Deriving a Hamiltonian from dimensionless equations

[thrillhouse86]

Hey all,

(As I mentioned in my previous post) I am trying to derive the Hamiltonian for a aeroelastic system, where the dynamical equations of motion (determined by Newtonian Mechanics) are known.

My process has been to
1. "guess" a form of the Lagrangian, check that it recreates the equations of motion
2. determine the canonical momentum
3. Perform a Legendre transform from the Lagrangian to the Hamiltonian

I've been thinking recently though, that the equations of motion are in dimensionless form (i.e. the equations have been appropriately scaled so that they are unitless). My question is:
Is it acceptable to derive the Hamiltonian from the dimensionless equations of motion, or do I have to use the "undimensionalised" equations of motion ? -

My rationale for this question is that the Hamiltonian does have to give me the total energy of the system - and I'm not sure that working from dimensionless equations will give me that ...

Cheers,
Thrillhouse

 

[Almsco]

Hi Thrillhouse,

It sounds like you're on the right track and have done a lot of work already. I think it is better to use the "undimensionalised" equations of motion when deriving the Hamiltonian, as that will give you the total energy of the system. The dimensionless equations might not provide the correct results. Good luck with your research!

 

[charng]

Hi Thrillhouse,

Thank you for sharing your process for deriving the Hamiltonian for your aeroelastic system. It seems like you have a good understanding of the steps involved.

To answer your question, yes, it is acceptable to derive the Hamiltonian from dimensionless equations of motion. In fact, working with dimensionless equations can often simplify the derivation process and make the resulting Hamiltonian more elegant. The Hamiltonian is a mathematical construct that represents the total energy of a system, and it is not dependent on the units of the equations of motion. As long as the equations of motion are correctly scaled, the resulting Hamiltonian will accurately represent the total energy of the system.

However, it is important to note that the Hamiltonian may have different numerical values when calculated from dimensionless equations compared to undimensionalized equations. This is because the scaling factors used in the dimensionless equations can affect the magnitude of the Hamiltonian. But as long as the relative values of the Hamiltonian components remain the same, the resulting dynamics of the system will be the same.

In summary, it is perfectly acceptable to derive the Hamiltonian from dimensionless equations of motion. Just make sure that the equations are correctly scaled and that the resulting Hamiltonian accurately represents the total energy of your system.

Best of luck with your research!