PHY 301: MATH AND NUM TECH Chapter 6: Eigenvalue Problems 1.Motivation 2.Differential Equations 3.Other Equations

PHY 301: MATH AND NUM TECH 6.1 Motivation 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 chapter, we’ll first work out the solution to the generic problem and then apply the method to 2 concrete examples: an vibrating string and an electrical circuit..

PHY 301: MATH AND NUM TECH 6.2. Differential Equations 6.2-A Linear Differential Equations: General Approach 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:

PHY 301: MATH AND NUM TECH 6.2. Differential Equations 6.2-A Linear Differential Equations: General Approach 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:

PHY 301: MATH AND NUM TECH 6.2. Differential Equations 6.2-A Linear Differential Equations: General Approach 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:

PHY 301: MATH AND NUM TECH 6.2. Differential Equations 6.2-A Linear Differential Equations: General Approach

PHY 301: MATH AND NUM TECH 6.2. Differential Equations 6.2-B Linear Differential Equations: Example 1 Electrical Circuit Writing that the voltage around each loop of current i 1 and i 2, is zero we get: In the electrical circuit below the capacitance C 1 =1microFarad and C 2 =2microF are initially charged at V AB =5Volts (say V A >V B ). At t=0 the switch S is closed. Find the charge on the capacitors as a function of time. Take R 1 =R 2 =100  Problem set up: In addition we can write a relationship between q’s and i’s: and So that the equations become: We massage the equations a little to put them in our generic form: Which finally yields:

PHY 301: MATH AND NUM TECH 5.2. Differential Equations 6.2-C Linear Differential Equations: Example 1 cont’d

PHY 301: MATH AND NUM TECH 5.2. Differential Equations 6.2-C Linear Differential Equations: Example 1 cont’d

PHY 301: MATH AND NUM TECH 5.2. Differential Equations 6.2-C Linear Differential Equations: Example 2. Vibrating String & Normal Modes Consider the following 3 masses m attached to a string under tension T. The string is fixed at the ends and massless. The distances are indicated on the drawing. The masses are taken to oscillate in a plane (2dim) and the tension constitutes the net force on the masses (i.e. no gravity is present –outer space - or mg and normal force cancel by letting the masses oscillate in a horizontal plane like a table top) The tension in the rope is assumed to be constant, say T.

PHY 301: MATH AND NUM TECH 5.2. Differential Equations 6.2-C Linear Differential Equations: Example 2. Vibrating String & Normal Modes

PHY 301: MATH AND NUM TECH 5.2. Differential Equations 6.2-C Linear Differential Equations: Example 2. Vibrating String & Normal Modes

PHY 301: MATH AND NUM TECH 5.2. Differential Equations 6.2-C Linear Differential Equations: Example 2. Vibrating String & Normal Modes E6.2-5 Solve the vibrating string problem for 2 masses on a string under tension T. The masses are located at a distance d from the fixed ends of the string, and are separated by a distance 2d. Find the general solution from the position of the masses as a function of time and find the normal modes of the system.