Lecture 9: Eigenvalues and Eigenvectors in Classical Mechanics (See Section 3.12 in Boas)

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1 Lecture 9: Eigenvalues and Eigenvectors in Classical Mechanics (See Section 3 in Boas) As suggested in Lecture 8 the formalism of eigenvalues/eigenvectors has many applications in physics, especially in mechanics The basic issue is the choice of coordinates to describe a given system An arbitrary choice of coordinates (ie, of degrees of freedom) will typically lead to a complicated description of the system due to the coupling (ie, interaction) between these coordinates Choosing the coordinates to correspond to the eigenvectors of the system (ie, choosing the basis vectors or basis-functions to be the eigenvectors) ensures that there is no coupling In the eigen-basis the dynamics is diagonal, and the time evolution of the eigenvectors is very simple As we have earlier argued most systems near equilibrium can be described in terms of harmonic oscillators Generally the naïve choice of the coordinates to describe these oscillators will lead to coupling between the oscillators We want to illustrate these points by discussing some simple examples (see also those in the text and the examples in Appendix B) As a first example we consider Exercise 959 in Boas (Edition 3) The mechanical system consists of two pendulums of equal lengths (l ), with equal masses ( m ), whose naïve coordinates are the angles of the two pendulums, and, measured from the vertical in the same direction (see the figure to the right) If we first ignore the spring (eg, set k = ), the two coordinates will evolve independently in time (ie, just independent pendulums) For small angular displacements (, ) we can use our knowledge of series expansions to approximate the physics (sin ) and obtain linear equations of motion In this (small angle) approximation the motion of each pendulum is harmonic with the same frequency gl, ie, two harmonic oscillators with the same frequency To obtain a more interesting (and realistic) problem we now connect the two masses with a spring, which has spring constant k For notational simplicity we choose the spring constant to satisfy the equation kl mg We will assume that the spring is un-stretched at the point of lowest gravitational energy, Thus the energy stored in the spring as a function of and is given by (start with the full result and then take the small angle limit) Physics 7 Lecture 9 Autumn 8

2 k k Vk, x y l sin l sin l cos l cos kl kl cos kl cos,, (9) while the gravitational energy is (normalized to vanish at the lowest point) mgl Vg, mgl cos cos, (9) Thus for small angles (low total energy) the total potential energy of the system looks like V kl mgl (93) kl V,,, We can rewrite this expression in vector/matrix notation as T kl T V, A, kl A, (94) The essential feature is that, in the basis of the naïve coordinates and, the potential energy is not diagonal, corresponding to explicit coupling between the two oscillators The kinetic energy can be written in the same notation as T m m ml T, l l ml T, (95) Physics 7 Lecture 9 Autumn 8

3 which we note is proportional to the unit matrix and, in that sense, trivial To simplify the analysis of this system (always our goal) we want to find its eigenvalues, which are the characteristic (or normal) frequencies of oscillation, and the corresponding eigenvectors, which are the characteristic (or normal) modes of oscillation Since the kinetic energy is proportional to the unit matrix, it does not distinguish the different modes and we proceed by solving the eigen-problem corresponding to the potential energy in Eq (94) For now we can ignore the overall dimensionful factor kl and write A det A 3 Tr A 4 3 (96) To obtain these eigenvalues easily we have used the results of Eqs (85) and (87) (in Lecture 8) The actual frequencies come from including the ratio of the dimensionful factors, kl ml k m g l, kl ml 3k m 3g l 3 To see the connection to (the more familiar) Newton s equation first recall the situation for a single (small angle) pendulum (see the Appendix) where we have (97) ml mgl T, V dv d Newton: ml mgl g l (98) i t Employing our usual (complex) Ansatz t e we find g i cos t, e, l (99) where the (conserved) total energy is given by Physics 7 Lecture 9 3 Autumn 8

4 m E T V sin cos l t gl t mgl a constant (9) Note that in the equation of motion (Newton) the first term is the angular acceleration times the moment of inertia and the second terms is (minus) the torque due to gravity A formal treatment of this system in terms of scalar quantities follows from the techniques of Lagrange as described in Chapter 9 in Boas and in Lecture Looking ahead we will make use of the examples of the simple and coupled pendulums to introduce the new notation here The special scalar quantity is the Lagrangian defined as the kinetic energy minus the potential energy For the simple pendulum the Lagrangian (for small angular displacement) is given by ml mgl L T V (9) As we will derive in Lecture this scalar quantity is connected to the equation of motion via Lagrange s equation d L L d ml mgl dt dt g l (9) For the system of coupled pendulums the corresponding Lagrangian is (still in the small angle approximation) ml L T V kl (93) Now there is a Lagrange equation for each independent variable yielding two equations of motion (in the naïve coordinates), Physics 7 Lecture 9 4 Autumn 8

5 d L L V dt ml kl, d L L V dt ml kl (94) Again using our standard exponential ( e it,, ) Ansatz we obtain the coupled (matrix) equations (divide through by ml ) k (95) m Comparing to Eq (96) we diagonalize to find the eigen-frequencies in Eq (97),, 3 To find the eigenvectors for the two pendulums we solve the eigenvector equations, 3, 3 (96) where we have chosen a specific (but arbitrary) phase and to normalize as unit vectors Thus the first normal mode corresponds to the two pendulums moving in phase with the same amplitude, It should be no surprise that with this motion the spring plays no role ( x y ) and the eigen-frequency is the frequency of each pendulum separately For the second mode the pendulums have the same amplitude but opposite phases, Thus the spring plays a role to resist the oscillation and increase the frequency of oscillation, 3 Note that as expected for a Hermitian (real and symmetric) matrix the eigenvectors are orthogonal, T The matrix that diagonalizes the potential energy matrix is (with our choice of phases) Physics 7 Lecture 9 5 Autumn 8

6 C, (97) which, by comparison with Eq 843, corresponds to a rotation of the basis vectors by 4 We have C C a rotation by, 4 kl kl 3 AC kl kl kl D, C (98) Noting that the kinetic energy is still diagonal (as expected, all matrices commute with the unit matrix) ml ml C C, (99) we can write the energy in the new coordinates as T kl V, D 3, ml T ml T, ml kl E 3 (9) The corresponding equations of motion (Newton) are (obtained either from Newton or Lagrange as above, but with, as the coordinates) Physics 7 Lecture 9 6 Autumn 8

7 k g cos,, m l ml kl t t 3k 3g 3 cos, m l ml kl t t (9) Note the essential point that, once we have gone to the normal mode basis (the eigenbasis), the dynamics is diagonal (no cross terms) and we need only solve the familiar simple harmonic oscillator problem (twice) By a sensible choice of basis we have greatly simplified the problem We are lazy but smart! In terms of the original (intuitive) basis we have C t cos t cos t t cos t cos t (9) To complete the analysis of this system we solve for a specific choice of initial conditions For example, let s assume that one pendulum is displaced and then released from rest,,,, sin,, sin, Physics 7 Lecture 9 7 Autumn 8

8 , t cos t t cos t t cos t cos t k m 3k m k 3k m m k 3k m m t cos t cos t (93) The normal modes oscillate at the (fixed and different) characteristic frequencies while the physical modes oscillate as linear combinations of these two frequencies The time behavior of the two angles is plotted in the next figure for The solid (red) line is the angle and the dashed (blue) line is the angle Note that the initial energy in the one pendulum is first passed to the other pendulum and then passed back (This is exactly the mathematics of neutrino oscillation where it is probability that is passed back and forth) t Physics 7 Lecture 9 8 Autumn 8