CLASSICAL MECHANICS
SOLUTIONS MANUAL
R. Douglas Gregory
November 2006
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2
Contents 1 The algebra and calculus of vectors
3
2 Velocity, acceleration and scalar angular velocity
27
3 Newton’s laws of motion and the law of gravitation
62
4 Problems in particle dynamics
76
5 Linear oscillations and normal modes
139
6 Energy conservation
179
7 Orbits in a central field
221
8 Non-linear oscillations and phase space
276
9 The energy principle
306
10 The linear momentum principle
335
11 The angular momentum principle
381
12 Lagrange’s equations and conservation principles
429
13 The calculus of variations and Hamilton’s principle
473
14 Hamilton’s equations and phase space
505
15 The general theory of small oscillations
533
16 Vector angular velocity
577
17 Rotating reference frames
590
18 Tensor algebra and the inertia tensor
615
19 Problems in rigid body dynamics
646
Chapter One The algebra and calculus of vectors
c Cambridge University Press, 2006
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Chapter 1 The algebra and calculus of vectors
Problem 1 . 1
In terms of the standard basis set fi ; j ; kg, a D 2 i c D i 5 j C 3 k.
j
2k, b D 3 i
4 k and
Find 3 a C 2 b 4 c and j a b j2 . Find j a j, j b j and a b. Deduce the angle between a and b. Find the component of c