Canonical Momentum & M

Question 1

If we assume V(q) (not qi), what is the equation for p(k), the canonical momentum?

Answer 1

p(k) = dL/dq(k)’ = dT/dq(k)’: sub in T and take out constants etc

Question 2

What do we do after finding the final equation for p(k)?

Answer 2

Sub in the kronecker delta 𝛿jk = dqj’/dqk’, and set j = k to set 𝛿jk = 1

Question 3

What is the final found equation for p(k) after 𝛿jk?

Answer 3

p(k) = sum over i of Mik*qi’

Question 4

What is a good example to put this equation for p(k) to use?

Answer 4

2 independent masses so (m1, 0, 0, m2) matrix, and get p1 = m1*x1’, which makes sense

Question 5

How do we find H in terms of Mij?

Answer 5

H = um over i of qi’pi - L = sum over i,j of qi’Mij*qj’ - (T-V) = T+V

Question 6

What is the equilibrium step to find the force/stiffness matrix?

Answer 6

qk’ = 0, p = 0, to stay in equil, qk’’ = 0, p’ = 0.

From Hamiltons equations: pk’ = 0dH/dqk. If p = 0, T = 0, so pk’ = 0dV/dqk = 0

Question 7

What do we do for a small displacement for the force/stiffness matrix?

Answer 7

• Write position x relative to q(0)
• xi = qi-q(0)i, xi’ = qi’
• dV/dxi = 0

Question 8

What is the equation for V(x) after expanding away from equilibrium (taylor series)?

Answer 8

V(x) = V0 + sum over i of xi dV/dxi at 0 + sum over i,j f 1/2xixj*d^2V/dxidxj at 0

Question 9

What is the equation for the canonical force Fi?

Answer 9

Fi = -dV/dxi = -1/2sum over i,j of d^2V/dxidxj at 0 * d/dxi(xixj)

Question 10

How do we expand and simplify d/dxi(xi*xj)?

Answer 10

=dxj/dxj *xj + xi *dxj/dxi, where the second derivative = 𝛿ij, so set i=j and simplify to get 2xj

Question 11

What is the final equation for the canonical force Fi?

Answer 11

Fi = -sum over j of d^2V/dxidxj at 0 xj, or -sum over j of kijxj, where kij is the stiffness matrix = d^2V/dxidxj at 0

Question 12

How do we find the stiffness matrix for a two spring system connexting 2 masses at a wall?

Answer 12

V = 1/2k1x1^2 + 1/2k2(x2-x1)^2. Then k11 = d^2V/dx1^2 at 0 = k1+k2, k12 = d^2V/dx1dx2, etc

Question 13

What do we find is the stiffness matrix for the two spring with 2 mass problem?

Answer 13

K = (k1+k2, -k2, -k2, k2) matrix

Question 14

How do we find an equation of motion for small oscillations using pi and Fi?

Answer 14

Set Fi = pi’, and sub in their equations, then get rid of i and j, so have mx’’ = -kx, then solve by using x(t) = sum over λ of Re(A(λ)X(λ)*exp(iω(λ)t)

Question 15

How do we find normal modes using mx’’ = -kx(t)?

Answer 15

Use x(t) = Axexp(iωt), and rearrange so have zero on one side. Get |K-M*ω^2| = 0: scalar equation and solve this to get frequencies of normal modes

Question 16

For 2 independent masses connected to a wall by a spring, what does the Lagrangian L equal?

Answer 16

L = 1/2 m1x1’^2 + 1/2m2x2’^2 -(1/2k1x1^2 + 1/2k2x2^2)

Question 17

How do we find ω^2 for the 2 independent masses?

Answer 17

Use the scalar equation: |K-Mω^2| = 0 -> get 2x2 matrix of (k1-m1ω^2, 0, 0, k2-m2ω^2) = 0, determinant of this matrix is (k1-m1ω^2)(k2-m2*ω^2) = 0

Question 18

What do we look at in the equation we found from the determinant of the 2x2 matrix?

Answer 18

Either one of the brackets equals zero or the other does, so rearrange the brackets to make ω the subject

Question 19

What equation do we get for ω1 and ω2?

Answer 19

ω1 = sqrt(k1/m1), ω2 = sqrt(k2/m2)

Question 20

How do we find the eignevectors for the 2 independent mass problem?

Answer 20

Find the direction such that (K-Mω^2)*X = 0

Question 21

For mode 1, how do we find the eigenvector X1?

Answer 21

• Try X=matrix(a, b), and multiply this by matrix found before from scalar equation
• Set first part of matrix to 0 and find 0a + (k2-m2k1/m1)*b = 0, so b = 0 and a can have any value.

Question 22

What do we find are the eignevectors X1 and X2?

Answer 22

X1 = matrix(1, 0): m1 moving, m2 still, X2 = matrix (0, 1): m2 moving, m1 still

Question 23

What is the Lagrangian for a diatomic molecule (2 masses connected)?

Answer 23

L = 1/2m1x1’^2 + 1/2m2x2’^2 - 1/2k(x2-x1)^2

Question 24

What is the matrix M equal to for the diatomic molecule?

Answer 24

M = matrix(m1, 0, 0, m2)

Question 25

How do we find the matrix K?

Answer 25

Kij = d^2V/dqidqj, where V = 1/2*k(x2-x1)^2

Question 26

What do we find that the matrix K is equal to for the diatomic molecule?

Answer 26

K = matrix(k, -k, -k, k)

Question 27

How do we find the normal modes for the diatomic molecule?

Answer 27

Use |K-Mω^2|=0 to get a matrix then find determinant and then find equations for ω1 and ω2

Question 28

For the diatomic molecule, what do we find for ω1 and ω2?

Answer 28

ω1^2 = 0, ω2^2 = k(m1+m2)/m1m2 = k(1/m1+1/m2) = k/μ, where μ is the reduced mass

Question 29

How do we get the directions for the diatomic molecule for mode 1?

Answer 29

Again use: (K-Mω^2)X = 0, and try X = martix(a, b), find that a = b, so X1 = matrix(1, 1)

Question 30

What do we find after finding the eigenvalues for the diatomic molecule?

Answer 30

That X1 and X2 move in same direction, with same phase and move same distance

Question 31

How can we use ω2^2 = k/μ for the second mode in the diatomic molecule problem?

Answer 31

Sub it into the matrix and rearrange to find a and b. Find that b/a = -m1/m2, so X2 = matrix(1, -m1/m2) with a=1, or X2 = matrix(m2, -m1) -> masses moving in opposite directions i.e. oscillating

Question 32

What do ω1 and ω2 represent?

Answer 32

ω1 = translational mode, ω2 = vibrational mode