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These notes were written during the Fall, , and Winter, , terms. They are indeed lecture notes – I literally lecture from these notes. They combine. CLASSICAL MECHANICS. THIRD EDITION. Herbert Goldstein. Columbia University. Charles Poole. University of South Carolina. John Safko. University of . According to classical physics, “reality” takes place in a product space . LMT class in mechanics, and suppose we change our system of units.

Motion in 1 dimension, Motion in 3 dimension, Conservation of energy, Newton's laws of motion, Conservation of momentum, Circular motion, Rotational motion, Angular momentum, Statics, Oscillatory motion, Orbital motion and Wave motion. In this text, the author constructs the mathematical apparatus of classical mechanics from the beginning, examining all the basic problems in dynamics, including the theory of oscillations, the theory of rigid body motion, and the Hamiltonian formalism.

Currently this section contains no detailed description for the page, will update this page soon. About Us Link to us Contact Us. Free Classical Mechanics Books. Classical Mechanics Books This section contains free e-books and guides on Classical Mechanics, some of the resources in this section can be viewed online and some of them can be downloaded.

Classical Mechanics by Radovan Dermisek This note covers the following topics: Radovan Dermisek NA Pages.

Tom Kirchner Pages. Variational Principles In Classical Mechanics The goal of this book is to introduce the reader to the intellectual beauty, and philosophical implications, of the fact that nature obeys variational principles that underlie the Lagrangian and Hamiltonian analytical formulations of classical mechanics.

Douglas Cline Pages. Classical Mechanics a Critical Introduction The first draft of this book was composed many years ago and was intended to serve either as a stand-alone text or as a supplementary tutor for the student.

Michael Cohen Pages. Eric D Hoker Pages. Iain Stewart This lecture note covers Lagrangian and Hamiltonian mechanics, systems with constraints, rigid body dynamics, vibrations, central forces, Hamilton-Jacobi theory, action-angle variables, perturbation theory, and continuous systems.

Iain W. Stewart NA Pages. Classical Mechanics Konstantin Likharev This note covers the following topics: Konstantin Likharev Pages. Joel Franklin NA Pages. Classical Mechanics by Matthew Hole Classical mechanics is the abstraction and generalisation of Newton's laws of motion undertaken, historically, by Lagrange and Hamilton.

Matthew Hole NA Pages. Classical Mechanics and Dynamical Systems This note explains the following topics: Martin Scholtz Pages. Classical Mechanics by Robert L. Dewar This note covers the following topics: Robert L. Dewar Pages. Classical Mechanics by Tom W. Kibble and Frank H. Berkshire This book is designed for students with some previous acquaintance with the elementary concepts of mechanics, but the book starts from first principles, and little detailed knowledge is assumed.

Tom W. Berkshire NA Pages. Classical Mechanics by Charles B. Thorn This book explains the following topics: Charles B. Thorn 76 Pages. Classical Mechanics by H. Rosu This note covers the following topics: Rosu Pages. Lecture Notes on Classical Mechanics The level of this note is appropriate for an advanced under graduate or a first year graduate course in classical mechanics.

Author Corner

Daniel Arovas Pages. Classical Mechanics a Critical Introduction This book covers the standard topics and the fundamental concepts of mechanics. The Outer Product The algebra of scalars and vectors based on the rules just mentioned has been so widely accepted as to be routinely employed by mathematicians and physicists today.

As it stands, however, this algebra is still incapable of providing a full expression of geometrical ideas. Yet there is nothing close to a consensus on how to overcome this limitation. Rather there is a great proliferation of different mathematical systems designed to express geometrical ideas — tensor algebra, matrix algebra, spinor algebra — to name just a few of the most common.

It might be thought that this profusion of systems reveals the richness of mathema- The Outer Product 21 tics. On the contrary, it reveals a widespread confusion — confusion about the aims and principles of geometric algebra.

The intent here is to clarify these aims and principles by showing that the preceding arguments leading to the invention of scalars and vectors can be continued in a natural way, culminating in a single mathematical system which facilitates a simple expression of the full range of geometrical ideas.

The principle that the product of two vectors ought to describe their relative directions presided over the definition of the inner product. But the inner product falls short of a complete fulfillment of that principle, because it fails to express the fundamental geometrical fact that two non-parallel lines determine a plane, or, better, that two non-collinear directed line segments determine a parallelogram.

But to make this possibility a reality, the notion of number must again be generalized. A parallelogram can be regarded as a directed plane segment. Just as vectors were invented to characterize the notion of a directed line segment, so a new kind of directed number, called a bivector or 2-vector, can be introduced to characterize the notion of directed plane segment Figure 5. Like 22 Origins of Geometric Algebra a vector, a bivector has magnitude, direction and orientation, and only these properties.

Just as the direction of a vector corresponds to an oriented straight line, so the direction of a bivector corresponds to an oriented flat plane. The distinction between these two kinds of direction involves the geometrical notion of dimension or grade.

Accordingly, the direction of a bivector is said to be 2-dimensional to distinguish it from the 1-dimensional direction of a vector. And it is sometimes convenient to call a vector a 1-vector to emphasize its dimension. Also, a scalar can be regarded as a 0-vector to indicate that it is a 0-dimensional number.

Since, as already shown, the only directional property of a scalar is its orientation, orientation can be regarded as a 0-dimensional direction. Thus the idea of numbers with different geometrical dimension begins to take shape. In ordinary geometry the concepts of line and plane play roles of comparable significance. It may be a good idea to point out that both line and plane, as commonly conceived, consist of a set of points in definite relation to one another.

It is the nature of this relation that distinguishes line from plane. A single vector completely characterizes the directional relation of points in a given line.

A single bivector completely characterizes the directional relation of points in a given plane. Thus, the notion of a plane as a relation can be separated from the notion of a plane as a point set.

After the directional properties of planes and lines have been fully incorporated into an algebra of directed numbers, the geometrical properties of point sets can be more easily and completely described than ever before, as we shall see.

Now return to the problem of giving algebraic expression to the relation of line segments to plane segments. Note that a point moving a distance and direction specified by a vector a sweeps out a directed line segment.

And the points on this line segment, each moving a distance and direction specified by a vector b, sweep out a parallelogram, Figure 5. The bivector is said to be the outer product of vectors a and b.

This is expressed by saying that the outer product is anticommutative. The relation of vector orientation to bivector orientation is fixed by the rule This rule, like the others, follows from the correspondence of vectors and bivectors with oriented line segments and plane segments. This formula expresses the relation between vector magnitudes and bivector magnitudes. The relation to is given in 5. Scalar multiplication can be defined for bivectors in the same way as it was for vectors.

For bivectors C and B and scalar the equation means that the magnitude of B is dilated by the magnitude of that is, and the direction of C is the same as that of B if is positive, or opposite to it if is negative. This last stipulation can be expressed by equations for multiplication by the unit scalars one and minus one: Bivectors which are scalar multiples of one another are said to be codirectional.

Scalar multiplications of vectors and bivectors are related by the equation For this is equivalent to Equation 5. For positive Equation 5. Note that, by 5. Adopting the principle, already applied to vectors, that a directed number is zero if and only if its magnitude is zero, it follows that if and only if Hence, the outer product of nonzero vectors is zero if and only if they are collinear, that is, if and only if Note that if 5.

The Outer Product 25 The relation of addition to outer multiplications is determined by the distributive rule: The corresponding geometrical construction is illustrated in Figure 5. Note that 5. So the algebraic properties and the geometrical interpretation of bivector addition are completely determined by the properties and interpretation already accorded to vector addition.

For example, the sum of two bivectors is a unique bivector, and again, bivector addition is associative. For 26 Origins of Geometric Algebra bivectors with the same direction, it is easily seen that the distributive rule 5. Both the inner and outer products are measures of relative direction, but they complement one another.

Relations which are difficult or impossible to obtain with one may be easy to obtain with the other. To illustrate the point, reconsider the vector equation for a triangle, which was analyzed above with the help of the inner product. Take the outer product of successively with vectors a, b, c, and use the rules 5.

It is convenient to write the first two equations on a single line, like so: Here are three different ways of expressing the same bivector as a product of vectors. This gives three different ways of expressing its magnitude: Using 5.

We shall see in Chapter 2 that all the formulas of plane and spherical trigonometry can be easily derived and compactly expressed by using inner and outer products. The theory of the outer product as described so far calls for an obvious generalization. Thus, the points on an oriented parallelogram specified by the bivector moving a distance and direction specified by a vector c sweep out an oriented parallelepiped Figure 5,5 , which may be characterized by a new kind of directed number T called a trivector or 3-vector.

The properties of T are fixed by regarding it as equal to the outer product of the bivector with the vector c. So write The study of trivectors leads to results quite analogous to those obtained above for bivectors, so the analysis need not be carried out in detail.

But one The Outer Product 27 new result obtains, namely, the conclusion that outer multiplication should obey the associative rule: The geometric meaning of associativity can be ascertained with the help of the following rule: This is an instance of the general rule that the orientation of a product is reversed by reversing the orientation of one of its factors.

Repeated applications of 5. Accordingly, write This equation provides a simple algebraic way of saying that 3 lines with directions denoted by vectors a, b, c lie in the same plane, just as Equation 5. Like any other directed number, a 3-vector has magnitude, direction and 28 Origins of Geometric Algebra orientation, and only these properties. The dimensionality of a 3-vector is expressed by the fact that it can be factored into an outer product of three vectors, though this can be done in an unlimited number of ways.

The magnitude of is denoted by and is equal to the volume of the parallelepiped determined by the vectors a, b and c. The orientation of a trivector depends on the order of its factors. The anticommutation rule together with the associative rule imply that exchange of any pair of factors in a product reverses the orientation of the result. For instance, Thus the idea of relative orientation is very easily expressed with the help of the outer product.

Without such algebraic apparatus the geometrical idea of orientation is quite difficult to express, and, not surprisingly, was only dimly understood before the invention of vectors and the outer product. The essential aspects of outer multiplication and the generalized notions of number and direction it entails have now been set down.

No fundamentally new insights into the relations between algebra and geometry are achieved by considering the outer product of four or more vectors. But it should be mentioned that if vectors are used to describe the 3-dimensional space of ordinary geometry, then displacement of the trivector in a direction specified by d fails to sweep out a 4-dimensional space segment.

So write The parenthesis is unnecessary because of the associative rule 5. Equation 5. This is a simple way of saying that space is 3-dimensional. Note the similarity in form and meaning of Equations 5. It should be clear that 5. By supposing that the outer product of four vectors is not zero, one is led to an algebraic description of spaces and geometries of four or more dimensions, but we already have what we need to describe the geometrical properties of physical space.

The outer product was invented by Hermann Grassmann, and, following a line of thought similar to the one above, developed into a complete mathematical theory before the middle of the nineteenth century. His theory has been accorded a prominent place in mathematics only in the last forty years, and it is hardly known at all to physicists.

Grassmann himself was the only one to use it during the first two decades after it was published. There are several reasons for this.

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He was the first person to arrive at the modern conception of algebra as a system of rules relating otherwise undefined entities. He realized that the nature of the outer product could be defined by specifying the rules it obeys, especially the distributive, associative, and anticommutive rules given above. He rightly expounded this momentous insight in great detail. And he proved its significance by showing, for the first time, how abstract algebra can take us The Outer Product 29 beyond the 3-dimensional space of experience to a conception of space with any number of dimensions.

Unfortunately, in his enthusiasm for abstract developments, Grassmann deemphasized the geometric origin and interpretation of his rules.

A rectangle is the geometric product of its base and height, and this product behaves in the same way as the arithmetic product.

The Greeks made frequent use of the correspondence between the product of numbers and the construction of a parallelogram from its base and height.

Classical Mechanics

For example, Euclid represented the distributive rule of algebra as addition of areas and proved it as a geometrical theorem. This correspondence between arithmetic and geometry was rejected by Descartes and duly ignored by the mathematicians that followed him. However, as already explained, Descartes merely associated arithmetic multiplication with a different geometric construction. But the truly significant advance, from the idea of a geometrical product to its full algebraic expression by outer multiplication, was made by his son.

Hermann Grassmann completed the algebraic formulation of basic ideas in Greek geometry begun by Descartes. The Greek theory of ratio and proportion is now incorporated in the properties of scalars and scalar multiplication.

The Greek idea of projection is incorporated in the inner product. And the Greek geometrical product is expressed by outer multiplication. The invention of a system of directed numbers to express Greek geometrical notions makes it possible, as Descartes had already said, to go far beyond the geometry of the Greeks.

It also leads to a deeper appreciation of the Greek accomplishments. It corresponds roughly to the distinction between scalar and vector. Only 30 Origins of Geometric Algebra in the work of Grassmann are the notions of direction, dimension, orientation and scalar magnitude finally disentangled. But his great accomplishment would have been impossible without the earlier vague distinction of the Greeks, and perhaps without its reformulation in quasi-arithmetic terms by his father.

Synthesis and Simplification Grassmann was the first person to define multiplication simply by specifying a set of algebraic rules.

By systematically surveying various possible rules, he discovered several other kinds of multiplication besides his inner and outer products. Nevertheless, he overlooked the most important possibility until late in his life, when he was unable to follow up on its implications.

There is one fundamental kind of geometrical product from which all other significant geometrical products can be obtained. All the geometrical facts needed to discover such a product have been mentioned above.

New Foundations for Classical Mechanics

It has already been noted that the inner and outer products seem to complement one another by describing independent geometrical relations. This circumstance deserves the most careful study. The simplest approach is to entertain the possibility of introducing a new kind of product ab by the equation Here the scalar has been added to the bivector At first sight it may seem absurd to add two directed numbers with different grades.

That may have delayed Grassmann from considering it. It is a kind of mathematical taboo — its real justification unknown or forgotten. It is supposedly obvious that you cannot add apples and oranges or feet and square feet. On the contrary, it is only obvious that addition of apples and oranges is not usually a practical thing to do — unless you are making a salad.

Absurdity disappears when it is realized that 6. All that mathematics really requires is that the indicated relations and operations be well defined and consistently employed. The mathematical meaning of adding scalars and bivectors is determined by specifying that such addition satisfy the usual commutative and associative rules.

With this understood, it now can be shown that the properties of the new product are almost completely determined by the obvious requirement that they be consistent with the properties already accorded to the inner and outer products. Synthesis and Simplification 31 The commutative rule together with the anticommutative rule imply a relation between ab and ba. Thus, Comparison of 6. However, if then And if then It should not escape notice that to get 6.

The product ab inherits a geometrical interpretation from the interpretations already accorded to the inner and outer products. It is an algebraic measure of the relative direction of vectors a and b. Thus, from 6. But more properties of the product are required to understand its significance when the relative direction of two vectors is somewhere between the extremes of collinearity and orthogonality.

To give due recognition to its geometric significance ab will henceforth be called the geometric product of vectors a and b. From the distributive rules 4.

Equation 6. The distributive rules 6. To derive them, the distributive rules for both the inner and outer products were needed. The relation of scalar multiplication to the geometric product is described by the equations 32 Origins of Geometric Algebra This is easily derived from 4. It says that scalar and vector multiplication are mutually commutative and associative. If the commutative rule is separated from the associative rule, it takes the simple form Now observe that by taking the sum and difference of equations 6.

Instead of regarding 6. It is curious, then, to note that by 6. The algebraic properties of the geometric product of two vectors have already been ascertained.

It should be evident that the corresponding properties of the inner and outer products can be derived from the definitions 6. The next task is to examine the geometric product of three vectors a, b, c. It is certainly desirable that this product satisfy the associative rule for that greatly simplifies algebraic manipulations.

But it must be shown that this rule reproduces established properties of the inner and outer products. This can be done by examining the product of a vector with a bivector. The quantity is something new; as the notation suggests, it is to be regarded as a generalization of the inner product of vectors.

Note that 6. The sign in 6. To this end, it is sufficient to show that 6. Utilizing the definitions 6. Now to understand the significance of let Use the definitions as before to eliminate the dot and wedge: To this, add and collect terms to get Thus This shows that inner multiplication of a vector with a bivector results in a vector. So Equation 6. Axioms for Geometric Algebra Let us examine now what we have learned about building a geometric algebra.

To begin with, the algebra should include the graded elements 0-vector, 1-vector, 2-vector and 3-vector to represent the directional properties of points, lines, planes and space.

We introduced three kinds of multiplication, the scalar, inner and outer products, to express relations among the elements. But we saw that inner and outer products can be reduced to a single geometric product if we allow elements of different grade or dimension to be added. However, now we aim to improve the precision of our language with an axiomatic formulation of the basic concepts.

Thus, 7. Any element of the geometric algebra can be called a multivector, because it can be represented in the form 7. For example, a vector a can be expressed trivially in the form 7. Note the k-vectors which are not scalars are denoted by symbols in boldface type. Now another simplification becomes possible. It will be noted that the geometric product of vectors which we have just considered has, except for commutivity, the same algebraic properties as scalar multiplication of vectors Axioms for Geometric Algebra 35 and bivectors.

In particular, both products are associative and distributive with respect to addition. Rather than regard them as two different kinds of multiplication, we can regard them as instances of a single geometric product among different kinds of multivector.

Thus, scalar multiplication is the geometric product of any multivector by a special kind of multivector called a scalar. The special geometric nature of the scalars is expressed algebraically by the fact that they commute with every other multivector. Thus, we define addition and multiplication of multivectors by the following familiar rules: For multivectors A, B, C,.

It is hardly necessary to discuss the significance of the above axioms, since they are familiar from the elementary algebra of scalars. They can be used to manipulate multivectors in exactly the same way that numbers are manipulated in arithmetic. For example, axiom 7. To complete our system of axioms for geometric algebra, we need some axioms that characterize the various kinds of k-vectors.

First of all, we assume that the set of all scalars in the algebra can be identified with the real numbers, and we express the commutivity of scalar multiplication by the axiom for every scalar and multivector A.

Vectors are characterized by the following axiom. It will be most convenient to do that after we introduce a couple of definitions. For a vector and any k-blade we define the inner product by and the outer product by adding these equations, we get Note that 7.

Using the definitions 7. According to 7. To assure this, we need one more axiom: For every vector a and 3-vector By virtue of the definition 7.

Finally, to assure that the whole algebraic system is not vacuous, we must assume that nonzero multivectors with all grades actually exist.

This completes our formulation of the axioms for geometric algebra. We have neglected some logical fine points e. We have chosen a notation for geometric algebra that is as similar as possible to the notation of scalar algebra.

This is a point of great importance, for it facilitates the transfer of skills in manipulations with scalar algebra to manipulations with geometric algebra. Let us note exactly how the basic operations of scalar algebra transfer to geometric algebra.

Axioms 7. Except for the absence of a general commutative law for multiplication, they are identical to the axioms of scalar algebra. Therefore, multivectors can be equated, added, subtracted and multiplied in Axioms for Geometric Algebra 37 exactly the same way as scalar quantities, provided one does not reorder multiplicative factors which do not commute.

Division by multivectors can be defined in terms of multiplication, just as in scalar algebra. But we need to pay special attention to notation on account of noncommutivity, so let us consider the matter explicitly.

In geometric algebra, as in elementary algebra, the solution of equations is greatly facilitated by the possibility of division. We can divide by a multivector A if it has a multiplicative inverse.

It should be noted, however, that some multivectors do not have multiplicative inverses see Exercise 7. Exercises Hints and solutions for selected exercises are given at the back of the book. Specify the justification for each step in the proof. The proofs in geometric algebra are identical to those in elementary algebra. Let where is a scalar and a is a nonzero vector.

What conditions on and a imply that does not exist? It can be proved that every multivector which does not have an inverse has an idempotent for a factor. Prove that every left inverse is also a right inverse and that this inverse is unique. Chapter 2 Developments in Geometric Algebra In Chapter 1 we developed geometric algebra as a symbolic system for representing the basic geometrical concepts of direction, magnitude, orientation and dimension. In this chapter we continue the development of geometric algebra into a full-blown mathematical language.

The basic grammar of this language is completely specified by the axioms set down at the end of Chapter 1.

But there is much more to a language than its grammar! To develop geometric algebra to the point where we can express and explore the ideas of mechanics with fluency, in this chapter we introduce auxiliary concepts and definitions, derive useful algebraic relations, describe simple curves and surfaces with algebraic equations, and formulate the fundamentals of differentiation and integration with respect to scalar variables.

Further mathematical developments are given in Chapter 5. Basic Identities and Definitions In Chapter 1 we were led to the geometric product for vectors by combining inner and outer products according to the equation Then we reversed the procedure, defining the inner and outer products in terms of the geometric product by the equations This did more than reduce two different kinds of multiplication to one.

It made possible the formulation of a simple axiom system from which an unlimited number of geometrical relations can be deduced by algebraic manipulation. In this section we aim to improve our skills at carrying out such deductions and establish some widely useful results.

The inner and outer products appear frequently in applications, because they have straightforward geometrical interpretations, as we saw in Chapter 39 40 Developments in Geometric Algebra 1.

For this reason, it is often desirable to operate directly with inner and outer products, even though we regard the geometric product as more fundamental. To make this possible, we need a system of algebraic identities relating inner and outer products. We derive these identities, of course, by using the geometric product and the axioms of geometric algebra set down in Section The different products are most easily related by the equation which generalizes 1.

To illustrate the use of 1. Beginning with the associative rule for the geometric product, we use 1. Since vectors are distinct from trivectors, we can separately equate vector and trivector parts on each side of the equation.

By equating trivector parts, we get the associative rule And by equating the vector parts we find an algebraic identity which we have not seen before, For more about this identity, see Exercise 1. This derivation of the associative rule 1. That derivation was considerably more complicated, because it employed a direct reduction of the outer product to the geometric product. Basic Identities and Definitions 41 Note the general structure of the method: an identity involving geometric products alone is expanded into inner and outer products by using 1.

This will be our principal method for establishing identities involving inner and outer products. As another example, note that the method immediately gives the distributive rules for inner and outer products. Thus, if a is a vector and and are r-blades, then by applying 1. These examples show the importance of separating a multivector or a multivector equation into parts of different grade. So it will be useful to introduce a special notation to express such a separation.

Accordingly, we write to denote the r-vector part of a multivector A. For example, if this notation enables us to write for the trivector part, for the vector part, while the vanishing of scalar and bivector parts is described by the equation According to axiom 7. A multivector A is said to be even odd if when r is an odd even integer. Obviously every multivector A can be expressed as a sum of an even part and an odd part Thus where 42 Developments in Geometric Algebra We shall see later that the distinction between even and odd multivectors is important, because the even multivectors form an algebra by themselves but the odd multivectors do not.

According to 1. We adopted this condition in Chapter 1 to express the fact that physical space is three dimensional, so we will be assuming it in our treatment of mechanics throughout this book.

However, such a condition is not essential for mathematical reasons, and there are other applications of geometric algebra to physics where it is not appropriate. For the sake of mathematical generality, therefore, all results and definitions in this section are formulated without limitations on grade, with the exception, of course, of 1.

This generality is achieved at very little extra cost, and it has the advantage of revealing precisely what features of geometric algebra are peculiar to three dimensions. Before continuing, it will be worthwhile to discuss the use of parentheses in algebraic expressions. The two interpretations give completely different algebraic results.

To remove such ambiguities without using parentheses, we introduce the following precedence convention: If there is ambiguity, indicated inner and outer products should be performed before an adjacent geometric product. Thus This convention eliminates an appreciable number of parentheses, especially in complicated expressions. The most useful identity relating inner and outer products is, of course, its simplest one: We derived this in Section before we had established our axiom system for geometric algebra.

Now we can derive it by a simpler method. First we use 1. By the same method we can derive the more general reduction formula where a and b are vectors while is an r-blade. We use 1. By iterating 1. Equation 1. Our definitions 1. It will be useful to generalize these definitions to apply to blades of any grade.More about Momentum of Rotation cont. Classical Mechanics a Critical Introduction The first draft of this book was composed many years ago and was intended to serve either as a stand-alone text or as a supplementary tutor for the student.

A few words are in order about the unique treatment of these two topics, namely, rotational dynamics and celestial mechanics. They are not part of algebra, yet ordinary algebra cannot be applied to geometry without them. The accuracy of the text has been improved by the accumulation of many corrections over the last decade. Mathematics today is an immense and imposing subject, but there is no reason to suppose that the evolution of a mathematical language for physics is complete.

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