Deriving Hamilton's Equations for a Particle: A Non-Physicist's Guide

[gumby55555]

I'm a PhD student (not in physics) working on a research problem where I need to use the Lagrangian/Hamiltonian approach for a problem. Suppose I have a particle/object that I can track the location/trajectory of; is it possible for me to derive or enumerate Hamilton's equations for that object? If so, how would one go about doing it?

The end goal would be to get a trajectory in phase space... I've been trying to read quite a few books but it's been tough going and I can't figure out how to solve the actual problem or how to approach it. Is it possible to take a trajectory from (x,y,t) and get a trajectory in phase space (q,p)?

If you could outline how or refer me to a source that makes it easy for those of us who are dummies in physics, I'd really appreciate it! :)

 

[jostpuur]

When you have a trajectory $t\mapsto x(t)$ in the coordinate space, and a Lagrange's function $(x,\dot{x})\mapsto L(x,\dot{x})$ fixed, the corresponding trajectory in a phase space is $t\mapsto (x(t),p(t))$, where

$$p(t) = \frac{\partial L(x(t),\dot{x}(t))}{\partial \dot{x}}.$$

So if you have only a trajectory in the coordinate space, but not a Lagrange's function, you cannot map this trajectory into phase space.

Furthermore, if you have only a one trajectory in the coordinate space, there can be several Lagrange's functions, which would imply this trajectory with Euler-Lagrange equations. So absolutely, there will not be a unique trajectory in phase space for any single trajectory in a coordinate space.

You will need to know something to get the Lagrange's function. For example, if you are only interested in Lagrange's functions of certain form, then a fixed trajectory could fix the Lagrange's function too.

 

[gumby55555]

Thanks so much for your reply, jostpuur... I really appreciate it! I hope you'll excuse my ignorance about some of this but couldn't I use the trajectory, x(t), and compute the $$\dot{x}$$(t) from that and put that into the Lagrangian to get the phase space trajectory?

I was very intrigued by what you said here, btw:

You will need to know something to get the Lagrange's function. For example, if you are only interested in Lagrange's functions of certain form, then a fixed trajectory could fix the Lagrange's function too.

Could I please inquire as to what the "something" is that might help fix the Lagrange's function? Also, what would be the various forms of the Lagrange's functions? Would they just be the various KE and PE terms for various situations plugged into T - V? Or would they involve some coordinates' transformations, as well?

I'm sorry to ask all these questions and I really appreciate your help! If you'd like to point me to some resources that might help a struggling non-physicist, I'd very much appreciate that, too! I've tried plowing through Goldstein but find that very tough going and it seems like I'd need months to get a handle on this problem via that route. I also got the Schaum's Lagrangian Mechanics problem solver but got overwhelmed with the number and variety of problems and didn't see where they discussed something similar to this. But thanks again... I REALLY appreciate your help!

 

[jostpuur]

Thanks so much for your reply, jostpuur... I really appreciate it! I hope you'll excuse my ignorance about some of this but couldn't I use the trajectory, x(t), and compute the $$\dot{x}$$(t) from that and put that into the Lagrangian to get the phase space trajectory?

Since you first only mentioned the trajectory, but not a Lagrangian, I thought that a Lagrangian would not be available in the problem. But if you know what Lagrangian you are using, then you can get the path in the phase space simply by using the definition of the canonical momentum, which is $p=\frac{\partial L}{\partial \dot{x}}$. It sounds like this is what you mean by your question, so the answer is yes, that is how it goes.

If the problem was merely to recall the definition of the canonical momentum (or the phase space), then it could be I understood incorrectly what kind of problem you have. Try not to get distracted by all of my comments.

 

[gumby55555]

I think the problem might have been my inadequate description, Jostpuur; I guess I should be more specific! We essentially have some tracking data of a particle's position over time in 2-d (actually, of a couple of objects) and I I'd like to transform it to a Lagrangian/Hamiltonian format. I'm trying to see if this is an approach I'd like to pursue or not: i.e., assuming I have the trajectory (x,t) of a particle in two dimensions, how can I get the phase space trajectories and then compare those trajectories. I was sort of assuming, based on the Lagrangian/Hamiltonian approach, that once I have (x, x_dot, t), I should be able to derive the Lagrangian and then use that to come up with the Hamiltonian equations, which should give me the phase space trajectory. But I just cannot figure out how to do the actual problem of deriving the Lagrangian from the trajectory (including going from (x,t) to (x_dot,t)) and I'm not sure where to read or research more about this (I've basically been reading Goldstein and Schaum's Lagrangian Mechanics problem solver).

The end goal would be to get a trajectory in phase space (q,p) (ostensibly using the Lagrangian/Hamiltonian approach) for each of the particles/objects and then to compare them to see what their relationships are in phase space. Is it possible for me to derive or enumerate Hamilton's equations for these objects? If so, how would one go about doing it? I've been trying to read quite a few books but it's been tough going and I can't figure out how to solve the actual problem or how to approach it. Is it possible to take a trajectory from (x,t) and get a trajectory in phase space (q,p)?

The difficulties I see so far are not having a Potential Energy (which we can assume to be zero or gravitational), not having the masses (but we can assume them to be unit mass), and not having any forces or constraints. But I think it should be possible to use the (x,t) to get (x_dot,t), no? And from that, the Lagrangian? Finally, I'm not sure about which change of coordinates to do, if any, for this kind of a problem...

But, if it helps, the data would be something like this for 1-d (I just made up the numbers):

time, x_position
0, 3
1, 5
2, 8
3, 15
4, 11
5, 17