"The facts are relative but the law is absolute"Special theory of relativity | General theory of relativity | Tensor calculus
Quantum mechanicsSchrodinger's equation | Matrix mechanics
Lagrange has perhaps done more than any other to give extent and harmony to such deductive researches by
showing that the most varied consequences … may be derived from one radical formula, the beauty of the method so suiting the dignity of the results as to make his great work a kind of scientific poem.
W. R. Hamilton
Classical physics is based on Newton's laws of motion and universal laws of gravitation. The laws of motion are alaways applicable where motion and accerelation are involved. It is very surprising that some fundamental processes are able to describe our universe completely or at least some parts of it. Newton's laws of motion are some such fundamental facts , which can be explained in a simple way with little mathematics. All the physical laws are more or less related to these laws. Einstein was able to discover relativity because he mastered all the other thoeries prior to him. The laws of motion are as follows:
a) If no external force is applied then a moving body will always move in a stright line and static body will remain static forever. This law holds everywhere and everything obeys it. But we will see later that some modification of this law is needed as Einstein 's theory of relativity suggested. Straight lines will need to be generalized to be geodesics. Ok let us break it down :
Newton thought that a material object is inertial object when no force acting on it so it's inertia opposes the change of its momentum or velocity. So he generalized his first law accordingly. But gravity seems to rule the universe and is everywhere. Force of gravity affect s every object. So it is better to represent gravity as a property of spacetime. So the ide a of geodesic arose. Geodesic is nothing but straight line in curved spacetime. For more details you can find discussion about it in General relativity.
If you like my writing pls click on google ads on this page or by visiting this pageb) The magnitude of the applied force is proprtional to rate of change of momentum of a body and the change of momentum occurs in the direction where the force is applied. This law is very fundamental like other three laws. The concept of force and mass are just defined using this law.
c) Every action has equal and opposite reaction. This is the case with everything. when we sit on a chair, the chair pushes up upward with a force equal and opposite of the force that we exert on the chair. When we fire a rifle the bullet and rifle experiences forces that are equal and opposite of each other. There are numerous other examples that we can give to demonstrate the third law.
Based on these laws of motion Newton gave the idea of his ClockWork universe. If a ball is thrown into space it will fall to earth after some time. If we know the intial position and velocity of the ball we can completely determine its trajectory in space. So Newton thought that if positions of every particle and forces acting on them are known, an intelligent being , given sufficient time, will able to calculate the state of the universe in later time. Past , present and future will be completely determined. This is also known as determinism. The evolution of our universe can be traced forward and bakward in time with certainty. There is no randomness in such determinism. With the advent of quantum mechanics determinism started to fall apart. The physical phenomena in atomic level become random like throwing a dice. A long discussion is needed to understand quantum mechanics, which at the moment best left. Although some kind of determinism is still present in quantum theory. The claasical phenomena is nothing but the statistical average of a large class of quantum phenomena. This fact is exactly a reflection of Bond's correspondence principle.
"Did no one come to save me because they miss me?"
Another revolution happened in classical physics when Kepler developed his laws of planetary motion. Kepler's law can be derived using Newton's law of gravitation. Kepler first gave mathematical descriptions of planetary motion. He explained planetary motion is three laws :
1) Every planet revolves around the sun in elliptic orbit.
2) Each planet sweeps equal area in equal time. That is, in the plane on which the planets orbit the sun the areas covered by the planet in equal times are equal.
3) Square of the period of the orbit ( T^2 ) is proportional to the cube of the distance D^3. D is the semi-major axis of the ellipse.
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Concepts in hamiltonian Mechanics
Classical hamiltonian mechanics is also known as hamiltonian mechanics. William hamilton first formulated a new mechanics which is now called Hamiltonian mechanics , starting from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. In Newtonian mechanics, the time evolution is obtained by computing the total force being exerted on each particle of the system, and from Newton's second law, the time-evolutions of both position and velocity are computed. In contrast, in Hamiltonian mechanics, the time evolution is obtained by computing the Hamiltonian of the system in the generalized coordinates and inserting it in the Hamilton's equations. Hamiltonian and Lagrangan are denoted by H and L which represent energy of a physical system. Hamiltonian H is related to Lagrangian L by certain relation, which are both reformulation of classical mechanics with more flexibility. Hamiltonian is a function of generalized coordinates like position q, momentum p and time t. It is the total energy of a system, which is algebric sum of potential and kinetic energy.
These generalized coordinates p and q represent degrees of freedom a system can have. Degrees of freedom is number of independent parameter that uniquely
characterize a system. Generalized coordinates fix the configuration of a system so that dynamics of the system can be determined from them.
For example, a pendulum swinging in one direction can be modeled with a single degree of freedom, namely, the angle
θ. A ball which is rolling on the floor has two degrees of freedom : one is the direction x of translation and other is related to angular velocity of the ball, namely θ . So in this case the system has two types of energy : one is kinetic energy and other is rotational energy.
Time evolution of a system can be determined from two equations related to Hamiltonian H:
The total differential of Hamiltonian is given by the relation
Thus , total differential is the algebric sum of sum of changes of individual components that changes due to change in each coordinate and total change in time coordinate. In the relaion above we used Legendre transform
which gives the relation between hamiltonian and Laglangian. Lagrangian L is a function of position q, derivative of position dq/dt and time t. And it is straight forward to find relationship between Hamiltonian and Lagrangian. Hamiltonian and Lagrangian are both dependable on kinetic and potential energy.
"A wise guy is always right. Even when he is wrong , he is right.."
In Newtonian mechanics, the time evolution is attained by computing the total force being exerted on each particle of the system, and from Newton's second law, the time-evolutions of both position and velocity are computed. In contrast, in Hamiltonian mechanics, the time evolution is obtained by computing the Hamiltonian of the system in the generalized coordinates and plugging it in the Hamilton's equations. This approach is equivalent to the one used in Lagrangian mechanics. In fact, as is shown below, the Hamiltonian is the Legendre transform of the Lagrangian when holding q and t fixed and defining p as the dual variable, and thus both method give the same equations for the same generalized momentum. The main inspiration to use Hamiltonian mechanics instead of Lagrangian mechanics comes from the symplectic structure of Hamiltonian systems.
While Hamiltonian mechanics can be used to describe simple systems such as a bouncing ball, a pendulum or an oscillating spring in which energy changes from kinetic to potential and back again over time, its strength is shown in more complex dynamic systems, such as planetary orbits in celestial mechanics. The more degrees of freedom the system possesses, the more complicated its time evolution is and, in most cases, it becomes chaotic.
In quantum mechanics, Hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system . It is usually denoted by H, also Ȟ or Ĥ. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of supreme importance in many formulations of quantum theory.
"Have ever have the feeling that you are not sure whether you are awake or dreaming..?"
The concept of phase space is related to Hamiltonian mechanics. The phase space plot specifies the state of a dynamical system at any given moment. Phase space is the space which axes are determined by the generalized coordinates ( p, q) . So if a system which has six degrees of freedom has N particles, the phase space will have 6N ( 3 position q coordinates and 3 momentum p coordinates for each particle) dimensions. Each state of any such dynamical system is a unique point in phase space. One dimensional phase space is called phase line and two dimensional phase space is called phase plane.A 6n dimensional phase space of Hamiltonian mechanics will look like this :
The total number of particles is n. If there is 10 particles the phase space will be 6X10 = 60 dimensional. If 20 , 120 dimensional and so on.
ActionAction is a physical quatity. It is usually expressed as the integral over time when a system evolve through a path in the time interval over which integral is taken. It is numerically equivalent to energy multiplied by time.
L is the lagrangian of the system. Action has many applications in both claasical and modern physics. It can be used to derive many laws of nature.
The most useful application is called priciple of least action. This principle states that for a physical process the action is minimum. As an example, when a body moves through two events in spacetime interval the body sees as the time between the events. For a clock which moves without constraints between two events in spaceitme, the time between the events that the clock shows becomes maximum. If the clock were constrainted to travel by some other route and were also present at the same two events, the time would be lesser. This is kind of cosmic laziness. Our universe is , according to this principle, very lazy. This is kind of cosmic boredoom. Everything in the universe tried to follow a path which takes least energy.
The reason that the time is maximum , not minimum is that spacetime interval is always timelike in relativity. Before going into more details slight different formulation of action can be stated: In this formulation action is a functional which takes a function as an argument and returns a scalar:
q(t) is generalized coordinate of the system. It is a function of time itself.
Generalised coordinates can be called degrees of freedom too. The Lagrangian is a function of q , dq/dt and t. Principle of least action is the condition when the variation of Lagrangian vanishes. In the language of calculus this is similar to finding maximum or minimum values of a function. The maximum or minimum values of a function occurs when it becomes stationary at the value of independent variable. The condition is that first order change of action is zero. In case of ordinary one variable function, the slope of curve becomes zero at the point where the function becomes extreme.
When the value of action is minimum for a path that an object follows, the actions corresponding to small paths that make up the whole path are also minimum. Otherwise the total action would not be minimum. Same interpretation can be given for maximum or minimum value for a functional. In case of functional we are interested in the function that extremizes the functional. This is called Hamiltonian Principle. The mathematical expression is
So the condition is that variation of action is zero. First order change of action is zero , to be precise. This condition leads to an interesting equation known as Euler-Lagrange equations. The latter has many applications relating to calculus of variation. For the moment I just mention the equation , which is necessarily a differential equation. The reason is that the equation contains lagrangian which is a function of a function.
Where f is a function of Y`, and Y. Y , on the other hand is a function of independent variable x. Euler - lagrange equation plays a very important role in deducing many theories of physics. Brachistochrone problem can be solved using this equation. This is a problem of calculus of variation , which states "what is the curve of a bead sliding on a frictionless wire under influence of gravity, that minimizes the time". So the problem is to find the curve of quickest descent.
If you like my writing pls click on google ads on this page or by visiting this pageJohann Bernoulli's solution divides the plane into strips and he assumes that the particle follows a straight line in each strip. The path is then piecewise linear. The problem is to determine the angle of the straight line segment in each strip and to do this he appeals to Fermat's principle , namely that light always follows the shortest possible time of travel. If v is the velocity in one strip at angle α to the vertical and u in the velocity in the next strip at angle β to the vertical then, according to the usual sine law
In the limit, as the strips become infinitely thin, the line segments tend to a curve where at each point the angle the line segment made with the vertical becomes the angle the tangent to the curve makes with the vertical. If v is the velocity at (x, y) and α is the angle the tangent makes with the vertical then the curve satisfies
The cycloid x(t) = h(t - sin t), y(t) = h(1 - cos t) satisfies this equation. To see this note that
= h(1 - cos t + sin2t/(1-cos t))
= h((1 - cos t)2+ sin2t)/(1 - cos t)
= h(2 - 2cos t)/(1-cos t) = 2h
ApplicationsSimple harmonic oscillator can be analyzed through principles of hamiltonian mechanics. The hamiltonian will be the sum of potential and kinetic energy.
The action of such operation on some function u describing the harmonic motion will produce an energy eigen value.
NotesEvery triangle satisfies some identities involving lengths of it's and sines and cosines of the angles that it posseses. The law involving sines is called sine law and that involving cosine is called cosine law.
It would be generally agreed that physics is an empirical science, as contrasted with logic and pure mathematics. I want, now to define in what this difference consists.
We may observe, in the first place, that many philosophers in the past have denied the distiction. Thorough-going rationalists have believed that the facts which we regard as only discoverable by observation could really be deduced from logical and and metaphysical principles; thorough-going empiricists have believed that the premisses of pure mathematics are obtained induction from experience. Both views seem to me false and are , I think , rarely held in present day; neverthless, it will be as well to examine the reasons for thinking that there is an epistomological distinction between pure mathematics and physics, before trying to discover its exact nature.
There is a traditional distinction between necessary and contingent propositions, and another between analytic and synthetic propositions. It was generally held before kant that necessary propositions were the same as analytics propositions, and contingent propositions were the same as synthetic propositions. But even before kant the two distinctions were different, even if they effected the same division of propositions. It was held that every proposition is necessary, assertoric, or possible, and that sense are ultimate notions, comprised under the head of "modality".
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FUNCTIONSIn mathematics, a function was originally the idealization of how a varying quantity depends on another varying quantity. For example, the position of a planet is a function of time parameter. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of function was formalized at the end of the 19th century in terms of Cantor's set theory, and this greatly enlarged the domains of application of the concept.
A function is a process or a relation that assigns each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the converse domain of the function. If the function is called f, this relation is denoted y = f(x) (read f of x), the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f The symbol that is used for representing the input is the variable of the function (one often says that f is a function of the variable x).
A function is uniquely represented by its graph which is the set of all ordered pairs (x, f(x)). When the domain and the codomain are sets of numbers, each such pair may be considered as the Cartesian coordinates of a point in the plane. In general, these points form a curve, which is also called the graph of the function. This is a practical representation of the function, which is commonly used everywhere. For example, graphs of functions are commonly used in newspapers for representing the evolution of price indexes and stock market indexes
Functions are widely used in science, and in most fields of mathematics. Their role is so important that it has been said that they are "the central objects of investigation" in most fields of mathematics.
Schematic depiction of a function described metaphorically as a "machine" or "black box" that for each input yields a corresponding output The red curve is the graph of a function, because any vertical line has exactly one crossing point with the curve. A function that associates any of the four colored shapes to its color. The sole purpose of explaining function theory is that hamiltonian mechanics is based on function named hamiltonian.
a function is a relation and conversely a relation is said to be a function. Every relation has a domain and a converse domain. The class of terms which has to some relation with a term or others is called the domain of a relation. The class of terms which has some relation to some term or others is called the converse domain. Both the domain and the converse domain are together called the field of the relation. For example spouse is a relation which domain consists of all the man as its re ferents and all the woman as its converse domain. The term from which a relation proceeds is called the referent and the term to which the relation proceeds is called relata. The relational arithmetic is vital to mathematical logic. Som e terms and properties must now be explained:
A relation is said to be reflextive when there is a term x such that xRx or the relation R hold between the term x with itself.
A relation is symmetrical when there are terms x and y such that if xRy , then yRx
A relation is transitive when a term y exists between x and z such that if xRy and yRz the xRz.
In every kind of relation xRy means relation R hold between x and y. The converse relation R^ holds between y and x. With the defintion of this relation R we can find the relative product of two relations. The relative product of two relation is defined as the logical product of two relations , which is itself a transitive relative. So when xRy and yRz then xRz. For example Grandfather is the relative product of both the relations son and father. So z is grandfather of x means there is some person y so x is the son of y and z is the father of y. A very useful relation in pure mathematics is the one-to-one relation. We all know what this relation is. To put it simply, a relation is one-to-one when there is two terms x and x` such that f(x) != f(x`) . Using one-to-one relation we can define a progression.
A progression is a one-to-one relation so that just one term belongs to domain but not to converse domain and all the domain is just the posterity of this one term.
Much been written concerning the laws of motion, the possibility
of dispensing with Causality in Dynamics, the relativity of motion and
other kindred questions. But there are several preliminary questions, of great
difficulty and importance, concerning which little has been said. Yet these
questions, speaking logically, must be settled before the more complex problems
usually discussed can be attacked with any hope of success. Most of the
relevant modern philosophical literature will illustrate the truth of these
remarks: the theories suggested usually repose on a common dogmatic basis,
and can be easily seen to be unsatisfactory. So long as an author confines
himself to demolishing his opponents, he is irrefutable; when he constructs
his own theory, he exposes himself, as a rule, to a similar demolition by the
next author. Under these circumstances, we must seek some different path,
whose by-ways remain unexplained. “Back to Newton” is the watchword of
reform in this matter. Newton’s scholium to the definitions contains arguments
which are unrefuted, and so far as I know, irrefutable: they have been
before the world two hundred years, and it is time they were refuted or
accepted. Being unequal to the former, I have adopted the latter alternative.
The concept of motion is logically subsequent to that of occupying a place
at a time, and also to that of change. Motion is the occupation, by one entity,
of a continuous series of places at a continuous series of times. Change is the
difference, in respect of truth or falsehood, between a proposition concerning
an entity and a time T and a proposition concerning the same entity and
another time T', provided that the two propositions differ only by the fact that
T occurs in the one where T' occurs in the other. Change is continuous when
the propositions of the above kind form a continuous series correlated with a
continuous series of moments. Change thus always involves (1) a fixed entity,
(2) a three-cornered relation between this entity, another entity, and some
but not all, of the moments of time. This is its bare minimum. Mere existence
at some but not all moments constitutes change on this definition. Consider
pleasure, for example. This, we know, exists at some moments, and we may
suppose that there are moments when it does not exist. Thus there is a
relation between pleasure, existence, and some moments, which does not
subsist between pleasure, existence, and other moments. According to the
definition, therefore, pleasure changes in passing from existence to nonexistence
or vice versâ. This shows that the definition requires emendation, if it
is to accord with usage. Usage does not permit us to speak of change except
where what changes is an existent throughout, or is at least a class-concept
one of whose particulars always exists. Thus we should say, in the case of
pleasure, that my mind is what changes when the pleasure ceases to exist. On
the other hand, if my pleasure is of different magnitudes at different times,
we should say the pleasure changes its amount, though we agreed in Part III
that not pleasure, but only particular amounts of pleasure, are capable of
existence. Similarly we should say that colour changes, meaning that there are
different colours at different times in some connection; though not colour,
but only particular shades of colour, can exist. And generally, where both the
class-concept and the particulars are simple, usage would allow us to say, if a
series of particulars exists at a continuous series of times, that the classconcept
changes. Indeed it seems better to regard this as the only kind of
change, and to regard as unchanging a term which itself exists throughout a
given period of time. But if we are to do this, we must say that wholes
consisting of existent parts do not exist, or else that a whole cannot preserve
its identity if any of its parts be changed. The latter is the correct alternative,
but some subtlety is required to maintain it. Thus people say they change
their minds; they say that the mind changes when pleasure ceases to exist in
it. If this expression is to be correct, the mind must not be the sum of its
constituents. For if it were the sum of all its constituents throughout time, it
would be evidently unchanging; if it were the sum of its constituents at one
time, it would lose its identity as soon as a former constituent ceased to exist
or a new one began to exist. Thus if the mind is anything, and if it can
change, it must be something persistent and constant, to which all constituents
of a psychical state have one and the same relation. Personal identity
could be constituted by the persistence of this term, to which all a person’s
states (and nothing else) would have a fixed relation. The change of mind
would then consist merely in the fact that these states are not the same at
The Infinitesimal Calculus is the traditional name for the differential
and integral calculus together, and as such I have retained it; although, as
we shall shortly see, there is no allusion to, or implication of, the infinitesimal
in any part of this branch of mathematics.
The philosophical theory of the Calculus has been, ever since the subject
was invented, in a somewhat disgraceful condition. Leibniz himself—who,
one would have supposed, should have been competent to give a correct
account of his own invention—had ideas, upon this topic, which can only
be described as extremely crude. He appears to have held that, if metaphysical
subtleties are left aside, the Calculus is only approximate, but is justified
practically by the fact that the errors to which it gives rise are less than
those of observation.* When he was thinking of Dynamics, his belief in
the actual infinitesimal hindered him from discovering that the Calculus
rests on the doctrine of limits, and made him regard his dx and dy as
neither zero, nor finite, nor mathematical fictions, but as really representing
the units to which, in his philosophy, infinite division was supposed to
lead.† And in his mathematical expositions of the subject, he avoided
giving careful proofs, contenting himself with the enumeration of rules.‡
At other times, it is true, he definitely rejects infinitesimals as philosophically
valid;§ but he failed to show how, without the use of infinitesimals,
the results obtained by means of the Calculus could yet be exact, and
not approximate. In this respect, Newton is preferable to Leibniz: his
Lemmas* give the true foundation of the Calculus in the doctrine of limits,
and, assuming the continuity of space and time in Cantor’s sense, they give
valid proofs of its rules so far as spatio-temporal magnitudes are concerned.
But Newton was, of course, entirely ignorant of the fact that his Lemmas
depend upon the modern theory of continuity; moreover, the appeal to time
and change, which appears in the word fluxion, and to space, which appears
in the Lemmas, was wholly unnecessary, and served merely to hide the fact
that no definition of continuity had been given. Whether Leibniz avoided
this error, seems highly doubtful: it is at any rate certain that, in his first
published account of the Calculus, he defined the differential coefficient by
means of the tangent to a curve. And by his emphasis on the infinitesimal,
he gave a wrong direction to speculation as to the Calculus, which misled
all mathematicians before Weierstrass (with the exception, perhaps, of
De Morgan), and all philosophers down to the present day. It is only in the
last thirty or forty years that mathematicians have provided the requisite
mathematical foundations for a philosophy of the Calculus; and these foundations,
as is natural, are as yet little known among philosophers, except in
France.† Philosophical works on the subject, such as Cohen’s Princip der
Infinitesimalmethode und seine Geschichte,‡ are vitiated, as regards the constructive
theory, by an undue mysticism, inherited from Kant, and leading to such
results as the identification of intensive magnitude with the extensive infinitesimal.
§ I shall examine in the next chapter the conception of the infinitesimal,
which is essential to all philosophical theories of the Calculus hiterto
propounded. For the present, I am only concerned to give the constructive
theory as it results from modern mathematics.
The differential coefficient depends essentially upon the notion of a continuous function of a continuous variable. The notion to be defined is not purely ordinal; on the contrary, it is applicable, in the first instance, only to series of numbers, and thence, by extension, to series in which distances or stretches are numerically measureable. But first of all we must define a continuous function.
We have already seen (Chap. 32) what is meant by a function of a variable, and what is meant by a continuous variable (Chap. 36). If the function is one-valued, and is only ordered by correlation with the variable, then, when the variable is continuous, there is no sense in asking whether the function is continuous; for such a series by correlation is always ordinally similar to its less than a, will differ by less than ε; in popular language, the value of the function does not make any sudden jumps as x approaches a from the left. Under similar circumstances, f(x) will have a limit as it approaches a from the right. But these two limits, even when both exist, need not be equal either to each other or to f(a), the value of the function when x = a. The precise condition for a determinate finite limit may be thus stated:*
Finite and Infinite
The purpose of the present chapter is not to discuss the philosophical
difficulties concerning the infinite, which are postponed to Part V.
For the present I wish merely to set forth briefly the mathematical theory of
finite and infinite as it appears in the theory of cardinal numbers. This is its
most fundamental form, and must be understood before the ordinal infinite
can be adequately explained.*
Let u be any class, and let u' be a class formed by taking away one term x
from u. Then it may or may not happen that u is similar to u' . For example, if u
be the class of all finite numbers, and u' the class of all finite numbers
except 0, the terms of u' are obtained by adding 1 to each of the terms of u,
and this correlates one term of u with one of u' and vice versâ, no term of either
being omitted or taken twice over. Thus u' is similar to u. But if u consists of all
finite numbers up to n, where n is some finite number, and u' consists of all
these except 0, then u' is not similar to u. If there is one term x which can be
taken away from u to leave a similar class u' , it is easily proved that if any other
term y is taken away instead of x we also get a class similar to u. When it is
possible to take away one term from u and leave a class u' similar to u, we say
that u is an infinite class. When this is not possible, we say that u is a finite class.
From these definitions it follows that the null-class is finite, since no term can
be taken from it. It is also easy to prove that if u be a finite class, the class
formed by adding one term to u is finite; and conversely if this class is finite,
so is u. It follows from the definition that the numbers of finite classes other
than the null-class are altered by subtracting 1, while those of infinite classes
are unaltered by this operation. It is easy to prove that the same holds of the addition of 1. 118. Among finite classes, if one is a proper part of another, the one has a smaller number of terms than the other. (A proper part is a part not the whole.) But among infinite classes, this no longer holds. This distinction is, in fact, an essential part of the above definitions of the finite and the infinite. Of two infinite classes, one may have a greater or a smaller number of terms than the other. A class u is said to be greater than a class v, or to have a number greater than that of v, when the two are not similar, but v is similar to a proper part of u. It is known that if u is similar to a proper part of v, and v to a proper part of u (a case which can only arise when u and v are infinite), then u is similar to v; hence “u is greater than v” is inconsistent with “v is greater than u”. It is not at present known whether, of two different infinite numbers, one must be greater and the other less. But it is known that there is a least infinite number, i.e. a number which is less than any different infinite number. This is the number of finite integers, which will be denoted, in the present work, by α0.* This number is capable of several definitions in which no mention is made of the finite numbers. In the first place it may be defined (as is implicitly done by Cantor†) by means of the principle of mathematical induction. This definition is as follows: α0 is the number of any class u which is the domain of a one-one relation R, whose converse domain is contained in but not coextensive with u, and which is such that, calling the term to which x has the relation R the successor of x, if s be any class to which belongs a term of u which is not a successor of any other term of u, and to which belongs the successor of every term of u which belongs to s, then every term of u belongs to s. Or again, we may define α0 as follows. Let P be a transitive and asymmetrical relation, and let any two different terms of the field of P have the relation P or its converse. Further let any class u contained in the field of P and having successors (i.e. terms to which every term of u has the relation P) have an immediate successor, i.e. a term whose predecessors either belong to u or precede some term of u; let there be one term of the field of P which has no predecessors, but let every term which has predecessors have successors and also have an immediate predecessor; then the number of terms in the field of P is α0. Other definitions may be suggested, but as all are equivalent it is not necessary to multiply them. The following characteristic is important: Every class whose number is α0 can be arranged in a series having consecutive terms, a beginning but no end, and such that the number of predecessors of
any term of the series is finite; and any series having these characteristics has the number α0. It is very easy to show that every infinite class contains classes whose number is α0. For let u be such a class, and let x0 be a term of u. Then u is similar to the class obtained by taking away x0, which we will call the class u1. Thus u1 is an infinite class. From this we can take away a term x1, leaving an infinite class u2, and so on. The series of terms x1, x2, . . . is contained in u, and is of the type which has the number α0. From this point we can advance to an alternative definition of the finite and the infinite by means of mathematical induction, which must now be explained. 119. If n be any finite number, the number obtained by adding 1 to n is also finite, and is different from n. Thus beginning with 0 we can form a series of numbers by successive additions of 1. We may define finite numbers, if we choose, as those numbers that can be obtained from 0 by such steps, and that obey mathematical induction. That is, the class of finite numbers is the class of numbers which is contained in every class s to which belongs 0 and the successor of every number belonging to s, where the successor of a number is the number obtained by adding 1 to the given number. Now α0 is not such a number, since, in virtue of propositions already proved, no such number is similar to a part of itself. Hence also no number greater than α0 is finite according to the new definition. But it is easy to prove that every number less than α0 is finite with the new definition as with the old. Hence the two definitions are equivalent. Thus we may define finite numbers either as those that can be reached by mathematical induction, starting from 0 and increasing by 1 at each step, or as those of classes which are not similar to the parts of themselves obtained by taking away single terms. These two definitions are both frequently employed, and it is important to realize that either is a consequence of the other. Both will occupy us much hereafter; for the present it is only intended, without controversy, to set forth the bare outlines of the mathematical theory of finite and infinite, leaving the details to be filled in during the course of the work.
THEORY OF FINITE NUMBERS
Having now clearly distinguished the finite from the infinite, we
can devote ourselves to the consideration of finite numbers. It is customary,
in the best treatises on the elements of Arithmetic,* not to define number or
particular finite numbers, but to begin with certain axioms or primitive
propositions, from which all the ordinary results are shown to follow. This
method makes Arithmetic into an independent study, instead of regarding
it, as is done in the present work, as merely a development, without new
axioms or indefinables, of a certain branch of general Logic. For this reason,
the method in question seems to indicate a lesser degree of analysis than that
adopted here. I shall nevertheless begin by an exposition of the more usual
method, and then proceed to definitions and proofs of what are usually taken
as indefinables and indemonstrables. For this purpose, I shall take Peano’s
exposition in the Formulaire,† which is, so far as I know, the best from the
point of view of accuracy and rigour. This exposition has the inestimable
merit of showing that all Arithmetic can be developed from three fundamental
notions (in addition to those of general Logic) and five fundamental
propositions concerning these notions. It proves also that, if the three notions
be regarded as determined by the five propositions, these five propositions
are mutually independent. This is shown by finding, for each set of four
out of the five propositions, an interpretation which renders the remaining
proposition false. It therefore only remains, in order to connect Peano’s
theory with that here adopted, to give a definition of the three fundamental
notions and a demonstration of the five fundamental propositions. When once this has been accomplished, we will know with certainty that everything in the theory of finite integers follows. Peano’s three indefinables are 0, finite integer* and successor of. It is assumed, as part of the idea of succession (though it would, I think, be better to state it as a separate axiom), that every number has one and only one successor. (By successor is meant, of course, immediate successor.) Peano’s primitive propositions are then the following. (1) 0 is a number. (2) If a is a number, the successor of a is a number. (3) If two numbers have the same successor, the two numbers are identical. (4) 0 is not the successor of any number. (5) If s be a class to which belongs 0 and also the successor of every number belonging to s, then every number belongs to s. The last of these propositions is the principle of mathematical induction. 121. The mutual independence of these five propositions has been demonstrated by Peano and Padoa as follows.† (1) Giving the usual meanings to 0 and successor, but denoting by number finite integers other than 0, all the above propositions except the first are true. (2) Giving the usual meanings to 0 and successor, but denoting by number only finite integers less than 10, or less than any other specified finite integer, all the above propositions are true except the second. (3) A series which begins by an antiperiod and then becomes periodic (for example, the digits in a decimal which becomes recurring after a certain number of places) will satisfy all the above propositions except the third. (4) A periodic series (such as the hours on the clock) satisfies all except the fourth of the primitive propositions. (5) Giving to successor the meaning greater by 2, so that the successor of 0 is 2, and of 2 is 4, and so on, all the primitive propositions are satisfied except the fifth, which is not satisfied if s be the class of even numbers including 0. Thus no one of the five primitive propositions can be deduced from the other four. 122. Peano points out (loc. cit.) that other classes besides that of the finite integers satisfy the above five propositions. What he says is as follows: “There is an infinity of systems satisfying all the primitive propositions. They are all verified, e.g., by replacing number and 0 by number other than 0 and 1. All the systems which satisfy the primitive propositions have a one-one correspondence with the numbers. Number is what is obtained from all these systems by abstraction; in other words, number is the system which has all the properties enunciated in the primitive propositions, and those only.” This observation appears to me lacking in logical correctness. In the first place, the question arises: How are the various systems distinguished, which agree in satisfying the primitive propositions? How, for example, is the system beginning with * Throughout
Variation of hamiltonian
Stationary values of a smooth real-valued function f of several variables. Illustrated is the case of a function f(x,y) of two variables. This is stationary where its graph (a 2-dimensional surface) is horizontal (df/dx = df/dy = 0). This occurs (a) where F has a minimum, but also in other situations such as (b) at a saddle point and (c) at a maximum. In the case of Hamilton’s principle —or a geodesic connecting two points a, b—the Lagrangian L takes the place of f, but the specification of a path requires inWnitely many parameters, rather than just x and y. Again, L may not be a minimum, though a stationary point of some kind.
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Reference materials:Lagrangian Mecahnics and Hamiltonian
A brief history of time by S. Hawking
Grand Design by Stephen Hawking