From Newsgroup: sci.physics.research
Hendrik van Hees <
hees@itp.uni-frankfurt.de> wrote or quoted:
In the Lagrange formalism you deal with curves x^j(t) and
d_t x^j(t)=\dot{x}^j
obviously transform like vector components, and the Lagrangian should be
a scalar. since the \dot{x}^j are vector components, and thus
p_j = \partial L/\partial \dot{x}^j
are the components of a co-vector.
Now I see that this explanation is also given in [1], where
the author writes in "2.3c. The Phase Space in Mechanics",
|For the present we wish merely to draw attention to certain
|basic aspects that seem mysterious when treated in most
|physics texts, largely because they draw no distinction there
|between vectors and covectors.
and
|Hamilton was led to define the functions
|p_i(q,q.):=dL/dq.^i (2.26)
. . .
|Equation (2.26) is then to be considered
|not as a change of coordinates in TM but
|rather as the local description of a map
|p: TM --> T*M
|/from the tangent bundle to the cotangent
|bundle/.
after he has shown that the components of "p" transform
like the components of a covector.
[1] "The Geometry of Physics" (2001) - Theodore Frankel.
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