Basic Theoretical Physics: A Concise Overview P8. This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I . Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers’ insight and. | 64 10 Coupled Small Oscillations all eigenvalues are 0 except for the above-mentioned six exceptional cases where they are zero the six so-called Goldstone modes . Writing 0 i Xi xi ui and neglecting terms of third or higher order in ui we obtain l E mrua - 2 E vuu - v 0 . a fl Here all masses ma can be replaced by 1 in the original equation if one adds a symbol . by the substitution ua maua. Thus we have 3N i2 3N L E u. - 2 E Vfuu - V 0 . a 1 a fi 1 Here V . v va p mamp is again a symmetric matrix which can also be diagonalized by a rotation in R n and now the rotation leaves also the kinetic energy invariant . The diagonal values eigenvalues of the matrix are positive with the abovementioned exception so they can be written as x with xa 0 for a 1 . 3N including the six zero-frequencies of the Goldstone modes. The xa are called normal frequencies and the corresponding eigenvectors are called normal modes see below . One should of course use a cartesian basis corresponding to the diagonalized quadratic form . to the directions of the mutually orthogonal eigenvectors. The related cartesian coordinates Qv with v 1 . 3N are called normal coordinates. After diagonalization1 the Lagrangian is apart from the unnecessary additive constant V 0 3N L - - Q2 2Q2 L 2 y Qv XV QV . v 1 Previously the oscillations were coupled but by rotation to diagonal form in R3N they have been decoupled. The Hamiltonian corresponds exactly to L the difference is obvious 3N H 2 E P - . v 1 Here Pv is the momentum conjugate with the normal coordinate Qv. 1 The proof of the diagonalizability by a suitable rotation in R. N w including the proof of the mutual orthogonality of the eigenvectors is essentially self-evident. It is only necessary to know that a positive-definite quadratic form Pi N 1 Va gwawp describes a 3A-dimensional ellipsoid in this Euclidean space which can be diagonalised by a rotation to the principal axes of the ellipsoid. A Typical Example Three Coupled Pendulums .
