Newtonian force as a covariant or contravariant quantity

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SUMMARY

The discussion centers on the classification of force in Newtonian mechanics as a covariant quantity rather than a vector. Burke's book, "Div, Grad, and Curl are Dead," posits that force operates as a 1-form, which is dual to vectors, and emphasizes that energy is a scalar in this context. Participants debate the implications of this classification, particularly regarding the symmetry between vectors and their duals, and the necessity of defining line integrals for 1-forms. The conversation also references the treatment of force in various texts, including works by Bergmann and Soper, highlighting the ongoing discourse in the physics community about the nature of force and momentum.

PREREQUISITES
  • Understanding of differential forms and their applications in physics.
  • Familiarity with Newtonian mechanics and the concept of force.
  • Knowledge of covariant and contravariant quantities in tensor analysis.
  • Basic principles of line integrals and their significance in physics.
NEXT STEPS
  • Study the concept of 1-forms and their role in physics, particularly in relation to force.
  • Explore Burke's "Div, Grad, and Curl are Dead" for insights on the duality of vectors and 1-forms.
  • Investigate the treatment of force and momentum in Hamiltonian mechanics.
  • Learn about metric-independent formulations of electromagnetism and their implications for physical quantities.
USEFUL FOR

Physicists, mathematicians, and students of theoretical physics interested in the foundational aspects of force, momentum, and the mathematical structures underlying classical mechanics and general relativity.

  • #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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