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G Hamiltonian Mechanics
Similarly, it must be possible to express the Cartesian coordinates in terms of the
generalized coordinates as
x i = x i (q 1 , q 2 , · · · , q 3N ; t), y i = y i (q 1 , q 2 , · · · , q 3N ; t), z i = z i (q 1 , q 2 , · · · , q 3N ; t) .
(G.1.2)
If one set of these equations can be constructed, it can be inverted (solved) to
obtain the other set. As you will know from linear algebra and calculus courses
(see also Sect. B.2 of Appendix B), such a solution is only possible if the Jacobian
determinant
∂(q 1 , · · · , q 3N )
∂(x 1 , y 1 , z 1 , · · · , z N )
≡
∂q 1
∂x 1
∂q 2
∂x 1
· · ·
∂q 3N
∂x 1
∂q 1
∂y 1
∂q 2
∂y 1
· · ·
∂q 3N
∂y 1
. . .
. . .
. . .
. . .
∂q 1
∂z 1
∂q 2
∂z 1
· · ·
∂q 3N
∂z 1
does not vanish at any point. We shall, however, treat this condition here only as a
formal requirement.
Extended Example Two-dimensional motion in a plane expressed in terms of
Cartesian and plane polar coordinates for comparison.
As you will recall, Cartesian and plane polar coordinates are related via
x = r cos θ , y = r sin θ ,
r = (x
2
+ y
2 )
1
2 , θ = tan
−1 y
x
.
We shall employ unit vectors i and j in the Cartesian system and unit vectors n and
l in the polar coordinate system. Remember that the unit vectors i and j, defined
in the directions of increasing x and y, respectively, are constant vectors, and are
thus independent of their location, while the unit vectors n and l associated with the
directions of increasing r and θ , respectively, are functions of θ , though not of r.
It may be deduced from Fig. G.1 that
n = i cos θ + j sin θ ,
l = −i sin θ + j cos θ ,
from which we see that
dn
dθ
= l ;
dl
dθ
= −n .
We shall now use the representation of r in the form r = rn(θ ) to study the motion
of our particle in plane polar coordinates. The velocity vector is given as