Lagrangian and External Forces

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Discussion Overview

The discussion revolves around the integration of external forces within the framework of Lagrangian mechanics. Participants explore how external forces, particularly non-gravitational ones, can be represented or accounted for in Lagrangian formulations, contrasting this with systems governed solely by kinetic and potential energies.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses uncertainty about how external forces fit into Lagrangian mechanics, noting that kinetic and potential energies alone do not fully describe systems influenced by such forces.
  • Another participant suggests that external forces can be analyzed using Lagrange multipliers, although they admit to forgetting some details.
  • A question is raised about the possibility of including external forces in the potential energy term of the Lagrangian, prompting further exploration of this idea.
  • One participant argues that non-gravitational external forces cannot be conveniently included in a Lagrangian formulation, suggesting that a work term might be necessary instead.
  • There is a discussion about the limitations of Lagrangian mechanics in representing forces like those from a hand pushing an object, which do not correspond to a potential gradient.
  • Another participant seeks resources or examples to better understand the application of Lagrange multipliers in this context, expressing confusion about their relevance.
  • It is reiterated that Lagrange multipliers are related to constrained optimization, with constraints often manifesting as forces in physical systems.

Areas of Agreement / Disagreement

Participants do not reach a consensus on how to incorporate external forces into the Lagrangian framework, with multiple competing views and uncertainties remaining regarding the appropriate methodology.

Contextual Notes

Participants express limitations in their understanding of how external forces can be integrated into Lagrangian mechanics, particularly regarding the need for additional terms or methods like Lagrange multipliers. There is also a lack of clarity on the applicability of these concepts to specific examples.

cmmcnamara
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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!
 
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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.
 
intuitively I know that kinetic and potential energies can't describe the systems as a whole

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

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.
 
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?
 
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.
 
Ok I see, thank you very much!
 

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