The Hamiltonian Flashcards
How can we use the Hamiltonian in the roller coaster example looked at before?
- Compute the Hamiltonian from the Lagrangian found before
- Since L(s,s’), dL.dt = 0, so dH/dt = -dL/dt = 0: E is conserved
How can we use the Hamiltonian for special relativity?
- Find Hamiltonian from special relativity version of the Lagrangian
- Find again that dH/dt = 0 so E is conserved
How do we determine how many coordinates we need?
If we have a particle in 3D, we need 3n coordinates.
What is the generalised version of the E-L equation?
dL/dqi - d/dt(dL/dq’i) = 0
What is the generalised version of the Hamiltonian?
H = sum over i of q’i * dL/dq’i - L
What would the Lagrangian be a function of for 2 particles in 1D?
L(x1, x2, x1’, x2’)
What is the canonical momentum and whta is its equation?
The generalised momentum: pi = dL/dq’i, where qi is a general coordinate
What is the equation for the canonical force?
Fi = dL/dqi, where i could be x, y, theta etc.
What is the equation for the kinetic energy of a mass with more than one coordinate?
T = 1/2 * m *(x’^2 + y’^2)
How do we allow for a small deviation from the optimum path in 2D?
(x(t), y(t)) -> (x(t), y(t)) + a(t), where a(t) is a small vector deviation
What are the boundary conditions for this vector deviation?
a(t0) = a(t1) = (0, 0)
What does the Lagrangian become a function of for more than 1 coordinate?
L(x+ax, y+ay, x’+ax’, y’+ay’)
How do we expand the Lagrangian for more than one coordinate?
1st order taylor series: L(x,y,x’,y’) + axdL/dx + aydL/dy + ax’*dL/dx’ + ay’ * dL/dy’
What is the equation for the change in action 𝛿A for multiple coordinates?
𝛿A = integral from t0 to t1 of dt[axdL/dx + aydL/dy + ax’dL/dx’ + dy’ * dL/dy’]
What do we do with the equation for 𝛿A?
Integrate the last 2 terms by parts and get 2 E-L equations.
What is the reciprocity relationship for partial derivatives?
dV/d1 at constant x2 * dx1/dx2 at constant V * dx2/dV at constant x1 = -1
How does the notation for q change when there is 2 particles in 1D or 1 particle in 2D?
For 1 particle in 2D, the q represents the coordinate index, whereas for 2 particles in 1D, q represents the particle index.
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?
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.
How can we get generalised velocities for polar coordinates?
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’)
What do we do after getting the general velocities for polar coordinates?
Sub into the equation for kinetic energy, solve and put into the Lagrangian (T-V).
Which 2 E-L equations do we need to compute for the polar coordinate problem?
In r and in Ф, so dL/dr - d/dt(dL/dr’) = 0, and the same for Ф.
What do the 2 E-L equations give us?
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.
What is the equation for the angular momentum?
L = rXp = mωr^2 = mФ’ *r^2
What does dV/dФ equal? When is it equal to zero?
dL/dt = -dV/dФ, and dV/dФ = 0 when we have a central force e.g. gravity -> angular momentum conserved
For a pendulum rotating in the x-y plane about the z axis, what are the equations for the different velocities?
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.
How do we incorporate these velocities into the Lagrangian? What does V equal in the Lagrangian?
Sub them in. V = -mgl*cosθ
What do we finally do for the pendulum problem?
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.
