The Hamiltonian

Question 1

How can we use the Hamiltonian in the roller coaster example looked at before?

Answer 1

• Compute the Hamiltonian from the Lagrangian found before

- Since L(s,s’), dL.dt = 0, so dH/dt = -dL/dt = 0: E is conserved

Question 2

How can we use the Hamiltonian for special relativity?

Answer 2

• Find Hamiltonian from special relativity version of the Lagrangian
• Find again that dH/dt = 0 so E is conserved

Question 3

How do we determine how many coordinates we need?

Answer 3

If we have a particle in 3D, we need 3n coordinates.

Question 4

What is the generalised version of the E-L equation?

Answer 4

dL/dqi - d/dt(dL/dq’i) = 0

Question 5

What is the generalised version of the Hamiltonian?

Answer 5

H = sum over i of q’i * dL/dq’i - L

Question 6

What would the Lagrangian be a function of for 2 particles in 1D?

Answer 6

L(x1, x2, x1’, x2’)

Question 7

What is the canonical momentum and whta is its equation?

Answer 7

The generalised momentum: pi = dL/dq’i, where qi is a general coordinate

Question 8

What is the equation for the canonical force?

Answer 8

Fi = dL/dqi, where i could be x, y, theta etc.

Question 9

What is the equation for the kinetic energy of a mass with more than one coordinate?

Answer 9

T = 1/2 * m *(x’^2 + y’^2)

Question 10

How do we allow for a small deviation from the optimum path in 2D?

Answer 10

(x(t), y(t)) -> (x(t), y(t)) + a(t), where a(t) is a small vector deviation

Question 11

What are the boundary conditions for this vector deviation?

Answer 11

a(t0) = a(t1) = (0, 0)

Question 12

What does the Lagrangian become a function of for more than 1 coordinate?

Answer 12

L(x+ax, y+ay, x’+ax’, y’+ay’)

Question 13

How do we expand the Lagrangian for more than one coordinate?

Answer 13

1st order taylor series: L(x,y,x’,y’) + axdL/dx + aydL/dy + ax’*dL/dx’ + ay’ * dL/dy’

Question 14

What is the equation for the change in action 𝛿A for multiple coordinates?

Answer 14

𝛿A = integral from t0 to t1 of dt[axdL/dx + aydL/dy + ax’dL/dx’ + dy’ * dL/dy’]

Question 15

What do we do with the equation for 𝛿A?

Answer 15

Integrate the last 2 terms by parts and get 2 E-L equations.

Question 16

What is the reciprocity relationship for partial derivatives?

Answer 16

dV/d1 at constant x2 * dx1/dx2 at constant V * dx2/dV at constant x1 = -1

Question 17

How does the notation for q change when there is 2 particles in 1D or 1 particle in 2D?

Answer 17

For 1 particle in 2D, the q represents the coordinate index, whereas for 2 particles in 1D, q represents the particle index.

Question 18

For two particles at x1 and x2 of mass m1 and m2 a distance x from eachother, how can we find the force on one due to the other?

Answer 18

Compute the lagrangian and then insert into E-L equation (with either x1 or x2 in it) to find the force on m1 due to m2.

Question 19

How can we get generalised velocities for polar coordinates?

Answer 19

Use the polar equation for x and y, and use the chain rule so x’ = dx/dt = dx/dr * dr/dt etc, and find the individual terms (where dr/dt = r’)

Question 20

What do we do after getting the general velocities for polar coordinates?

Answer 20

Sub into the equation for kinetic energy, solve and put into the Lagrangian (T-V).

Question 21

Which 2 E-L equations do we need to compute for the polar coordinate problem?

Answer 21

In r and in Ф, so dL/dr - d/dt(dL/dr’) = 0, and the same for Ф.

Question 22

What do the 2 E-L equations give us?

Answer 22

The r one gives is the motion in a circle with the radial component of force due to V, and the Ф one gives us the angular momentum with the tangential component of the force due to V.

Question 23

What is the equation for the angular momentum?

Answer 23

L = rXp = mωr^2 = mФ’ *r^2

Question 24

What does dV/dФ equal? When is it equal to zero?

Answer 24

dL/dt = -dV/dФ, and dV/dФ = 0 when we have a central force e.g. gravity -> angular momentum conserved

Question 25

For a pendulum rotating in the x-y plane about the z axis, what are the equations for the different velocities?

Answer 25

v(θ) = lθ’ = pendulum speed, v(Ф) = lsinθ * Ф’ = circular speed, these are perpendicular, where l is the length of the pendulum, θ is the angle away from the z-axis the pendulum is and Ф is the angle around the z-axis.

Question 26

How do we incorporate these velocities into the Lagrangian? What does V equal in the Lagrangian?

Answer 26

Sub them in. V = -mgl*cosθ

Question 27

What do we finally do for the pendulum problem?

Answer 27

Look at the E-L equation in Ф and find that dL/dФ = 0, so d/dt(dL/dФ’) = 0, and angular momentum about the z-axis is conserved.