Canonical Momentum & M Flashcards
If we assume V(q) (not qi), what is the equation for p(k), the canonical momentum?
p(k) = dL/dq(k)’ = dT/dq(k)’: sub in T and take out constants etc
What do we do after finding the final equation for p(k)?
Sub in the kronecker delta 𝛿jk = dqj’/dqk’, and set j = k to set 𝛿jk = 1
What is the final found equation for p(k) after 𝛿jk?
p(k) = sum over i of Mik*qi’
What is a good example to put this equation for p(k) to use?
2 independent masses so (m1, 0, 0, m2) matrix, and get p1 = m1*x1’, which makes sense
How do we find H in terms of Mij?
H = um over i of qi’pi - L = sum over i,j of qi’Mij*qj’ - (T-V) = T+V
What is the equilibrium step to find the force/stiffness matrix?
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
What do we do for a small displacement for the force/stiffness matrix?
- Write position x relative to q(0)
- xi = qi-q(0)i, xi’ = qi’
- dV/dxi = 0
What is the equation for V(x) after expanding away from equilibrium (taylor series)?
V(x) = V0 + sum over i of xi dV/dxi at 0 + sum over i,j f 1/2xixj*d^2V/dxidxj at 0
What is the equation for the canonical force Fi?
Fi = -dV/dxi = -1/2sum over i,j of d^2V/dxidxj at 0 * d/dxi(xixj)
How do we expand and simplify d/dxi(xi*xj)?
=dxj/dxj *xj + xi *dxj/dxi, where the second derivative = 𝛿ij, so set i=j and simplify to get 2xj
What is the final equation for the canonical force Fi?
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
How do we find the stiffness matrix for a two spring system connexting 2 masses at a wall?
V = 1/2k1x1^2 + 1/2k2(x2-x1)^2. Then k11 = d^2V/dx1^2 at 0 = k1+k2, k12 = d^2V/dx1dx2, etc
What do we find is the stiffness matrix for the two spring with 2 mass problem?
K = (k1+k2, -k2, -k2, k2) matrix
How do we find an equation of motion for small oscillations using pi and Fi?
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)
How do we find normal modes using mx’’ = -kx(t)?
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
For 2 independent masses connected to a wall by a spring, what does the Lagrangian L equal?
L = 1/2 m1x1’^2 + 1/2m2x2’^2 -(1/2k1x1^2 + 1/2k2x2^2)
How do we find ω^2 for the 2 independent masses?
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
What do we look at in the equation we found from the determinant of the 2x2 matrix?
Either one of the brackets equals zero or the other does, so rearrange the brackets to make ω the subject
What equation do we get for ω1 and ω2?
ω1 = sqrt(k1/m1), ω2 = sqrt(k2/m2)
How do we find the eignevectors for the 2 independent mass problem?
Find the direction such that (K-Mω^2)*X = 0
For mode 1, how do we find the eigenvector X1?
- 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.
What do we find are the eignevectors X1 and X2?
X1 = matrix(1, 0): m1 moving, m2 still, X2 = matrix (0, 1): m2 moving, m1 still
What is the Lagrangian for a diatomic molecule (2 masses connected)?
L = 1/2m1x1’^2 + 1/2m2x2’^2 - 1/2k(x2-x1)^2
What is the matrix M equal to for the diatomic molecule?
M = matrix(m1, 0, 0, m2)
