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On page-5 of this paper1 by E. Minguzzi titled "A geometrical introduction to screw theory", he writes:

Who adopts this point of view argues that it should also be adopted for forces in mechanics, which should be treated as 1-forms instead as vectors. This suggestion, inspired by the concept of conjugate momenta of Lagrangian and Hamiltonian theory, sounds more modern but would be geometrically well-founded only if one could develop mechanics without any mention to the scalar product.

I have some understanding of forms calculus but I'm not quite sure how the hamilton and lagrangian theories motivate the form's formulation of calling the conservative force as an exterior derivative of the potential. I.e: one form field associated with the potential.

Note: I have not actually done differential forms rigorously and all my understanding of it stems from this2 youtube playlist. So, I'm like for the big picture understanding behind what the author was trying to say there.

References:

1 E. Minguzzi, "A geometrical introduction to screw theory", Eur. J. Phys. 34 (2013) 613 - 632, arXiv:1201.4497.

2 Prof Ghrist Math, "Calculus Blue Multivariable Volume 4 : Fields", YouTube. [Playlist]. Available: here. [Accessed: Oct 7, 2020].

Urb
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tryst with freedom
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  • This might be a HSM question if you're asking for the historic development? – Charlie Oct 07 '20 at 17:22
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    I'm not quite sure, The idea I had whilst asking was how the mathematics of hamilton and lagrangian suggests the form formulation rather than vector one. – tryst with freedom Oct 07 '20 at 17:25
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    This is relevant (although there are more reasons than the ones discussed in the link): https://physics.stackexchange.com/q/131348/226902 ..for example hydrodynamics (both Newtonian and relativistic) is naturally formulated by thinking that momenta and forces are 1-forms. – Quillo Oct 07 '20 at 17:39
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    I think that the idea that the identification that we are dealing with some phase space in analytic (or geometric) mechanics paves the way for identifying force as a differential form, or more specifically, a covector. Because if your phase space has a symplectic structure, then one can use Hamilton's equation to find a phase flow on the geometry. For example, a double pendulum has a phase space of the torus which is indeed symplectic and hence one does uses the Hamiltonian to work. Force as treated in newtonian mechanics is the same as force as a covector in say Hamiltonian mechanics... – Yuzuriha Inori Oct 08 '20 at 20:29
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    ...(contd) because it's just related to the fact that the force is a map from a tangent space ($TM$) of our phase space (here, read manifold) to the tangent space of the cotangent bundle (I might be abusing terms here, but I mean $TT^M$) (manifold) and that this is isomorphic to $T^T^M$ which incorporates Hamiltonian mechanics (or possibly Lagrangian mechanics, I am just not sure at this moment, but one of them corresponds to $T^TM$ and the other to $T^T^M$).

    Maybe someone can expand on this?

     – Yuzuriha Inori Oct 08 '20 at 20:33

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