Physics 3210When expanding the potential energy about a minimum (at the origin), we have What can we say about the coefficients of the second term?We can write the Lagrangian near a minimum of the potential as What can we say about the matrices A=[ajk] and K=[kjk]?To solve the system of n linear, coupled, homogenous, 2nd order ODEs described by what form of solution is a good guess?For a non-trivial solution of the linear system to exist, what properties must the matrix M satisfy?For the problem of two coupled harmonic oscillators (as sketched), we have What is the correct matrix A=[ajk]?Slide 7Which of the following are rank-zero tensors (aka scalars)?Which of the following are rank-one tensors (aka vectors)?Which of the following is the correct part of the Levi-Civita antisymmetric tensor εij1?The vector v has components vx, vy, vz. How is this vector written in a new coordinate system in which the coordinate axes are inverted (reflected through the origin)?Which of the following are pseudovectors (rather than true vectors)?Slide 13Why is it useful to describe coupled oscillations using normal modes?The motion of a system of coupled oscillators can be described using normal coordinates: What equation is satisfied by the normal coordinate?A system of two coupled harmonic oscillators has general solution What is the second normal coordinate in terms of x1 and x2?Physics 3210Week 14 clicker questionsWhen expanding the potential energy about a minimum (at the origin), we have What can we say about the coefficients of the second term?A. B. C. ( )( )2j j j kj j kj j k0 0V VV q V 0 q q qq q q� �= + + +� � �� �L,j0V0 for all jq�=�j0V0 for some jq�=�j0V0 for some jq�>�D. E. j0V0 for all jq�>�j0V0 for all jq�<�We can write the Lagrangian near a minimum of the potential asWhat can we say about the matrices A=[ajk] and K=[kjk]?A. Both A and K are symmetric.B. Both A and K are antisymmetric.C. A is symmetric and K is antisymmetric.D. A is antisymmetric and K is symmetric.( )jk j k jk j kj k1T V a q q k q q2= - = -�&&,LTo solve the system of n linear, coupled, homogenous, 2nd order ODEs described bywhat form of solution is a good guess?A. Growing exponential.B. Decaying exponential.C. Linear function.D. Linear function times an exponential.E. Oscillating function.A K 0+ =q q&&For a non-trivial solution of the linear systemto exist, what properties must the matrix M satisfy?A. M must be invertible.B. M must be non-invertible.C. M must be symmetric.D. M must be antisymmetric.E. M must be orthogonal.( )20 0K A M- w = =q q 0For the problem of two coupled harmonic oscillators (as sketched), we have What is the correct matrix A=[ajk]?A. B. C. 2 21 2 jk j kj k1 1 1T mx mx a x x2 2 2= + =�& & &&,0 mAm 0� �=� �� �D. E. kx1m mkk12x2m 0A0 m� �=� �� �m 0A0 m� �=� �-� �0 mAm 0-� �=� �� �m mAm m� �=� �� �Physics 3210Wednesday clicker questionsWhich of the following are rank-zero tensors (aka scalars)?I. The dot product of two position vectors, u and v.II. The kinetic energy of a particle.III. The x component of a particle’s velocity.A. I only.B. I and II.C. I, II, and III.D. II and III.E. I and III.Which of the following are rank-one tensors (aka vectors)?I. The momentum of a particle.II. The angular velocity of a particle.III. The center of mass position of a system of particles.A. I only.B. I and II.C. I, II, and III.D. II and III.E. I and III.Which of the following is the correct part of the Levi-Civita antisymmetric tensor εij1?A. B. C. ij10 1 01 0 00 0 0� �� �e = -� �� �� �D. E. ij11 0 00 1 00 0 1� �� �e =� �� �� �ij10 1 11 0 01 0 0-� �� �e = -� �� �� �ij10 0 10 0 01 0 0-� �� �e =� �� �� �ij10 0 00 0 10 1 0� �� �e =� �� �-� �The vector v has components vx, vy, vz. How is this vector written in a new coordinate system in which the coordinate axes are inverted (reflected through the origin)?A. B. C. xyzvvv-� �� ��= -� �� �-� �vD. E. xyzvvv� �� ��=� �� �-� �vxyzvvv� �� ��=� �� �� �vxyzvvv-� �� ��= -� �� �� �vxyzvvv-� �� ��=� �� �� �vWhich of the following are pseudovectors (rather than true vectors)?I. The momentum of a particle.II. The angular velocity of a particle.III. The torque on a particle.A. I only.B. I and II.C. I, II, and III.D. II and III.E. I and III.Physics 3210Friday clicker questionsWhy is it useful to describe coupled oscillations using normal modes?A. The normal modes describe the transfer of energy between different oscillators.B. Each normal modes describes an oscillation at a single frequency.C. Only one normal mode can be excited at a time.D. It allows for greater mathematical complexity.The motion of a system of coupled oscillators can be described using normal coordinates:What equation is satisfied by the normal coordinate?( ) ( )ji tj0 j j0 jj jt e tw= a = h� �q q qA.B.j j j0h +wh =&&C.D.2j j j0h +w h =&&( )2j jK A 0- w h =( )j jK A 0- w h =A system of two coupled harmonic oscillators has general solutionWhat is the second normal coordinate in terms of x1 and x2? A.B.kx1m mkk12x2( ) ( ) ( )1 1 2t x t x th = +C.D.( )( )1 21 2i t i t1 1 2i t i t2 1 2x t e ex t e ew ww w=a +a=- a +a( ) ( ) ( )1 1 2t x t x th =- +( ) ( ) ( )1 1 2t x t x th = -( )( ) ( )1 21x t x tt2+h