(Hint: you might want to first work the case of a similar tensor defined using basis covectors rather than basis vectors.)
du dv + dv du.
Solution to the above two problems.
(ii) show that at every point of H the above vectors lie in the hypersurface, which means that the vectors can be "pulled back" to H;
(iii) show that the above vector fields are symmetries of H.
(ii) use coordinates (u,v), where u is the angle of the top rod from the vertical, and v is the angle of the second rod from the vertical. What ranges are appropriate for u and v?
(iii) Suppose gravity acts. What is the potential energy function? Sketch the force 1-form at a few points.
(iv) Suppose a force acts horizontally on the top bob. What is the potential energy function?
(v) Suppose a force acts horizontally on the bottom bob. What is the force 1-form? Figure it out from the principle of virtual work, and explain this in our language.
(vi) Find equilibria for the above two cases.
(v) (Optional) Show that the above two cases are consistent with the reciprocity theorem.