Book contents
- Frontmatter
- Contents
- Preface to the second edition
- Preface to the first edition
- 1 Preliminary algebra
- 2 Preliminary calculus
- 3 Complex numbers and hyperbolic functions
- 4 Series and limits
- 5 Partial differentiation
- 6 Multiple integrals
- 7 Vector algebra
- 8 Matrices and vector spaces
- 9 Normal modes
- 10 Vector calculus
- 11 Line, surface and volume integrals
- 12 Fourier series
- 13 Integral transforms
- 14 First-order ordinary differential equations
- 15 Higher-order ordinary differential equations
- 16 Series solutions of ordinary differential equations
- 17 Eigenfunction methods for differential equations
- 18 Partial differential equations: general and particular solutions
- 19 Partial differential equations: separation of variables and other methods
- 20 Complex variables
- 21 Tensors
- 22 Calculus of variations
- 23 Integral equations
- 24 Group theory
- 25 Representation theory
- 26 Probability
- 27 Statistics
- 28 Numerical methods
- Appendix Gamma, beta and error functions
- Index
9 - Normal modes
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface to the second edition
- Preface to the first edition
- 1 Preliminary algebra
- 2 Preliminary calculus
- 3 Complex numbers and hyperbolic functions
- 4 Series and limits
- 5 Partial differentiation
- 6 Multiple integrals
- 7 Vector algebra
- 8 Matrices and vector spaces
- 9 Normal modes
- 10 Vector calculus
- 11 Line, surface and volume integrals
- 12 Fourier series
- 13 Integral transforms
- 14 First-order ordinary differential equations
- 15 Higher-order ordinary differential equations
- 16 Series solutions of ordinary differential equations
- 17 Eigenfunction methods for differential equations
- 18 Partial differential equations: general and particular solutions
- 19 Partial differential equations: separation of variables and other methods
- 20 Complex variables
- 21 Tensors
- 22 Calculus of variations
- 23 Integral equations
- 24 Group theory
- 25 Representation theory
- 26 Probability
- 27 Statistics
- 28 Numerical methods
- Appendix Gamma, beta and error functions
- Index
Summary
Any student of the physical sciences will encounter the subject of oscillations on many occasions and in a wide variety of circumstances, for example the voltage and current oscillations in an electric circuit, the vibrations of a mechanical structure and the internal motions of molecules. The matrices studied in the previous chapter provide a particularly simple way to approach what may appear, at first glance, to be difficult physical problems.
We will consider only systems for which a position-dependent potential exists, i.e., the potential energy of the system in any particular configuration depends upon the coordinates of the configuration, which need not be be lengths, however; the potential must not depend upon the time derivatives (generalised velocities) of these coordinates. So, for example, the potential —qv. A used in the Lagrangian description of a charged particle in an electromagnetic field is excluded. A further restriction that we place is that the potential has a local minimum at the equilibrium point; physically, this is a necessary and sufficient condition for stable equilibrium. By suitably defining the origin of the potential, we may take its value at the equilibrium point as zero.
We denote the coordinates chosen to describe a configuration of the system by qi, i = 1, 2, …, N. The qi need not be distances; some could be angles, for example.
- Type
- Chapter
- Information
- Mathematical Methods for Physics and EngineeringA Comprehensive Guide, pp. 322 - 339Publisher: Cambridge University PressPrint publication year: 2002