- #1
thrillhouse86
- 80
- 0
Hi all,
I am in a bit of a dilly of a pickle of a rhubarb of a jam with determining the Hamiltonian of a specific system. For background information it is an 2-DoF aero-elastic system where I am (temporarily) neglecting the aerodynamic lift and moment terms.
Being an intrinsically engineering problem, the full equations of motion that include the lift and moment terms have been formulated using Newtonian mechanics. I have temporarily ignored these nonconservative terms and derived a Lagrangian which, upon applying the Euler-Lagrange equations reproduces the equations of motion.
Now I have determined the Hamiltonian by defining the canonical momentum as:
p_{i} = dL/dq{dot}_i and applied the Legendre transform (sure enough it is H = T + V) and I now have the Hamiltonian in terms of my generalised position and velocities - the problem is that I cannot for the life of me write the hamiltonian in terms of the generalised position and momentum.
So I guess my question is two-fold
1. Is there a systematic way of determining the Hamiltonian in terms of the generalized position & momentum given the Hamiltonian in terms of the generalised position and velocity ? Or is the only way a character building algebraic excersize ?
2. Is it possible to have a conservative system which has a Lagrangian but does not have a Hamiltonian ?
Cheeers,
Thrillhouse86
I am in a bit of a dilly of a pickle of a rhubarb of a jam with determining the Hamiltonian of a specific system. For background information it is an 2-DoF aero-elastic system where I am (temporarily) neglecting the aerodynamic lift and moment terms.
Being an intrinsically engineering problem, the full equations of motion that include the lift and moment terms have been formulated using Newtonian mechanics. I have temporarily ignored these nonconservative terms and derived a Lagrangian which, upon applying the Euler-Lagrange equations reproduces the equations of motion.
Now I have determined the Hamiltonian by defining the canonical momentum as:
p_{i} = dL/dq{dot}_i and applied the Legendre transform (sure enough it is H = T + V) and I now have the Hamiltonian in terms of my generalised position and velocities - the problem is that I cannot for the life of me write the hamiltonian in terms of the generalised position and momentum.
So I guess my question is two-fold
1. Is there a systematic way of determining the Hamiltonian in terms of the generalized position & momentum given the Hamiltonian in terms of the generalised position and velocity ? Or is the only way a character building algebraic excersize ?
2. Is it possible to have a conservative system which has a Lagrangian but does not have a Hamiltonian ?
Cheeers,
Thrillhouse86