Hamilton Flashcards
Why are Halmiton Equation useful
All first order, easier for simulation using computers as you cant solve second order
Better than lagrange for large number of particles
What is the difference between Hamilton and Lagrange
Hamilton System motion in terms of 1st order, time dependent equations of motion, number of initialconditions is still 2s
Lagrange is KE-PE (No physical meaning)
Hamilton is KE + PE physical meaning i.e total energy
What is the Hamilton formulation based on
momentum p = mx.
and q
Nothing else
H = H(qi, pi), function of anything else not a Hamilntonian
What is momentum equal to
pi = dL/dqi.
Partial differential of Lagrangian with respect to q.
What is the Lagrangian
L = T - U
What are the variables officially called in Hamilton
(q,p) conjugate or canonical variables
Generalised Momentum pi = dL/dqi.
What is the Lagrange equations of motion
d/dt(dL/dqi.) - dL/dqi = 0 where L = T - U
How is the hamiltonian defined
H = Sum to i of Pi*qi - L
In a conservative system how can the hamiltonian be written
H = T + U constant in conservative system so no external force
H is thus total mechanical energy
What does changing between the Lagrange formulation to the Hamiltonian correspond to
Changing variable from (qi, qi., t) (qi, qi. independent) to (qi, pi, t) (qi, pi independent)
Consider a free particle of mass m and velocity v what is the proper Hamilntonian
H = px. - L but conservative system to H = T + U
p = mv
KE = T = 0.5mv^2
No PE so U = 0
thus H = T + U => H = T
H = 0.5mv^2 - WRONG as v is q. so need to get rid of
p/m = v => H = p^2 / 2m
What are the cannonical equations of motion
qi = dH/dpi and Pi. = - dH/dqi
What is the cannonical equation of motion for Pi. equal to
Pi. = dH/dqi = dPi/dt = dmv/dt = mx..
What is the 5 step recipe for Hamiltonian Mechanics
- Set up the Lagrangian L = T - U
- Compute s number of conjugate momenta using Pj = dL/dqj.
- Form the Hamiltonian H = pq. - L (eliminate qi. from H to get proper Hamiltonian
- Use qi. = dH/dpi and Pi. = - dH/dqi
- Apply the result of 4 to go back to Newtons law equations of motion
Why is it useful to know its a conservative system for Hamiltonian mechanics
Skip many steps, is conservative H = T + U, write immediately, express T in terms of momenta pi, then go straight to Hamiltonian Dynamics without ever writting the lagrangian
