Modern Introductory Mechanics Walter Wilcox Download free books at WALTER WILCOX MODERN INTRODUCTORY MECHANICS Download free eBooks at bookboon.com Modern Introductory Mechanics 2nd edition © 2015 Walter Wilcox & bookboon.com ISBN 978-87-403-0855-6 Download free eBooks at bookboon.com ENDNOTES MODERN INTRODUCTORY MECHANICS 283 CONTENTS Mathematical Review Newtonian Mechanics 45 Linear Oscillations 89 Nonlinear Oscillations 139 Gravitation 178 Calculus of Variations 190 Lagrangian and Hamiltonian Mechanics 235 Endnotes 283 www.sylvania.com We not reinvent the wheel we reinvent light Fascinating lighting offers an ininite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges An environment in which your expertise is in high demand Enjoy the supportive working atmosphere within our global group and beneit from international career paths Implement sustainable ideas in close cooperation with other specialists and contribute to inluencing our future Come and join us in reinventing light every day Light is OSRAM Download free eBooks at bookboon.com Click on the ad to read more MATHEMATICAL REVIEW MODERN INTRODUCTORY MECHANICS MATHEMATICAL REVIEW TRIGONOMETRY Mathematics is the language of physics, so we must all have a certain luency he irst order of business is to remind ourselves of some basic relations from trigonometry opposite , hypotenuse adjacent cos θ = , hypotenuse opposite sin θ tan θ = = adjacent cos θ sin θ = For right now just think of a vector as something with both a magnitude and a direction: A A2 A1 Vector notation: A = (A 1, A ) Vector addition: A2 A+ B A B2 B A1 B1 (A + B)1 = A1 + B1 ; (A + B)2 = A2 + B2 Download free eBooks at bookboon.com MATHEMATICAL REVIEW MODERN INTRODUCTORY MECHANICS Unit Vectors: ^ e3 ^ e2 ^ e1 hese are handy guys which point along the 1,2 or directions with unit magnitude We always choose a right-handed coordinate system in this class he right-hand rule identiies to in the above igure; your such a system: curl the ingers of your right hand from direction thumb will point in the Matrices A matrix is a collection of entries which can be arranged in row/column form: columns rows A 11 A 21 A 31 A 12 A 22 A 32 A13 A 23 A 33 A single generalized matrix element is denoted: Aij row column + B11matrices): Addition of matrices (number of rows and columns the same for both A 12and A B A 11 A 12 + B11 B12 = A 11 + B11 A 12 + B12 (1.1) A 21 A 22 B21 B22 A 21 + B21 A 22 + B22 Download free eBooks at bookboon.com MATHEMATICAL REVIEW MODERN INTRODUCTORY MECHANICS In more abstract language (“index notation”) this is just (A + B)ij = Aij + Bij, (1.2) where and are taking on all possible values independently In the above equation and are said to be “free” indices he free indices on one side of an equality must always be the same on the other side Multiplication of matrices (Here we only require that the number of columns of A 12equal A + B11 ): the number of rows of B A 11 B1 + A12 B2 A A B 11 12 = (1.3) A 21 A 22 B2 A 21 B1 + A 22B2 Another example: A 11 A 21 A 12 B11 A 22 B21 B12 A 11 B11 + A 12B21 B22 = A 21 B11 + A 22B21 A11 B12 + A 12B22 A 21 B12 + A 22B22 (1.4) Notice that the result has the same number of rows as A and the same number of columns + B11 In index language, these two examples can be written much more compactly as: as B (1.5) (1.6) Note that dummy indices are ones which are summed over all of their values Unlike free indices, which must be the same on both sides of an equation, dummy indices can appear on either side Also notice that dummy indices always appear twice on a given side of an equation hese rules trip up many beginning students of mechanics For reference, here is a summary of the understood “index jockey” rules for index manipulations: “Dummy” indices are those which are summed Each such index always appears exactly twice One can interpret this sum as matrix multiplication only if the indices can be placed directly next to each other Separate summation symbols must be used for independent summations Download free eBooks at bookboon.com MATHEMATICAL REVIEW MODERN INTRODUCTORY MECHANICS In general, one can not change the order of indices on an object, such as Aij (Occasionally one knows efect of interchanging indices; see later comments on symmetric and antisymmetric matrices.) Free indices are those that are unsummed In general, each free index appears once on both sides of a given equation Identity matrix (3 × context): 1 0 0 0 0 1 he identity matrix is often simply written as the number “1”, or is absent altogether in contexts where its presence is unambiguous (Physicists must learn to read behind the lines for the meaning!) We will need three additional matrix operations AA-1 = 1 Inverse: (1.7) he “1” on the right hand side here means the identity matrix A heorem from linear -1 = implies A-1A = (Can you prove this?) Finding algebra establishes that AA general is fairly complicated For most of the matrices we will encounter, inding A-1Ain = -1 A Awill= be easy (I’m thinking of rotation matrices, which will follow shortly.) Notice: (AB)-1(AB) = 1, (AB)-1A = B-1, => (AB)-1 = B-1A-1 Transpose: (1.8) T = A Aij ji (1.9) Examples of the transpose operation: B1 B = , B2 "column matrix" A A 11 A = A 21 BT = (B1 B2) "row matrix" A A A 12 A 11 T A 22 , A = A 12 A A 21 A 22 Download free eBooks at bookboon.com MATHEMATICAL REVIEW MODERN INTRODUCTORY MECHANICS Also (AB) Tij = (AB)ji , = ∑A jk Bki , k = ∑B T ik T , A kj k = (BTAT)ij , => (AB)T = BTAT (1.10) Determinant: (1.12) Note that and that and We’ll ind a more elegant deinition of the determinant later An important is not zero point to realize is that the inverse of a matrix, A, exists only if An important point about linear algebra will also be called upon in later chapters A system of linear (only x1,2,3 appear, never (x1,2,3)2 or higher powers) homogeneous (the right hand side of the following equations are zero) equations, A x + A x + A x = 0, 12 13 11 A 21x1 + A 22x2 + A 23x3 = 0, A31x1 + A32x2 + A33x3 = 0, (1.13) has a nontrivial ( x1,2,3 are not all zero!) solution if and only if A 11 det A 21 A 31 A 12 A 22 A 32 A13 A 23 = A 33 (1.14) Download free eBooks at bookboon.com ...WALTER WILCOX MODERN INTRODUCTORY MECHANICS Download free eBooks at bookboon.com Modern Introductory Mechanics 2nd edition © 2015 Walter Wilcox & bookboon.com... 978-87-403-0855-6 Download free eBooks at bookboon.com ENDNOTES MODERN INTRODUCTORY MECHANICS 283 CONTENTS Mathematical Review Newtonian Mechanics 45 Linear Oscillations 89 Nonlinear Oscillations... Download free eBooks at bookboon.com Click on the ad to read more MATHEMATICAL REVIEW MODERN INTRODUCTORY MECHANICS MATHEMATICAL REVIEW TRIGONOMETRY Mathematics is the language of physics, so