Newtonian force as a covariant or contravariant quantity

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Discussion Overview

The discussion revolves around the classification of force as either a covariant or contravariant quantity within the framework of Newtonian mechanics. Participants explore the implications of this classification on concepts such as energy, work, power, and the relationship between vectors and their duals, with references to both classical and modern interpretations.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants reference Burke's assertion that force is a 1-form, arguing that this perspective aligns with the definition of work as a scalar derived from force acting on a displacement vector.
  • Others question whether energy can be considered a true scalar in nonrelativistic mechanics, suggesting it may be a scalar density instead.
  • There is a contention regarding the classification of force as a contravariant vector based on Newton's second law, with some arguing that this contradicts Burke's claims.
  • Participants discuss the identification of dual vectors and vectors in the context of Euclidean space, with some asserting that while they are isomorphic, they are not the same.
  • Some argue that the choice between covariant and contravariant representations for physical quantities should depend on the intended mathematical operations, such as line integrals.
  • A historical reference is made to Bergmann's treatment of force and momentum in his work on special and general relativity, noting inconsistencies in the treatment of 4-momentum.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the classification of force and related quantities, with no consensus reached on whether force should be considered a covariant or contravariant quantity.

Contextual Notes

Participants highlight the importance of definitions and the context in which quantities are considered, noting that the discussion relies on interpretations that may vary based on mathematical frameworks and physical applications.

  • #151
Ok, I actually can't understand the Lewis notes I linked to in #127.

These guys http://www.google.com/url?sa=t&rct=...uLu-lMgYA63SlGua6NB5lOA&bvm=bv.41867550,d.b2I do it differently. In proposition 7.8.1 they do define a force via contraction with a vector field. However they contract with a symplectic 2 form, not a metric. I think the relevant definition of the 2 form is on p161.

Incidentally the Lewis notes say ad hoc "motivation" is that the Euler-Lagrange equations transform as the components of a one form. He say this is hand-waving because the EL equations are not a one-form.
 
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  • #152
I guess the Lewis notes http://www.mast.queensu.ca/~andrew/teaching/math439/pdf/439notes.pdf are fine after all. Force is a one-form. What I was confused by is their notation F: R X TQ -> T*Q. All they mean is that a force is an assignment of a one form to every velocity at every position. Just as they say a vector field is V: Q -> TQ by which they mean a vector field is an assignment of a vector at every position.
 
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