Lagrangian and External Forces

cmmcnamara

Hi all,

Doing some self-study on Lagrangian uses on the internet and I'm getting it pretty well thus far, but I'm just not sure how external forces fit in exactly. Up until now I've only tackled problems with gravity and constraints involved but intuitively I know that kinetic and potential energies can't describe the systems as a whole since external forces don't factor in that way. Is there something that shows a derivation/example (where I imagine work fits in) that someone might link me to? Thank you!

Matterwave

External forces like constraint forces can be analyzed in Lagrangian mechanics using Lagrange multipliers.

Sadly, I have forgotten some of the details, so I can't help you here off the top of my head.

StationZero

Why not? Why can't the external forces just be included in the potential energy side of the equation?

cmmcnamara

The reason I don't think it can describe it full is because it takes kinetic and potential energies into account but not the influence of a non-gravitational external force; I'm thinking a work term would be needed. I'm not sure, I'm still learning the methodology here. Could you provide me with some examples perhaps?

Matterwave

The two fundamental macroscopic forces of nature, gravity and electromagnetism, can be described using a Lagrangian framework. Although, in the case of a magnetic field, the Lagrangian is not L=T-V since there is no "potential" (V) for a magnetic field.

If you are talking about "external force" like "my hand pushing on this object", where the force applied is not expressible as a gradient of a potential, then there is no really convenient way to include that into a Lagrangian formulation. The closest you can get is the forces of constraint using the Lagrange multiplier formalism.

cmmcnamara

Sorry to be seemingly in resourceful but are there any links to something like that? Googling gets me Langrangian multipliers which I'm not sure is what I need, last I recalled that was a multi variable optimization method. Or is this indeed what I should be looking at?

Matterwave

Lagrange multipliers, like I mentioned before, deal with constrained optimization. The constraints, for a physical system, usually come in the form of constraint forces (e.g. the normal force), which are about as close to "external forces" in the sense that you are talking that I can think of.

cmmcnamara

Ok I see, thank you very much!