Hamilton's Equations and more examples

Question 1

What are Hamiltons equations?

Answer 1

qi’ = dH/dpi, pi’ = -dH/dqi

Question 2

What do we start with for the change in Lagrangian dL?

Answer 2

Sum over i of dL/dqi dqi + dL/dqi’ dqi’ = sum over i of dpi/dt dqi + pi dqi’

Question 3

What do we do with this version of the Lagrangian?

Answer 3

• d(sum over i of pi*qi’) = sum over i of dpi qi’ + sum over i of pi dqi’
• sum over i of pi dqi’ = dL - sum over i of dpi/dt dqi

Question 4

What is the third step after rearranging the Lagrangian again?

Answer 4

• dL = sum over i of (pi’ dqi 0 qi’ dpi) + d(sum over i of pi qi’)
• Rearrange so d(sum over i of pi qi’ - L) = sum over i of (qi’ dpi - pi’ dqi), so dH = sum over i of qi’ dpi - pi’ dqi
• Need H(pi, qi), so dh = sum over i of dH/dpi dpi + sum over i of dH/dqi dqi
• Compare thee two equations for dH

Question 5

For the harmonic oscillator example, how to we convert H(x, v) to H(px, x)?

Answer 5

• use v = px/m to re-write H
• apply hamiltons equations
• solve the equations we get from these

Question 6

What is phase space?

Answer 6

• For n qi’s, imagine space that has 2n dimensions
• Position in phase space is (p1, p2……pn; q1, q1,……..qn)
• A point in phase space gives us the current state of the system
• Path/line tells us the evolution of the system

Question 7

For the 1D harmonic oscillator example, how can we use the Hamiltonian to plot px against x?

Answer 7

H = px^2/2m + 1/2kx^2 = const, which is equation for an ellipse, so can plot an ellipse with known axis-crossing points

Question 8

What is the equation for the Lagrangian for a pendulum?

Answer 8

L = T-V = 1/2ml^2 *θ’^2 - mgl(1-cosθ)

Question 9

What does p(θ) equal and what can we use this for?

Answer 9

• p(θ) = dL/dθ’ = ml^2θ’
• Can use this in the Hamiltonian: H = θ’ *dL/dθ’ - L = 1/2 ml^2 *θ’^2 + mgl(1-cosθ)
• Then sub in p(θ) to get H(p(θ), θ) = E

Question 10

What do we do after finding H(p(θ), θ)?

Answer 10

• Use Hamilton’s equations: p(θ)’ = -dH/dθ = -mglsinθ, and θ’ = dH/dp(θ) = p(θ)/ml^2
• Can use this to plot phase space picture

Question 11

What do we find in terms of the phase space for small values of E?

Answer 11

• Small oscillations so sinθ ~ θ, cosθ ~ 1-θ^2/2
• Sub in θ for sinθ in hamiltons equations, and then set p(θ) = 0 for one solution and θ=0 for the other solutions
• Plot θ against p(θ) for phase space graph.

Question 12

What do we find in terms of the phase space for very large values of E? (enough for pendulum to rotate completely)

Answer 12

• sinθ = 0 when θ=0, π, 2π etc
• sketch p(θ) - θ curve by considering θ=0 in Hamiltons equations and θ=π in Hamiltons equations
• Get graph symmetric in θ axis looks like very shallow sine curve going from +π to -π

Question 13

For a particle, mass m, moving on surface of a cylinder radius R, with F = -kr, what does V equal?

Answer 13

V = 1/2k(x^2 +y^2 + z^2) = 1/2k(R^2 +z^2)

Question 14

For a particle, mass m, moving on surface of a cylinder radius R, with F = -kr, what does T equal?

Answer 14

T = 1/2m(R^2 *Ф’^2 + z’^2), where R term in KE in x-y and z term in kinetic energy in z

Question 15

What do we do after determining the Lagrangian for the particle moving on the surface of a cylinder problem?

Answer 15

• Get p(Ф) and p(z): p(Ф) = dL/dФ’, p(z) = dL/dz’

- H = sum over i of qi’*pi - L = Ф’ p(Ф) + z’ p(z) - L

Question 16

How do we get an equation for H(p(Ф), p(z), Ф, z)?

Answer 16

Substitute for Ф’ and z’ using equations found before for p(Ф) and p(z)

Question 17

What do we do after finding H(p(Ф), p(z), Ф, z)? What does this mean?

Answer 17

-Apply Hamiltons equations: p(Ф)’ = -dH/dФ = 0, so p(Ф) = const
p(z)’ = -dH/dz = -kz
-Since p(Ф) = const, if it is rotating, it will keep moving (p(Ф) is conserved)