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Consider a mechanical system that has N degrees of freedom. Assume also that the system is close to one of its stable equilibrium points. We will show that this system acts like N independent SHOs, usually with N different frequencies. One or more of these independent oscillations can be present depending on the initial conditions. In a state where only a single oscillation frequency is excited, the N different degrees of freedom move synchronously at a common mode frequency. The ratios between the different displacements for each degree of freedom, known as the mode displacement ratios, are an intrinsic characteristic of the normal mode that is oscillating. The amplitude of any particular mode is known as the normal coordinate. Each normal coordinate oscillates in time like a single SHO. All possible movements of the system, for sufficiently small displacements from the equilibrium point, can be described as a linear combination of modes.
Why do we concentrate on “small” vibrations for such a system? By definition, if the differential equations of motion are linear, the system is then said to be a linear system. Taylor's theorem guarantees that most systems are linear if the displacements are small enough. The motion can then be approximately described by a set of linear differential equations very similar to the equation for a simple harmonic oscillator. […]
Nature is found to conspire in just such a way that the time integral of the Lagrangian is smallest if the motion obeys Newton's Laws. Mechanics can be based on the single principle: Minimize the time integral of the Lagrangian. Three laws of motion can be condensed into one universal principle!
The mathematical language needed to provide the framework for this is called variational calculus. The variational calculus can be used as a powerful tool in solving mechanics problems with explicit constraints. It is also the most general means of solving nonholonomic problems with constraints on the velocities such as for rolling motion. This type of problem cannot be solved by choosing coordinates equal to the number of degrees of freedom but must be embedded in a higher-dimensional space.
The well-known theoretical physicist E. P. Wigner refers to the “unreasonable effectiveness of mathematics in theoretical physics.” Mathematical beauty is and should be the chief guiding principle of theorists, according to P. A. M. Dirac, one of the inventors of quantum mechanics. Although it is hard to define exactly what mathematical beauty is, the search for beauty was the guiding principle in the invention of two major advances in physics in the twentieth century: relativistic quantum mechanics and general relativity. In this chapter, we will discover an elegant formulation of classical mechanics. The mathematical techniques uncovered here are not only beautiful, but they have become the language of modern theoretical physics. […]
Joseph Louis Lagrange reformulated Newton's Laws in a way that eliminates the need to calculate forces on isolated parts of a mechanical system. Any convenient variables obeying the constraints on a system can be used to describe the motion. If Lagrangian mechanics rather than Newtonian mechanics is used, it is only necessary to consider a single function of the dynamical variables that describe the motion of the entire system. The differential equations governing the motion are obtained directly from this function without any vector force diagrams. Lagrangian mechanics is extremely efficient: There are only as many equations to solve as there are physically significant variables.
Lagrange did not introduce new physical principles to mechanics. The physical concepts are due to Newton and Galileo. But he succeeded in giving a more powerful and sophisticated way to formulate the mathematical equations of classical mechanics, an approach that has spread its influence over physics far beyond the purely mechanical problems.
We will begin by solving some examples that lead us toward this new formulation of mechanics. We plan to use the concept of virtual work to derive this. We will consider extended rigid bodies to be made up of collections of massive point particles. Summing over the constituent particles will lead to an efficient and general method for obtaining the differential equations of motion for any frictionless mechanical system. […]
Of the order of the discussion, which requires that we speak first of the eternal punishment of the lost in company with the devil, and then of the eternal blessedness of the saints
We come next to the nature of the punishment which is to be visited upon the devil and all who belong to him when the two cities – the City of God and the city of the devil – have reached their deserved ends through Jesus Christ our Lord, the Judge of the living and dead. I shall in this book discuss this question more diligently, as far as God's help enables me to do so.
I have adopted this order, and preferred to deal with the felicity of the saints later, because, though both the saved and the damned will then be united with their bodies, it seems more incredible that bodies will endure in everlasting torments than that they will remain without any pain in eternal blessedness. Thus, when I have demonstrated that such punishment ought not to be thought unbelievable, this will be of great help to me; for it will make it easier to believe in the immortality of the bodies of the saints, which are delivered from all pain.
Moreover, this order is not at variance with the divine writings. For, in such writings, the blessedness of the good is sometimes put first, as in the words, ‘They that have done good, unto the resurrection of life; and they that have done evil, unto the resurrection of damnation.’
It is clear, then, that felicity consists in the full attainment of all desirable things. It is not a goddess, however, but a gift of God. Therefore no god should be worshipped by men except one who is able to bestow felicity on them. Hence, if felicity itself were a goddess, we might fairly say that Felicity alone would be the proper object of worship. But only God can confer those blessings which can be received even by men who are not good, and who therefore do not have felicity. Now, therefore, let us consider why He willed that the Roman empire should be so great and so enduring. For we have already argued at length that the great number of false gods whom the Romans worshipped did not accomplish this, and we shall continue to say this wherever it seems appropriate to do so.
That the cause of the Roman empire, and of all kingdoms, is not mere chance; nor does it consist in the position of the stars
According to the judgment or opinion of some, things happen by ‘chance’ when they have no cause, or no cause arising from a rational order, and by ‘fate’ when they come about not by the will of God or men, but as a result of a necessary sequence. The cause of the greatness of the Roman empire is therefore neither chance nor fate; for it is beyond doubt that human kingdoms are established by divine providence.
Preface: The plan and argument of the work here undertaken
Most glorious is the City of God: whether in this passing age, where she dwells by faith as a pilgrim among the ungodly, or in the security of that eternal home which she now patiently awaits until ‘righteousness shall return unto judgment’, but which she will then possess perfectly, in final victory and perfect peace. In this work, O Marcellinus, most beloved son – due to you by my promise – I have undertaken to defend her against those who favour their own gods above her Founder. The work is great and arduous; but God is our helper.
I know, however, what efforts are needed to persuade the proud how great is that virtue of humility which, not by dint of any human loftiness, but by divine grace bestowed from on high, raises us above all the earthly pinnacles which sway in this inconstant age. For the King and Founder of this City of which we are resolved to speak has revealed a maxim of the divine law in the Scriptures of His people, where it is said, ‘God resisteth the proud but giveth grace unto the humble.’ But the swollen fancy of the proud-spirited envies even this utterance, which belongs to God, and loves to hear the following words spoken in its own praise: ‘To spare the humble and subdue the proud.’
As I promised in the last book, this final book of the whole work will contain a discussion of the eternal blessedness of the City of God. This City is called ‘eternal’ not because its existence is extended through many ages but will nonetheless at some time come to an end, but in the sense intended in the Gospel, where it is written that ‘of His kingdom there shall be no end’. Nor will that City be like an evergreen tree, where the same greenness seems to persist because the appearance of dense growth is preserved by the emergence of fresh leaves in the place of those which wither and fall: it will not present a mere appearance of perpetuity by new members arising to succeed those who die. Rather, all the citizens of that city will be immortal; for men also will obtain that which the angels have never lost. This will be brought about by God, the most almighty Founder of that City. For He has promised it, and He cannot lie; and He has shown His good faith by doing many things that He has promised, and many, indeed, that He has not promised.
For He it was Who in the beginning made the world and filled it with all good things, both visible and intelligible. Among these things, He created nothing better than those spirits to whom He gave intelligence, making them capable of contemplating and apprehending Him.
When I began to speak of the City of God, I thought it necessary first of all to answer its enemies, who pursue earthly joys and long only for fleeting things. They rail against the Christian religion, which is the one saving and true religion, for whatever sorrows they suffer in respect of these things. And they do this even though they suffer rather through the mercy of God in admonishing them than from His severity in punishing.
Among those enemies there are many ignorant men whose hatred of us is all the more grievously inflamed by the authority of the learned. For the former believe that the extraordinary events which have occurred in their own day did not occur at all in times gone by; and they are supported in this belief even by those who know it to be false, but who conceal their knowledge in order to seem to have just cause for murmuring against us. It was necessary, therefore, to demonstrate from the books in which their own authors have recorded and published the history of times gone by, that matters are far other than the ignorant suppose. At the same time, it was necessary to teach that the false gods whom once they worshipped openly, and still worship in secret, are most vile spirits and malignant and deceitful demons: so much so that they take delight in crimes which, whether real or fictitious, are nonetheless their own, and which they have desired to have celebrated for them at their own festivals.
Whether any families are to be found in the period after the Flood from Noah to Abraham whose members lived according to God
Was the progress of the Holy City continuous from the time of the Flood onwards, or was it so disrupted by ungodliness that, at times, not one man existed who was a worshipper of the one true God? It is difficult to find in Scripture any clear statement as to this question. But, from the time of Noah, who, with his wife and three sons and their wives, was found worthy to be saved in the Ark from the devastation of the Flood, we do not find, until the time of Abraham, anyone in the canonical books whose godliness is proclaimed by the divine eloquence. The exception to this is when Noah commended his sons Shem and Japheth in his prophetic blessing; for he knew and foresaw what was to happen far in the future. Hence it was also that he cursed his middle son – that is, the one younger than the first-born but older than the last – who had sinned against his father. Noah did not curse Ham in his own person, but in the person of his son, in these words: ‘Cursed be Canaan; a servant of servants shall he be unto his brethren.’ Now Canaan was the son of Ham, who did not cover his sleeping father, but, rather, drew attention to his nakedness.
Of the two orders of the human race, going forward from the beginning towards their different ends
Many opinions have been held, and many things said and written, of the felicity of Paradise: of Paradise itself, the life there of the first human beings, and their sin and its punishment. In previous books, I too have dealt with these things, speaking either according to what I have read in the Holy Scriptures, or according to my own inferences from them consistent with their authority. If a more detailed treatment were to be sought, this would give rise to a great number and variety of arguments requiring more volumes than we here have either need or time for. I do not have time to dwell on all the questions which may be raised by idle and pedantic persons who are more ready to ask questions than capable of understanding the answers.
I think, however, that I have now dealt sufficiently with the great and most difficult questions concerning the beginning of the world and of the soul and of the human race itself. I divide the human race into two orders. The one consists of those who live according to man, and the other of those who live according to God. Speaking allegorically, I also call these two orders two Cities: that is, two societies of men, one of which is predestined to reign in eternity with God, and the other of which will undergo eternal punishment with the devil.
Of the need to discuss the natural theology with those philosophers who are more distinguished in knowledge than the rest
We now have need of a far greater effort of mind than was necessary in solving and explaining the questions raised in our previous books; for it is not with ordinary men that we are now to discuss the theology which the Romans call natural. This is unlike both the mythical theology and the civil – that is, the theatrical theology and the urban, the one of which displays the crimes of the gods, while the other demonstrates their still more criminal desires. Thus, it is with the philosophers that we must here have converse, whose very name, if it is translated into Latin, attests to their love of wisdom.
Moreover, if God, by Whom all things were made, is wisdom, as the divine authority and truth have shown, then the true philosopher is a lover of God. But the thing itself whose name this is does not reside in all who glory in that name; for it does not follow that those who are called philosophers are lovers of true wisdom. Clearly, then, we must select from all those whose written opinions we have been able to study those with whom this question may be not unworthily discussed. For I have not undertaken this work in order to refute the vain opinions of all the philosophers, but only those whose opinions have to do with theology (which Greek word we understand to signify reason or discourse concerning divinity); and not, indeed, all of those.
This translation has been made from the critical edition of B. Dombart and A. Kalb, published in the Corpus Christianorum series (2 vols., Turnhout, 1955). Despite numerous – and inevitable – misprints, this is the best available edition. I have also from time to time consulted the Corpus Scriptorum Ecclesiasticorum Latinorum edition of E. Hoffmann (2 vols., Vienna, 1899–1900). In view of the great length of the text, and in keeping with the general principles of the series of which this translation is a part, footnotes have been kept to a minimum. For very extensive annotation, the reader is referred to the edition of J. E. C. Welldon (see Bibliographical Note). No abbreviations have been used which are not standard or self-explanatory.
Of the fall of the first human beings, through which mortality was incurred
Now that we have dealt with the most difficult questions concerning the origin of our world and the beginning of the human race, the proper order of the discussion that we have here undertaken requires that we next discuss the fall of the first man – or, rather, of the first human beings – and the origin and propagation of human death. For God did not make men like the angels: that is, in such a way that, even if they sinned, they could not by any means die. Rather, if they discharged the duty of obedience, the reward of an angelic immortality and a blessed eternity was to follow without the intervention of death; but if they disobeyed, they were to be most justly punished with the sentence of death. But of this also we have already spoken in the preceding book.
Of that death which can befall the soul even though it is to live for ever, and of that to which the body is subject
I see, however, that I must speak somewhat more carefully of the kind of death with which we are here concerned. For although the human soul is truly said to be immortal, it nonetheless also has a certain kind of death of its own. The soul is called immortal, then, because, at least to some extent, it never ceases to live and feel; whereas the body is called mortal because it can be deprived of all life, and cannot, of itself, live at all.
The point reached in the foregoing argument, and what is left of the question to be discussed
Some have held the opinion that there are both good and bad gods. Others, thinking better of the gods, have ascribed so much honour and praise to them that they have not dared to believe that any god is bad. But those who have said that some gods are good and some bad have included the demons under the name ‘gods’; and sometimes, though more rarely, they have called the gods demons. Indeed, they have pointed out that Jupiter himself, whom they wish to say is the king and prince of the rest of the gods, is called a demon by Homer.
But those who assert that all the gods are good, and that they are far superior to those men who are deemed to be good, are with good reason troubled by the deeds of the demons. They cannot deny them, nor can they attribute them to the gods, who they say are all good; and so they are compelled to make a distinction between gods and demons. Thus, whatever rightly displeases them in the depraved deeds of occult spirits, and in the desires through which they manifest their power, they believe to be due to demons, not gods. They believe also that, because no god has dealings with mankind, these same demons are appointed as mediators between men and the gods, to carry our prayers to them and to bring their answers back.
I am here endeavouring most diligently to uproot and extirpate depraved and ancient opinions which the long-continued error of the human race has implanted deeply and tenaciously in the dark places of the soul; for these opinions are hostile to the truth of godliness. In performing this task, my own small ability is aided by the co-operation of the grace of the true God. Those whose intellects are nimbler and superior, for whom the previous books are sufficient – indeed, more than sufficient – must bear with me patiently and equably for the sake of others, and not deem superfluous what they now feel is not necessary for themselves. For we are here proclaiming a matter of the very first importance: namely, that the true and truly holy Divinity, even though He furnishes us with the help necessary for the frail life that we live now, should nonetheless be sought and worshipped not for the transitory vapour of this mortal life, but for the sake of the blessed life to come, which is nothing less than eternal.
Whether, since it is clear that deity is not to be found in the civil theology, we are to believe that it is to be found among the select gods
This Divinity, or, as we may call it, Deity (for this is a word which our Christian authors do not hesitate to use in order to render more accurately the Greek word theotes) – this Divinity or Deity does not exist in the civil theology as described by Marcus Varro in sixteen books.
We have learned that, in keeping with the promises made to him by God, it was from the seed of Abraham that the Israelite nation took its origin according to the flesh, while all nations take their origin from him according to faith; and the progress of the City of God through the ages will show how these promises are being fulfilled. Since, therefore, the previous book dealt with the period down to the end of David's reign, we shall now touch on other events which followed that reign, in so far as seems sufficient for the work that we have undertaken.
We come now, therefore, to the period extending from when the holy Samuel began to prophesy, down to the time when the people of Israel were led away captive into Babylon, and then to the point, seventy years later, when, after the return of the Israelites, the house of God was restored according to the prophecy of the holy Jeremiah. The whole of this time is the age of the prophets. We can, of course, without impropriety give the name of prophet to Noah himself, in whose days the whole earth was destroyed by the Flood; and to others also, both before and after him, down to the time when there began to be kings among the people of God.
That all men would have been plunged into an everlasting second death by the sin of the first man, had not God's grace redeemed many
As I have already said in the preceding books, God chose to create the human race from one single man. His purpose in doing this was not only that the human race should be united in fellowship by a natural likeness, but also that men should be bound together by kinship in the unity of concord, linked by the bond of peace. And the individual members of this race would not have been subject to death, had not the first two – one of whom was created from no one, and the other from him – merited it by their disobedience. So great was the sin of those two that human nature was changed by it for the worse; and so bondage to sin and the necessity of death were transmitted to their posterity.
Now the sway of the kingdom of death over men was so complete that all would have been driven headlong, as their due punishment, into that second death to which there is no end, had not some of them been redeemed by the unmerited grace of God. Thus it is that, though there are a great many nations throughout the world, living according to different rites and customs, and distinguished by many different forms of language, arms and dress, there nonetheless exist only two orders, as we may call them, of human society; and, following our Scriptures, we may rightly speak of these as two cities.