1. Ch11: Normal Modes 1. Review diagonalization and eigenvalue problem 2. Normal modes of Coupled oscillators (3 springs, 2 masses) 3. Case of Weakly coupled oscillators 4. General Case 5. Double Pendulum 6. Three Coupled Pendulums

2. Diagonalization and Eigenvalue PROBLEM

3. 1. Review: a) Diagonalization and Eigenvalue problem

4. An nxn matrix A is said to be diagonalizable if there exist a matrix P such that A=PDP -1 (or equivalently P -1 AP=D) where D is a diagonal matrix. Theorem: P exists if and only if there exist n independent eigenvectors of A; Consider the n independent eigenvectors of A: in that case P is made up of the eigenvectors of A, entered as column coefficients. D is in addition found to be made up of the corresponding eigenvalues of A Proof for n=3 (general case identical) 1. Review: a) Diagonalization and Eigenvalue problem

5. Now let’s compute PD where D is the diagonal matrix of the eigenvalues arranged in the same order as their corresponding eigenvectors in the matrix P. We get: But that’s exactly the same answer as we got on the previous page computing AP!!! So we can conclude that: AP=PD and thus, if P is invertible (i.e. its det is non-zero and thus the eigenvector are linearly independent!), we finally obtain: A=PDP -1 where P is the matrix of the eigenvectors of A in column placement and D is the diagonal matrix of the eigenvalues of A is in the same position as the eigenvectors in P: If some of the eigenvalues are degenerate one can carry out the same procedure as long as one can find n linearly independent eigenvectors. 1. Review: a) Diagonalization and Eigenvalue problem

6. Physical Systems are commonly described by ( usually differential) equations involving sets of coupled variables. If the equations are linear and homogeneous (no constant term), they can be looked at as linear transformations, and thus we can express them in matrix form. This, in turn allows us, using diagonalization, to decouple the equations in order to obtain simple solutions. In addition, the solutions to the diagonalized problem exhibit the fundamental characteristics of a physical system. In this section, we work out the solution to the generic problem 1. Review: b) Application to linear coupled differential eqns

7. Consider, for instance, the set of n differential equations at right: All dependent variables, x 1, …, x n, are all function of the independent variable t; in addition note that the equations are coupled together – variables are interrelated- but they are quite special because only one order of derivative appears – here noted “s”- Thus the approach, although quite general, works only for this type of diff. eq. Inserting into (1-2) we get: and since P is constant the left hand side can be re-written as: By calling P the matrix of the eigenvectors of a, we can define the new variable x such that: In general this is not so simple. What would make it easy is if the two equations in (1-1) were decoupled, i.e. the first would be an equation involving only x 1 and the second equation an equation involving only x 2. This can be clearly achieved by diagonalizing the matrix a. Furthermore, in dimensions higher than 2, the solution would be really horrendous, so that diagonalizing the matrix is even more important in that case. And finally, the properties of the system are obscured by the complication of the solution. Diagonalizing the matrix will exhibit the underlying characteristic of a system. And where the matrix a is: These equations can be re-written in matrix /vector form as: where: 1. Review: b) Application to linear coupled differential eqns

8. And thus finally the equations are decoupled. The solution is very easily found Now, multiplying both sides by on the left we get: Since we know that is the diagonal matrix of the eigenvalues, we have replaced our linear system of equations (1-1) by a decoupled linear system: 1. Review: b) Application to linear coupled differential eqns

9. This method can also handle derivatives of mixed order Consider for example: Now define: then the above equation becomes: So that now the system can be solved as previously. Note that every derivative adds an new variable (in our example the 2 nd derivative adds the variable x2 Which together with the definition of x 2 becomes a system of two equations: 1. Review: b) Application to linear coupled differential eqns

10. 2. Normal Modes and Coupled Oscillators

11. 2. Normal Modes and Coupled Oscillators: A) 2 masses 3springs general set-up

12. 2. Normal Modes and Coupled Oscillators: B) 2 masses 3 springs – special case same m’s & k’s

16. 3. Weakly Coupled: 2 masses 3 springs with same m’s & middle k<< outer k’s

18. 4. General case:

21. 5. Double Pendulum: small oscillations using method of General Case. A.Lagrangian to EoM assuming small oscillations (neglect terms of order ϕ 4 and higher. B.Matrix form of EoM the diagonalize C.Solve in Normal Modes basis D.Transform back to original coordinates

22. 5. Double Pendulum: small oscillations using method of General Case.

26. 6. Three pendulum with springs