Central Forces Flashcards
For planar motion in a central field, what is the angular momentum L equal to?
L = r x p, where r & p define a plane
What is the Lagrangian equal to for planar motion?
L = 1/2 m(r’^2 + r^2 *θ’^2) - V(r)
What does the canonical momenta p(θ) equal for the planar motion? What is the case for circular and elliptical orbit?
p(θ) = dL/dθ’ = mr^2 *θ’
Circular orbit: r fixed, θ’ const, elliptical orbit: r varies, θ’ varies to make r^2 *θ’ const
How do we find equations for the central and centrifugal forces for the planar motion?
- Use E-L equation in r, and rearrange to make m*r’’ the subject
- dV/dr term if the central force and mr*θ’^2 term is the centrifugal force (not real force, consequence of the rotational motion)
How can we write the centrifugal force in terms of the angular momentum?
Fc = |L|^2/(m*r^3) = -dUc/dr, where Uc(r) is a potential energy
How can we write the equation of motion for planar motion in terms of Uc(r)?
m*r’’ = -d/dr(Uc + V) = -d/dr Ueff, where Ueff is the effective potential energy.= Uc + V
What is the equation for the Hamiltonian in terms of Ueff for planar motion?
H = p(r)^2/2m + Uc(r) + V(r) = p(r)^2/2m + Ueff(r) = E
What is the equation for p(r)?
p(r) = sqrt(2m*(E-Ueff(r)))
What is the equation for E if we look at p(r) = 0 and what can we learn from this?
E-Ueff(r) = 0. From this we learn that there is no radial motion when p(r) = 0 - closest or furthest approach in an orbit
What does a sketch of Ueff look like?
Like the strong nuclear force graph - high at first then comes down past x axis then back up and tends to -0.
On the Ueff graph, where are E1, E2 and E3 and their corresponding radii?
y-axis is E, x-axis is r, E1 is at start, E2 is halfway between crossing x-axis and tunring point, and E3 is just after turning point. r values can be read off from that.
What happens at E = 0, E=E2 and E = E3?
E=0 is parabola, E=E2 has 2 rotts, r2 and r2’ where p(r) = 0, E=E3 is a circular orbit, here 2 repeated roots at r = r3
For 2 masses, what do we make the Lagrangian a function of?
The centre of mass coordinates: L(x0, y0, r, θ, x0’, y0’, r’, θ’), where θ is the angle from horizontal the line joining the masses is and (x0, y0) is the centre of mass
What is the equation for the kinetic energy of the centre of mass for the 2mass system?
T(CoM) = 1/2 (2m)(x0’^2 + y0’^2)
What is the equation for the potential energy of the 2 mass system?
V = 1/2 k(r-r0)^2, where r0 is the equilibrium value of r
What is the kinetic energy o the 2 mass system relative to the CoM?
T(rel) = 1/4mr^2 θ’^2 + 1/4m*r’^2
What is the equation for the Lagrangian of the 2 mass system?
L = m(x0’^2 + y0’^2) + 1/4m(r’^2 + r^2 θ’^2) - 1/2k(r-r0)^2
What do we find if we look at the E-L equation in x0 for the 2 masses connected by a spring system?
No x0 dependence in L, so total momentum along x-direction is conserved.
What do we find if we look at the E-L equation in y0 for the 2 masses connected by a spring system?
Momentum along y-direction is conserved.
What do we find if we look at the E-L equation in r for the 2 masses connected by a spring system?
Find equation of motion r’’ = -2k/m (r-r0) +rθ’^2
What do we find if we look at the E-L equation in θ for the 2 masses connected by a spring system?
Find that angular momentum is conserved
If we set the angular momentum θ’ to zero, what do we find for the 2 mass system?
Motion split into several ‘modes’: 2 translational modes up and across, 1 vibrational mode (side to side) and 1 rotational mode
What are normal modes?
A pattern of motion where all particles are moving with same frequency.
How can we build up the overall motion of the system?
By using a superposition of normal modes.
How do we determine how many normal modes we have?
If we have n generalised coordinates, we have n normal modes with n frequencies (in total).
What is a good example of normal modes?
2 coupled oscillators (2 masses, 3 springs, connect to walls either side.
What is the Lagrangian for the 2 coupled oscillator problem?
L = 1/2mx1’^2 + 1/2mx2’^2 - (1/2kx1^2 + 1/2kx2^2+1/2*k12(x1-x2)^2)
Which 2 variables for we apply the E-L equation to for the 2 coupled oscillator problem?
x1 and x2.
What solutions do we try for x1 and x2 for the 2 coupled oscillator problem?
x1(t) = A1exp(iωt), x2(t) = A2exp(iωt)
What do we do with the trial solutions for x1 and x2 for the 2 coupled oscillator problem?
Sub them into E-L equations and rearrange to make A1 the subject in each. Then set these equal to eachother and rearrange to fine ω1 and ω2.
How do we compare the motion of ω1 and ω2?
Use equation found for A1 (relative displacement of particle 1 & 2), and find that ω1 is symmetric mode (moves in same direction) whereas ω2 is anti-symmetric mode (moves towards eachother)
What is another expression for T to find the inertia matrix?
T = 1/2 sum over i,j of qi’mij*qj’, where mij = m = (m11, m12, m21, m22) = 4x4 matrix
What can we write mij as? What does this mean?
mij = d^2T/dqi’dqj’ = d^2T/dqj’dqi’ = mji -> matrix m is symmetric about diagonal: is the inertia matrix
What is a second example we can use for the inertia matrix?
Mass m2 moving at x2’ to a mass m1 (moving at x1’), so T = 1/2m1x1’^2 + 1/2m2(x2’-x1’)^2
How do we find the inertia matrix M from T?
Using d^2T/dqi’dqj’, so m11 = d^2T/dx1’^2, m12 = d^2T/dxq’dx2’, m22 = d^2T/dx2’^2
What does M equal for the two masses moving problem?
M = (m1+m2, -m2, -m2, m2) = matrix
