In the determination of the probability distribution of the Euclidean length between a fixed point and a second point picked at random from the surface of a sphere in n dimensions the following spherical coordinates were introduced. [These can be found stated explicitly for n = 4 in 17 and, in general for any n, in 4, pp. 593-595, 10, 13, pp. 435-437, 14 p. 11]
This particular choice of coordinates is but one example of a
family of "polyspherical coordinates" which can be used to specify
points in n dimensional space. A graphical scheme or tree for developing
such coordinates was given by Vilenkin, Kuznetsov and Smorodinskii in the
1960's , though G. N. Watson made implicit use of such coordinates
[16, pp 420-421] long before. Polyspherical coordinates seem to
be used primarily to solve problems that arise in mathematical physics
[2, 3, 10]. It might therefore be desirable to introduce
this transformation, which basically generalizes a topic from elementary
calculus, to a wider mathematical audience.
For and let be the transformation on in to in defined by
Suppose that for Now consider
Hence by an inductive argument equation (2) is established.
From equations (1) and (2), this simplifies to
From theorem A2, it follows that maps elements of to points in which lie on the dimensional "surface" of a hypersphere of radius centered at the origin.
be subsets of
defined as follows:
Now, let , then . Define the unique angle, ; since , , , and .
Now for , . Hence, , so is a unique angle in with , and .
Now suppose that for there are k unique angles , , in with . Then it follows that for each such m, .
Now, . But theorem A1 implies that Hence, , so is a unique angle in with , and So by an inductive argument there exist unique angles in with and for . Let r be the positive number given by . Then , but by theorem A1 this reduces to So the ordered pair lies on the unit circle and there is a unique angle in with and Thus, there exists a unique with for . This completes the proof.
For the transformation from to is obviously many to one since any choice of angles in gives for . However, even for the transformation is many to one as the following theorem makes explicit.
Case 1a: If , then for and or . Hence, and therefore any choice of the angles for gives for .
this is the only possible case.
Case 1b: If , then at least one for and with . However, , so is a unique angle in .
Case 2a: If , then and for and or . Hence, and therefore any choice of the angles for gives for .
Case 2b: If , then and at least one for and with . However, , so is a unique angle in .
Proceeding in this manner, suppose for it has been established that there are m unique angles , for and with
Now , so by theorem A1 . Therefore, is a unique angle in .
Case ma: If ,
then by theorem A1
Therefore, for and or . Hence, and any choice of the angles for gives for .
Case mb: If , then by theorem A1 and so at least one for and with . However, again by theorem A1 , so is a unique angle in . This case is only possible for or . Therefore, the maximum number of uniquely defined angles in is the maximum of , i.e., .
Hence, for any element of one of the following two cases must occur.
Case a: All for for some l with and there are uncountably many in with for .
Case b: There are
unique angles with
In particular, , but by theorem A1 . So and is either 0 or with . Thus, any angle in gives the correct results that
Therefore, there are uncountably many in with for .
From theorems A3 and A4 the map is onto but many to one. This degeneracy of the transformation is already seen in three dimensions when and for then the azimuthal angle is completely undetermined.
volume (or content) of
is 0 since are
held fixed. Therefore, the dimensional
equals the dimensional
volume of .
This "volume" (the "surface area" of a hypersphere of radius)
equals the dimensional
integral of the Jacobian of the
[11 p.186]. The n dimensional integral of this same Jacobian
over the domain
gives the volume in n dimensions of a hypersphere of radius a.
The direction in associated with a change of a given coordinate u is along the vector whose k' th rectangular component is given by with [5, pp. 212-215, 12, pp. 141-142]. For the transformation the k' th rectangular component of a vector which "points" in the direction of increasing is . Similarly, the k'th rectangular component of a vector in the direction of increasing is .
The transformation equations of (1) yield the following explicit expressions.
For let be an matrix defined as follows:
Thus, and .
While in general,
From equations (4) and (8)
From equations (5) and (9)
From equations (6), (7), and (9) for ,
To explore the properties of the matrix, ,
the following trigonometric identity will prove useful.
For the base case, and
Now assume that .
Consider . In the sum over k make the substitution . This gives
which is 1 by the induction hypothesis.
matrix B is orthogonal if and only if ,
the transpose of B [7, pp. 291-292, 312-313,
8, p. 93]. Thus, if B is orthogonal, where is the identity matrix.
These two products are expressed in the following two equations.
So an equivalent definition of an orthogonal matrix is an matrix whose n columns (or rows) form an orthonormal basis for the space of n dimensional column (or row) vectors. Finally, from the product rule for determinants, if B is an orthogonal matrix, [7, pp. 206-207]
Part I. Demonstrate that for ,
Case 1: j = 1.
From equation (8) and theorem A2
Case 2: .
From equation (9)
but by lemma 1 this reduces to .
Case 3: .
Case 4: .
Part II. Demonstrate that for and and ,
From the definition of in
equations (8) and (9) this reduces to considering the following nine cases.
Case 1: and for .
Case 2: and for .
By lemma 1 this is simply
Case 3: and for .
Case 4: and for .
Case 5: and for .
Case 6: and for .
Case 7: and for .
Case 8: and for .
Case 9: and for and .
from lemma 1.
The n columns of
are the n orthogonal unit vectors associated with the n variables
. This means that
constitute a system of orthogonal curvilinear coordinates
[6 pp. 21-27, 8, p. 219, 9 pp. 279-284, 18]. From equations (10) through (12) the "scale" factors of this curvilinear system are given as follows: [4, p 594, 13, p.436]
Since is an orthogonal matrix its determinant must be either 1 or . A more precise statement is made in the next theorem.
For , .
Suppose , then the bottom row is . Expanding the determinant about the bottom row gives , where is an minor determinant. The matrix M is defined as .
From the substitution for and equations (8) and (9) it follows tha t Thus, .
Suppose . Multiply the first column of by and add this product to the second column of to generate a new column, B .
The determinant is not changed by replacing the second column of by B . In this new matrix the only non-zero entry in the bottom row is in the position. Evaluating by expanding about this row gives
where is an minor determinant. The matrix N is defined as ,
but as before the substitution for and the definition of the matrix results in the simplified form:
However, this is just to the power .
Proof: From equations (10) through (12)
is given in terms of by
the following equations.
Now by factoring out the scale factor from each column of
An equivalent argument would be to use the result thatis
an orthogonal matrix, so that from the scale factors the volume element
in n dimensions is given by
For and , . The inverse function theorem [ 1 pp. 256-257, 5, pp. 150-151, 8, p. 205 ] then provides an additional argument that the map is injective (i.e., one to one).
The author thanks Professor Charles Goebel of the University of Wisconsin-Madison Physics Department for the reference to the work by Vilenkin, Kuznetsov and Smorodinskii. This proved invaluable in finding other relevant references.
1. R. Bartle, The Elements of Real Analysis, John Wiley, 1964.
2. X. Chapuisat and C. Iung, Vector parametrization of the N-body problem
in quantum mechanics: polyspherical
coordinates, Phys. Rev. A 45, 6217 (1992).
3. Ye.M.Hakobyan, G.S.Pogosyan, and A.N.Sissakian, On a Generalized
D-Dimensional Oscillator: Interbasis Expansions,
Phys. Atom. Nucl. 61, 1762-1767 (1998).
4. S. Hassani, Mathematical Physics: A Modern Introduction to its Foundations, Springer, 1999.
5. W. Kaplan, Advanced Calculus, 2nd ed., Addison-Wesley, 1973.
6. P. Morse and H. Feschbach, Methods of Theoretical Physics, Part I, McGraw Hill, 1953.
7. B. Noble, Applied Linear Algebra, Prentice-Hall, 1969.
8. L. Rade aqnd B. Westergren, Beta Mathematics Handbook, 2nd ed., CRC Press, 1990.
9. K. Riley, M. Hobson, and S. Bence, Mathematical Methods for Physics
and Engineering, Cambridge University Press,
10. M.A. Rodriguez and P.Winternitz, Quantum superintegrability and
exact solvability in n dimensions,
J. Math. Phys., 43, 1309-1322, (2002).
11. W. Rudin, Real and Complex Analysis, 2nd ed., McGraw-Hill, 1974.
12. M. Speigel, Schawm's Outline of Theory and Problems of Advanced Calculus, Schawm Publishing Company, 1963.
13. N. J. Vilenkin, Special Functions and the Theory of Group Representations:
Translations of Mathematical
Monographs, Vol 22, American Mathematical Society, 1968.
14. N. J. Vilenkin and A. U. Klimyk, Representation of Lie Groups
and Special Functions, Vol 2: Class I
Representations, Special Functions, and Integral Transforms, Kluwer Academic Press, 1993.
15. N. J. Vilenkin, G. I. Kuznetsov and Y. A. Smorodinskii, Eigenfunctions
of the Laplace operator realizing
representations of the groups U(2), SU(2), SO(3), U(3) and the symbolic method, Sov. J. Nucl. Phys 2, 645-655
16. G. N. Watson, Theory of Bessel Functions, 2nd edition, Cambridge University Press, 1944.
17. Wolfram Research: The hypersphere entry at the website http://mathworld.wolfram.com/Hypersphere.html .
18. Wolfram Research: The curvilinear coordinates entry at the website
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Document first posted 6/19/2002
Document last revised 9/27/2002