applied mathematics in his first masterpiece, which is indispensable in today's mathematical physics. The goal of many workers in some special branches of theoretical physics is to generalize the whole theory as the Hamilton principle. Fourteen years later, this outstanding work led Jacobi to declare at the British Society meeting in Manchester in 1842 that "Hamilton is the Lagrange of your country", which refers to the English-speaking people. Since Hamilton himself has taken great pains to describe the essence of his new method in a language that non-professionals can understand, we quote an abstract of his own paper submitted to the Royal Irish Academy of Sciences on April 23, 1827:

In optics, a light is considered to be a straight line or broken line or curve along which light travels; a ray system is considered to be an aggregation of these lines, which is due to some similarity in some common connection, origin, or production, in a word, bound together by some kind of optical unity. As a result, light from a luminous point forms an optical system. When they are reflected in the mirror, they form another optical system. To study the geometric relationship of light in a system in which we know its optical origin and history, to explore how they are configured, how they diverge or converge or become parallel, how they are tangent or tangent into what kind of surface or curve, how can they be combined into part of the beam, and how can each particular light be determined? And distinguish it from other light, which is necessary to study the ray system.

In order to popularize the study of this system, so that it can transition to the study of other systems without changing the program, and in order to determine general rules and some general method, to combine and coordinate these isolated optical devices through them, it is necessary to construct the theory of ray system. Finally, in order to do this with the power of modern mathematics, using functions instead of graphics and formulas instead of charts, it is necessary to construct the algebraic theory of these systems, that is, the application of algebra to optics.

In order to construct such an application, Cartesian methods must be used to apply algebra to geometry. In the general scientific progress, the three dimensions of space have obtained their three algebraic equivalents, as well as appropriate concepts and symbols. In this way, by taking the relationship between the three coordinates of any point on a plane or a surface as the equation of the plane or surface, the surface becomes algebraically defined; all the points are extended, so that a straight line or a curve can be expressed in the same way by specifying two such relations, which correspond to two surfaces, and the straight line can be regarded as the intersection of them. In this way, through the general study of the corresponding three variable equations and the general study of surfaces and curves, it is possible to find all the common properties; even if each geometric problem can not be answered immediately, they can at least be expressed in algebra, so the improvements or discoveries in each algebra can be applied or explained in this geometry.

The science of space and time is closely intertwined and inextricably linked. From then on, improving one discipline can improve another. The problem of drawing tangents to a curve leads to the discovery of flow number or differential calculus; those problems of length and quadrature lead to its inversion, that is, integral calculus; the study of surfaces and curvature requires partial differential calculus; isoperimetric problems lead to the formation of variational calculus. On the contrary, all these great steps in algebraic science have their direct application to geometry, which leads to the discovery of new relationships between points or lines or surfaces. So even if the application of this new method is not so diverse and important, there will still be the pleasure of thinking about it as a way to derive a high degree of wisdom.

The first important application of this algebraic method of coordinates to optical systems was made by Napoleon Malus, a French engineering officer in the Egyptian army, who was known in the history of physical optics as the discoverer of the polarization of reflected light. Malu submitted a profound mathematical work to the French Academy in 1807, which belongs to the type mentioned above and is entitled "Monographs on Optics". The method used in that paper can be described in this way-- the direction of a straight light in any optical system is thought to depend on the position of a particular point on that light and follow a law that characterizes that particular optical system and distinguishes it from other systems. This law can be expressed algebraically by determining the expression of three coordinates for any other point on the ray, that is, the function of the three coordinates of any point.

As a result, Malu used general symbols to represent these three functions (or at least three functions equivalent to these functions) and drew several important general conclusions through very complex but symmetrical calculations; many of these conclusions, along with many other conclusions, I got it myself later, and I didn't know what Malu had done at that time. I started my own attempt to apply algebra to optics in almost the same way. But my research soon led me to replace Malu's method with a very different (I believe I have proved) method that is more suitable for the study of optical systems. Since there is no need to use the three functions mentioned above, or at least their two ratios, only one function is sufficient for this method. I call this function the characteristic function, or the principal function. In this way, he made his reasoning by setting two equations of a ray, on the other hand, I established and used an equation of a system.

This function, which I introduced for this purpose as the basis of my deductive method in mathematical optics, seems to previous authors to be a very profound and extensive result of induction in that science. this known result is often called the law of minimum action, but it is sometimes also called the least time principle, which includes all the previously discovered rules that determine the form and position of light along their paths. And changes in the direction of light caused by ordinary or unusual reflections or refractions.

Light consumes a certain amount from any first point to any second point-in one physical theory it is action, in another it is time; if both ends of the path remain the same, then light takes the actual path, compared with any other path, it consumes the least amount; in professional language, it is zero. The mathematical novelty of my method is, first of all, to consider this quantity as a function of these endpoint coordinates, according to what I call the law of changing action, when the coordinates change, the action also changes. The second is to simplify all the research on the optical system of light into the study of this function: the simplification of mathematical optics from a new point of view, a simplification similar to (as I think) Descartes gives the simplification of algebra to geometry.

There is no need to add anything to Hamilton's description. It may be more rewarding when you read the whole abstract for the second time. In this great work on the X-ray system, Hamilton built something even better than he knew. Almost 100 years after the above abstract was written, it was found that the method that Hamilton introduced into optics was exactly the method needed by wave dynamics associated with modern quantum theory and atomic structure theory. It can be recalled that Newton preferred the emission or particle theory of light, while Huygens tried to use the wave theory of light to explain the phenomenon of light. The two viewpoints are combined in modern quantum theory and coordinated in the sense of pure mathematics. Modern quantum theory was formed in 1925. In 1843, at the age of 28, Hamilton realized his ambition to extend the principles of optics to the whole dynamics.

Hamilton's prediction of what is called conical refraction in optics is of a qualitative and quantitative type (such as Einstein's prediction of light deflection and Maxwell's prediction of radio waves). Based on his ray system theory, he mathematically predicted that a completely unexpected phenomenon would be found in connection with the refraction of light in biaxial crystals. While he was pondering his paper on light, the third supplement, he was surprised by a discovery, which he described as follows:

In some cases, in a biaxial crystal, there should be more than two, not three, or any finite number of refracted rays, corresponding to or generated from a single incident ray, but an infinite number of refracted rays, or a cone of refracted rays; in other cases, a single ray in such a crystal produces an infinite number of rays arranged in another cone. Therefore, he foresaw two new laws of light from the theory and named them inner cone refraction and outer cone refraction.

This prediction, and its confirmation by Humphrey Lloyd's experiments, earned Hamilton unlimited praise. According to some, this astonishing success was the culmination of Hamilton's career; after this great work on optics and dynamics, Hamilton declined. Others think that Hamilton's greatest work has not yet been done-this is the quaternion theory, which Hamilton himself thinks is his masterpiece, and the masterpiece that made him immortal.

The history of quaternions is too long to be fully introduced here, and even Gauss's prediction in 1817 was not the beginning of this field; Euler preceded Gauss with an isolated result, which is easiest to explain in quaternions. The origin of quaternions can be traced even further, because Augustus de Morgan once half-jokingly proposed to trace their history from ancient India to the Victorian era for Hamilton. However, we only need to look at the main parts of this discovery here.

English algebraist who established ordinary algebra on its own in the first half of the 19th century. They foresaw the steps currently taken in the cautious and rigorous development of any branch of mathematics and established algebra on the postulate. Previously, when it was assumed that all algebraic equations had roots, all kinds of numbers that entered mathematics-fractions, negative numbers, irrational numbers-were allowed to work on the same basis as ordinary positive integers. Because of habit, ordinary positive integers are so "old" that all mathematicians think they are "natural". It seems a bit foolish to build a system on blind, formal tricks in mathematical symbols and naively believe in its self-consistency.

This credulity culminates in the principle of formal eternity, which actually means that a set of rules that produce consistent results for a class of numbers, such as positive integers, when applied to any other type of number, such as imaginary numbers, will continue to produce consistency even when the results are not clearly explained. Trusting the perfection of meaningless symbols often leads to absurdity.

At this point, it must be borne in mind that algebra only deals with finite processes, and when infinite processes enter, for example, when the summation of infinite series, we go beyond algebra and enter another field. This point is emphasized because the "algebra" marked in the usual elementary textbooks contains many things that are not algebra in the modern sense-for example, infinite geometric series.

The nature of what Hamilton did in creating quaternions is clearer in the context of a set of postulations of ordinary algebra.

A field F is made up of elements for a _. A system consisting of a set S and two operations called addition and multiplication, which can be implemented on any two elements an and b of S, in this order, the unique definite element of S is produced.

So that the public Imurv can be satisfied. For simplicity, we write aplb and ab instead of the results of the above figure, and call them the sum and product of an and b, respectively. In addition, the element of S is called the element of F.

i. If an and b are any two elements of F, then aquib and ab are the only definite elements of F, and

Ⅱ. If a _ c is any three elements of F, then

There are two different elements in Ⅲ. F, marked as 0 and 1, so that if an is any element of F, then axiomorpha.

Ⅳ. No matter what the element an of F is, there is always an element x in F that makes a+x=0.

v. No matter what the element a (not zero) of F is, there is always an element y in F that makes ay=1.

The whole ordinary algebra is derived from these simple postulates. Such a set of postulates can be regarded as a concentration of experience. For centuries people have used mathematics and obtained useful results based on the laws of arithmetic-from experience, this practice has inspired most of the laws contained in these precise postulations, but once they understand the inspiration of experience, the explanations provided by experience (here ordinary arithmetic) are deliberately concealed or forgotten. The system defined by the postulate is developed abstractly in its own value by ordinary logic and mathematical wit.

If, as usual, I denotes the root sign-1, then a "plural" is a number of type α + bi, where Aline b is a "real number". Instead of regarding α + bi as a "number", Hamilton thought of it as an ordered pair of "numbers", which he wrote down as (a _ r _ b). One of the advantages of this new method of dealing with complex numbers is that the definition of sum and product of even numbers is regarded as an example of a general abstract definition of sum and product in a field. Therefore, if the consistency of the system defined by the axiom of a domain is proved, similar conclusions can be drawn about complex numbers and the general rules on which they are combined without further proof. Hamilton only needs to give the definition of sum and product in the theory that complex numbers are considered as even numbers (a _ r _ b), (c _ r _ d) and so on.

The sum of (a) and (c) is (a), and the product is (ac-bd,ad+bc). The 0BEI in the domain corresponds here to (0d0), (1pc0). With these definitions, it is easy to prove that Hamilton's number pairs satisfy all the axioms stated for a field, but they also conform to the formal rules for complex operations. Therefore, the two forms of "and" are (aforc) + I (bounded), corresponding to the number pair (aquarc). Therefore, the two forms of "and" correspond to the number pair (aformac) and (cmagine d) respectively. In addition, the formal product of axibiquist cymdi produces (ac-bd) + I (ad+bc), which corresponds to the number pair (ac-bd,ad+bc).

After dealing with the plural with an even number, Hamilton tried to extend his idea to triple or quaternion, otherwise, such work would be meaningless. Hamilton's aim is to invent an algebra that treats rotation in three-dimensional space as complex numbers or his number pairs do in two-dimensional space, both of which are like Euclidean spaces in elementary geometry. Now, you can think of a complex a+bi as a vector, that is, a line segment with both length and direction, which is obvious in the diagram.

But Hamilton was blocked for years by an unexpected difficulty in trying to symbolize the behavior of vectors in three-dimensional space in order to preserve the vector properties used in physics, especially in rotational systems. For a long time he did not even guess the true nature of the difficulty.

Hamilton opposes the purely abstract and public systematic exposition of algebra, and he tries to build algebra on some "more real" basis, for this meaningless cause, he took advantage of Kant's erroneous view refuted by the creation of non-Euclidean geometry, that is, space as "a pure form of sensory intuition". Indeed, Hamilton, who did not seem to know non-Euclidean geometry, followed Kant's belief that "time and space are the two great sources of knowledge, and all kinds of transcendental comprehensive understandings can be derived from them." In this regard, pure mathematics provides an excellent example of our understanding of space and its various relationships. Because they are pure form of sensory intuition, they reflect the a priori possibility of comprehensive propositions.

Of course, any mathematician who is not completely ignorant today knows that Kant's mathematical concept is wrong, but in the 1840s, when Hamilton was on his way to the quaternion, Kant's philosophy of mathematics still makes sense for those who have not heard of Robachevsky (the founder of non-Euclidean geometry). Hamilton applied Kant's theory to algebra with a seemingly bad mathematical pun and came to a strange conclusion: because geometry is the science of space, because time and space are "pure sensory forms of intuition". So the rest of mathematics must belong to time, and he wastes most of his time scrutinizing the bizarre theory that algebra is the science of pure time.

Hamilton's difficulty in trying to construct a vector and rotation algebra for three-dimensional space stems from his subconscious belief that the most important laws of ordinary algebra must continue to exist in the algebra he is looking for. How are the vectors multiplied together in three-dimensional space?

In order to understand the difficulty of this problem, it must be borne in mind that the ordinary complex number a+bi has given a simple explanation by rotation in a plane, and that the complex number obeys all the rules of ordinary algebra, especially the rate of multiplicative exchange: if A _ line B is an arbitrary complex number, then A × B × A, no matter whether A ~ B is interpreted in terms of algebra or rotation in the plane. In that case, it seems reasonable to predict that the same commutative law is also valid for the generalization of complex numbers representing rotation in three-dimensional space.

Hamilton's great discovery is a kind of algebra, a kind of rotational "natural" algebra in three-dimensional space, in which the commutative law is not established. In this quaternion Hamiltonian algebra, there is a multiplication in which A × B is not equal to B × A, but is equal to negative B × A, that is, A × B × B × A.

The ability to abandon the multiplicative commutative law and construct a compatible and practically useful algebraic system is a first-rate discovery that may be comparable to the formation of non-Euclidean geometric ideas. One day (October 16, 1843), when Hamilton and his wife went for a walk on a bridge, he (after 15 years of futile thinking) suddenly felt enlightened. So much so that the basic formula of the new algebra was engraved on the bridge deck. His great discovery still guides algebraists to other algebras to this day. In fact, mathematicians follow in the footsteps of Hamilton, creating various algebras at will by negating one or more postulates on a field and developing its results. Some of these "algebras" are extremely useful; the general theories that include many of these algebras also include Hamilton's great invention.

According to Hamilton's quaternion, various vector analyses that have been loved by the past two generations of physicists have emerged. Today, all this, including quaternions, as long as it is related to physical applications, has been put aside by the incomparably simple and universal tensor analysis that became popular with general relativity in 1915.

Finally, Hamilton's deepest tragedy is his stubborn belief that quaternions are the key to solving the mathematics of the physical universe. History has proved that he has sadly deceived himself. Never before has a great mathematician been so hopelessly wrong.

The last 22 years of Hamilton's life devoted almost exclusively to the detailed scrutiny of quaternions, including their applications to dynamics, astronomy and the wave theory of light. He died of gout on September 2, 1865, at the age of 61.

A person who loves labor and truth. I wish my epitaph were the same.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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