More Lagrangian Stuff & Intro to Hamiltonian

Question 1

Why use Lagrangian Methods?

Answer 1

• Concepts have applications outside mechanics

- Can easily change coordinate system to the easiest one to use

Question 2

Describe the problem with a mass on a cycloidal path.

Answer 2

Path defined by rolling a circle, and tracking the motion of a point on the circle.

Question 3

How can we plot the position of the point mass on a graph?

Answer 3

x-y axis with θ being the max and minimum at +/- pi

Question 4

What are the x and y coordinates equal to for this problem?

Answer 4

x = R*(θ+sinθ)
y = R*(1-cosθ)

Question 5

What is a good choice of coordinates for this cycloidal path problem?

Answer 5

Distance u from the minimum position (x=0, y=0) at θ=0

Question 6

What is the kinetic and potential energy equal to the the cycloidal path problem?

Answer 6

T = 1/2 * mu’^2, V = mgy = mgR(1-cosθ) = mgR2sin^2(θ/2)

Question 7

For the potential energy V, how can we convert V(θ) to V(u)?

Answer 7

u is the length on the path, so we need to find u = integral from θ = 0 to θ’ du

Question 8

What is the length of line element du equal to?

Answer 8

Use pythagoras: du^2 = dx^2 + dy^2 = ((dx/dθ)^2 + (dy/dθ)^2) (dθ)^2

Question 9

How can we simplify the equation for du?

Answer 9

Substitute in the equations for x and y in the differentials, then rearrange.

Question 10

What is the final equation for du?

Answer 10

du = 2R*cos(θ/2) dθ, so u = integral from θ=0 to θ’ of that.

Question 11

How can we use the equation for u to change V(θ) to V(u)?

Answer 11

Rearrange for sin^2(θ/2), and substitute this into the equation for V(θ)

Question 12

What do we find is the Lagrangian for the cycloidal path problem?

Answer 12

L = 1/2 * m*u’^2 - mgu^2/8R = L(u,u’)

Question 13

What do we do with the Lagrangian after finding it for the cycloidal path problem?

Answer 13

Solve the E-L equation to find the equation of motion for it.

Question 14

What is the equation for the Hamiltonian?

Answer 14

H = q’*dL/dq’ - L, for generalised coordinate q.

Question 15

How do we find if the Hamiltonian changes with time?

Answer 15

Find dH/dt, assuming L(q,q’,t)

Question 16

What is the first step in computing dH/dt?

Answer 16

-Use chain rule, so dH/dt = q’‘dL/dq’ + q’d/dt(dL/dq’) - (dL/dq * dq/dt + dL/dq’ * dq’/dt + dL/dt)

Question 17

What do we finally find for dH/dt?

Answer 17

dH/dt = q’*(d/dt(dL/dq’) - dL/dq) - dL/dt

This is E-L = 0, so dH/dt = -dL/dt

Question 18

Is the Hamiltonian conserved?

Answer 18

Yes

Question 19

What is the Hamtiltonian and how can we show this?

Answer 19

Is the total energy: show this by subbing in the Lagrangian and finding it is T+V rather than T-V.