Newtonian force as a covariant or contravariant quantity

[atyy]

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.

 

[atyy]

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.