Lecture 1 - Generalized Coordinates, Inverse Function Theorem, Implicit
Function Theorem
January 20, 2015
1
Generalized Coordinates
The goal of analytical mechanics is to translate the physical world into mathematical relations. This is done using
coordinates. The coordinates establish a one-to-one correspondence between numbers and points in physical space.
We live in a spatially three-dimensional universe, and so each point in space can be represented by a point in a
three-dimensional coordinate system. One such coordinate system is the Cartesian (or rectangular) coordinate system,
which is formed from three perpendicular axes x̂, ŷ, and ẑ. Every point in space is assigned three numbers (x, y, z)
which correspond to a unique point in the coordinate system. Another coordinate system is spherical coordinates,
and in this system, the same physical point will be assigned a different triple of number (r, θ, φ). A coordinate
transformation describes how one set of coordinates relates to another. In this example, we have the transformations
p
x2 + y 2 + z 2
z
θ = arccos p
2
x + y2 + z2
y
φ = arctan .
x
x = r cos φ sin θ
r=
y = r sin φ sin θ
z = r cos θ
(1)
When we have N particles, the system is described by 3N rectangular coordinates (xi , yi , zi )N
i=1 . For notation,
we will assume that some ordering of the rectangular coordinates (xi , yi , zi )N
is
chosen
so
that
{xi , yi , zi }N
i=1
i=1 =
{x1 , x2 , x3 , · · · , x3N }. Each of the xi will evolve in time, and knowing the 3N time-dependent functions xi (t) gives
us a complete description of the system’s behavior. In many situations, the easier task is to express the rectangular
coordinates as functions of other set of coordinates, q1 (t), q2 (t), · · · , q3N (t), just like spherical coordinates. The qi
are called generalized coordinates of the system,. The coordinate transformation from rectangular coordinates to
generalized coordinates is given by some function g : R3N → R3N with component functions gi : R3N → R. The
total transformation is
(x1 , y1 , · · · , yN , zN ) → (g1 (x), g2 (x), · · · , g3N (x))
so that
q1 = g1 (x1 , x2 , · · · , x3N )
q2 = g2 (x1 , x2 , · · · , x3N )
···
q3N = g3N (x1 , x2 , · · · , x3N ).
Here, we are letting x be shorthand for some tuple of rectangular coordinates x = (x1 , x2 , · · · , x3N .
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(2)
Not any set of qi will be permissible and g must be invertible so that we can return to the underlying rectangular
coordinates. In other words, there must be some function f = g −1 with component functions fi : q → xi so that
x1 = f1 (q1 , q2 , · · · , q3N )
x2 = f2 (q1 , q2 , · · · , q3N )
···
x3N = f3N (q1 , q2 , · · · , q3N ).
(3)
Under what conditions can such functions be found? This is given by the inverse function theorem from calculus. To
properly state it, first let us recall that a function g : R3N → R3N is called continuously differentiable or smooth at a
∂gi
point a ∈ R3N if all the partial derivatives ∂x
exist and are continuous in some open set containing a. To characterize
j
smoothness another way, let us introduce the derivative of g, which is the linear map Dg : R3N → R3N represented
by the matrix
 ∂g1
∂g1
∂g1 
· · · ∂x
∂x1
∂x2
3N
 ∂g2
∂g2
·
·
·
·
·
· 

 ∂x1
∂x2
(4)
Dg = 
.
 ···
··· 
∂g3N
∂g3N
· · · · · · ∂x
∂x3N
3N
Then g is smooth at a if the operator Dg is continuous at a. The real number det[Dg(a)] is called the Jacobian of g
evaluated at a.
Theorem (Inverse Function Theorem). Suppose that g : R3N → R3N is smooth in an open set containing a and
det[Dg(a)] 6= 0. Then there exists an open set A containing a and an open set B containing g(a) such that g : A → B
has an inverse g −1 : B → A which, for all b ∈ B, is smooth at b and satisfies det[Dg −1 (b)] 6= 0. The function g is
called locally invertible at a if it satisfies the assumptions of the theorem.
This theorem is extremely powerful since it provides conditions for when a set of generalized coordinates can be
inverted into rectangular coordinates. From now on, we will only consider generalized coordinates
q1 = g1 (x1 , x2 , · · · , x3N )
q2 = g2 (x1 , x2 , · · · , x3N )
···
q3N = g3N (x1 , x2 , · · · , x3N ),
(5)
with g being a smooth map from some open set A to its range B such that det[Dg(a)] 6= 0 for all a ∈ A.
Example. Consider a simple two-variable change of coordinates given by (q1 , q2 ) = g(x, y) = (ex cos y, ex sin y).
Note that the Jacobian is nonzero everywhere, but the function is clearly not 1-to-1 everywhere. It is only invertible
locally.
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Incorporating Kinematical Constraints
Kinematical constraints are physical constraints that can be expressed in terms of mathematical relations between the
coordinates and their derivatives. For example, if the motion of a particle is confined to surface of the unit sphere, then
its coordinates (x, y, z) must satisfy the relationship
x2 + y 2 + z 2 = 1.
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More generally, we can consider a collection of functions
1≤k≤m
φk (x) = 0
which the coordinates must satisfy. For the time being, we will only consider constraints that depend on the coordinates
x. More general constraints will be considered later.
With the m equations φk (x) = 0 for 1 ≤ k ≤ m, we wish to reduce the number of independent variables from
3N to n = 3N − m. This can often be accomplished by using the φk to solve for m of the variables. When can this
be done? An answer is given by the following, where we let Φ : R3N → Rm be the total constraint function with
component functions φk .
Theorem (Implicit Function Theorem). Suppose that Φ : Rn × Rm → Rm is smooth in an open set containing
a = (a1 , a2 ), where a1 ∈ Rn and a2 ∈ Rm . Suppose that Φ(a1 , a2 ) = 0, and further suppose that the last m columns
of the m × 3N matrix
 ∂φ1
∂φ1 
∂φ1
· · · ∂x
∂x1
∂x2
3N
 ∂φ2
∂φ2
·
·
·
·
·
· 
 ∂x1

∂x2
DΦ = 

 ···
··· 
∂φm
∂φm
· · · · · · ∂x
∂x3N
3N
form a submatrix whose Jacobian is non-vanishing at a2 . Then there exists an open set A ⊂ Rn containing a1 and a
unique smooth map Ψ : A → Rm such that Ψ(a1 ) = a2 and
Φ(x, Ψ(x)) = 0
for all x ∈ A.
Any collection of m smooth scleronomic constraints φk are said to be independent at a if DΦ(a) has an m × m
submatrix with a non-vanishing Jacobian. The point a is said to be nonsingular, and the system is then said to possess
n = 3N − m degrees of freedom.
Example. Suppose a particle is constrained to move so that its coordinates satisfy
φ1 (x, y, z) = x + 2y + 3z − 1 = 0
φ2 (x, y, z) = x3 + y 2 − z 2 = 0.
Inspection shows that Φ(−2, 3, −1) = 0. Let us use the Implicit Function Theorem to obtain 3 − 2 = 1 free variable
around the point (−2, 3, −1). We have
!
!
1
2
3 1 2 3
DΦ =
=
.
3x2 2y 2z (−2,3,−1)
12 6 −2
Every 2 × 2 submatrix has a non-vanishing determinant. So there exists a function, say Ψ, so that (x, y) = Ψ(z) for z
near −1 and Φ(ψ(z), z) = 0.
Suppose that we are given m independent constraints φk at some point a = (a1 , a2 ). Then for some open set
A ⊂ Rn with a1 ∈ A, the Implicit Function Theorem guarantees the existence of a function Ψ : A → Rm with
component functions ψi so that we can express the rectangular coordinates as
x1 = x1
xn+1 = ψ1 (x1 , x2 , · · · , xn )
x2 = x2
xn+2 = ψ2 (x1 , x2 , · · · , xn )
···
···
x3N = ψm (x1 , x2 , · · · , xn ).
xn = xn
3
By the Inverse Function Theorem, we can then perform a local coordinate change on the first n coordinates into
generalized coordinates. On the remaining m coordinates, we apply the constraint function Φ. Specifically, let g :
A → B such that for x = (x1 , x2 , · · · , xn ) ∈ A, we have
q1 = g1 (x)
qn+1 = φ1 (x, Ψ(x)) = 0
q2 = g2 (x)
qn+2 = φ2 (x, Ψ(x)) = 0
···
···
qn = gn (x)
q3N = φm (x, Ψ(x)) = 0.
(6)
There exists an inverse f = g −1 that will express the (x1 , · · · , xn ) ∈ A as a functions of (q1 , · · · , qn ) ∈ B. For the
other coordinates, the inverse transformation is given by substitution into the Ψ. Thus, the full inverse transformation
is
x1 = f1 (q1 , · · · , qn )
xn+1 = fn+1 (q1 , · · · qn ) := ψ1 ◦ f (q1 , · · · , qn )
x2 = f2 (q1 , · · · , qn )
xn+2 = fn+2 (q1 , · · · qn ) := ψ2 ◦ f (q1 , · · · , qn )
···
···
xn = fn (q1 , · · · , qn )
x3N = f3N (q1 , · · · qn ) := ψm ◦ f (q1 , · · · , qn ).
We have expressed the 3N rectangular coordinates in terms of n generalized coordinates around some point a. The
functions fk are smooth and the first n of them will have a non-vanishing Jacobian.
The key point is that for every (x1 , x2 , · · · x3N ) satisfying Φ(x1 , x2 , · · · x3N ) = 0, we can uniquely associate this
point with n of the xi , and these can be further associated with the n generalized coordinates:
(x1 , x2 , · · · , x3N ) ↔ (x1 , x2 , · · · , xn ) ↔ (q1 , q2 , · · · , qn ).
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