1. Introduction
In [1], the author proposed the computation of the first digits of π by counting the number of collisions of a system consisting of two balls and a wall. Extensive analysis of this problem has been done, see [2] and references therein.
In the present work, we analyze a similar formulation for a hypothetical hyperbolic situation. Our analysis is based on an original decomposition of the velocity for given values of hyperbolic kinetic energy and hyperbolic momentum. Thanks to a useful transformation, the basic geometry of the problem is reduced to reflections on a hyperbola. It follows that a sequence of interactions becomes a sequence of reflections. Then, for the possible sequences of interactions, it is easy to determine the end of the process and the total number of interactions. Finally for two particular ratios of the masses, the analysis leads to the computation of the first digits of any logarithmic constant
(where p is a prime number) [3].
2. Ideal Physical System
Suppose two points of masses M and m (
), noted pointM and pointm, interacting between each other in one dimension. Let us consider their respective speeds V and v. Suppose also that the point of mass m could interact with a wall. Two quantities will be considered: the h-momentum
and the h-kinetic energy
Let the matrix H and J be defined by
The h-momentum of the two point mass system can be written as
and velocities
having the same h-momentum are on a straight line with direction
because
. The h-kinetic energy
of the two point mass system is such that
and velocities
having the same h-kinetic energy are on a hyperbola.
The direction
will play a special role in the sequel. Let us note
the hyperbolic angle of this direction with a horizontal axis OV, so we have
We will consider two kinds of interactions. The first kind will be between the two points of masses M and m. It will be called h-elastic interactions which means that both h-momentum and h-kinetic energy remain constant. The second kind will be between the point of mass m and the wall. It will produce sign changes of the velocity of the point of mass m, so the h-momentum of the system changes while keeping constant its h-kinetic energy.
By following rules that ensure the alternation of the two kinds of interactions, we will consider the dynamics of the two point masses. We will look at the total number of interactions, counting interactions between the two point masses and interactions between a point mass and the wall. We will see that under certain conditions, the number of interactions corresponds to the first digits of
.
3. Direct Analysis: The Natural Coordinate System
In this section, we analyze the system with respect to its natural coordinate system VOv.
3.1. Observations
Let the velocity be given by
with
, such that
. Let us point out that
. The rule of the process is such that if
1)
, so
, there will be a pointm-wall interaction;
2)
, so
, there will be no interaction;
3)
, so
, there will be a pointM-pointm interaction.
Let us observe that for
3.2. Two-Point Mass System
For the pointM-pointm interaction, the velocity
of constant h-kinetic energy and constant h-momentum lends itself well to a decomposition. The possible velocities in this case are given by the intersection points of a hyperbola (h-kinetic energy) and a line (h-momentum). There are no more than two intersection points. This decomposition will be helpful to explain the transformation of the velocity during an interaction between the two point masses.
3.2.1. Decomposition of the Velocity
We will break down the velocity
using a h-orthogonal basis.
Theorem 1 The set
is a h-orthogonal basis with respect to the quadratic form used to define the hyperbola of constant h-kinetic energy.
Proof. Indeed we have
Moreover
and
We can now decompose the velocity
as follows.
Theorem 2 The velocity
can be written as
where
and
Corollary 1 We have
so
Thanks to this decomposition of
, we will see that a pointM-pointm interaction consists simply in making a change of sign of the coefficient S in this decomposition.
3.2.2. Compatibility Condition
Using the hyperbolic Cauchy-Bunyakovski-Schwarz inequality [4], we get
A more precise expression is given in the next theorem.
Theorem 3 The h-kinetic energy and the h-momentum of the pointM-pointm system are related by the relation
Proof. Using the decomposition of Theorem 2, to be on the hyperbola
must satisfy
so the result follows.
For given compatible E and Q, possible values of the velocity are given in the next theorem.
Theorem 4 Under the compatibility condition
if
1)
, so
, we have two possible velocities
2)
, so
, we have only one possible velocity
We can also obtain a decomposition of the h-kinetic energy and the velocity of the system. Let us introduce the following two average velocities
and
Theorem 5 The h-kinetic energy and the velocity of the system are decomposable as follows
and
3.2.3. Elastic Interaction
For a h-elastic interaction, h-momentum and h-kinetic energy remain constant. The velocity
is therefore one of the two points on the hyperbola described above.
Theorem 6 Let
be the velocity before the interaction, and
be the velocity after the interaction. Then
and
Moreover
Proof. From Theorem 4,
is the second point on the hyperbola obtained by changing the sign of the coefficient S, so
From Theorem 2, we get
Moreover, we have
so the last result follows.
Corollary 2
On the hyperbola of Figure 1, the velocity moves down left along the direction opposite to
from points above the line of direction
to points below the line of direction
, for example from
to
,
to
, and so on.
3.3. Point Mass and Wall System
The pointm-wall interaction is easier to analyze.
Theorem 7 Let
be the velocity before the interaction of massm with the wall and
be the velocity after the interaction. Let
to have a interaction with the wall, so we have
and
Figure 1.
displacements on the hyperbola
.
The h-momentum decreases at each interaction with the wall, and we have
Proof. When the pointm interacts with the wall, it bounces with opposite velocity of the same magnitude, i.e.
. Since the pointM doesn’t interact with the wall,
. For the coefficients R and S, using Theorem 2 we have
For the h-momentum we have
and since
, the h-momentum decreases at each interaction of the pointm with the wall.
Corollary 3
On the hyperbola of Figure 1, the velocity moves upward from points below the OV axis to points above the OV axis, for example from
to
,
to
, and so on.
3.4. Stopping Criterion and Trajectory on the Hyperbola
Suppose the velocity
is
with
, such that
. There will be no new interaction if
1) is the initial velocity and
;
2) is the velocity after a pointm-wall interaction (after moving up vertically) and
;
3) is the velocity after a pointM-pointm interaction (after moving down left) and
.
On Figure 1, the trajectory of the velocity
on the hyperbola is given for the process with at least one interaction. We see that it moves successively from
to
, to
, to
, and eventually up to the final point. There is alternation of moving down left and moving up vertically.
4. A Useful Transformation
The standard parametrization of the hyperbola suggests a way to transform the graph of the h-kinetic energy. Let us consider
. So we can write
where
We will suit to call
the velocity.
The expression for the h-kinetic energy becomes
which is such that this quadratic form coincides now with the hyperbolic standard inner product in
, say the matrix H is now the matrix J. The h-momentum is now
and the lines of constant h-momentum are of direction
.
We also have
5. On Hyperbolic Reflection and Rotation
Let us recall that
so we have
Moreover, for
, we have
Some useful results about hyperbolic rotations and reflections are now given. Let us consider any angle
. For the rotation matrix
of an angle
we have
and
For the reflection matrix
which represents a reflection with respect to a line which makes an angle
with the OW axis, we have
and
To complete this section let us present some identities whose proofs are simple and omitted.
Lemma 8 For any angle
, we have
Lemma 9 For two angles
and
, we have
1)
;
2)
;
3)
;
4)
.
6. Indirect Analysis: The Transformed Coordinate System
In this section we analyze the system with respect to the transformed coordinate system WOw.
6.1. Observations
Let the velocity be given by
such that
, and
. Now, the rule of the process is such that if
1)
, so
, there will be a pointm-wall interaction;
2)
, so
, there will be no more interaction;
3)
, so
, there will be a pointM-pointm interaction.
6.2. Two-Point Mass System
The velocity
can also be decomposed, and the possible velocities are given by the intersection points of a hyperbola (h-kinetic energy) and a line (h-momentum).
6.2.1. Decomposition of the Velocity
Let us express the velocity in terms of the new variables and an appropriate orthonormal basis.
Theorem 10 The set
is an h-orthogonal basis with respect to the quadratic form used to define the hyperbola of constant h-kinetic energy.
Proof. Indeed we have
Moreover
and
Theorem 11 The expression of the velocity
is
where
and
Moreover
6.2.2. Compatibility Condition
The condition remains the same, but we can rewrite the expressions for the velocity.
Theorem 12 Under the condition
1) if
, so
, there are two possible velocities
2) if
, so
, there is only one possible velocity
6.2.3. Elastic Interaction
For a h-elastic interaction, h-momentum and h-kinetic energy remain constant. The velocity
is therefore one of the two points on the hyperbola as described above.
Theorem 13 Let
be the velocity before the interaction, and
be the velocity after the interaction. The velocities are related by the relation
and the coefficients by the relation
Moreover
if and only if
Proof. From Theorem 12, we have
From the decomposition of Theorem 11, we get
Moreover a direct computation leads to
and the last result follows.
Remark We have the following decomposition for the matrix T of the linear system of Theorem 6
On the unit hyperbola of Figure 2, the velocity moves down left along the direction opposite to
from points above the line of direction
to points
below the line of direction
, for example from
to
,
to
, and so on.
6.3. Point Mass and Wall System
In the new coordinate system, for a pointm-wall interaction we have
and
.
Theorem 14 If
, is the velocity before the interaction of the pointm with the wall with
, we have
which is a reflection with respect to the OW axis, the line
. For the coefficients we have
Figure 2.
displacements on the hyperbola
.
The momentum decreases at each interaction with the wall, and we have
Proof. The first result is obvious. For the coefficients, we use Theorem 11 to get
For the momentum
with
.
On the hyperbola of Figure 2, the velocity moves upward from points below the OW axis to points above the OW axis, for example from
to
,
to
, and so on.
Remark Directly, or from Theorem 7 and Theorem 14, we have
6.4. Stopping Criterion and Trajectory on the Hyperbola
Suppose the velocity
is
with
, such that
. There will be no new interaction if
1) is the initial velocity and
;
2) is the velocity after a pointm-wall interaction (after moving up vertically) and
;
3) is the velocity after a pointM-pointm interaction (after moving down left) and
.
On Figure 2, the trajectory of the velocity
on the hyperbola is given after at least one interaction. We see that it moves successively from
to
, to
, to
, to
, and eventually up to the final point. There is alternation of moving down left and moving up vertically.
7. Sequence of Interactions
7.1. The Problem
To any point on the hyperbola
there exist an unknown angle
such that
Using this initial condition, two sequences of alternating interactions are analyzed. We will see that we get an approximation of
which depends on the choice of M and m (
). We will use the notation
for the velocity after the k-th interaction.
7.2. Two-Point Mass Interaction First
For the sequence of interactions starting with a pointM-pointm interaction followed by a pointm-wall interaction, we must have
, or
which also means that
.
Theorem 15 A sequence of two interactions, a pointM-pointm interaction followed by a pointm-wall interaction, is a rotation of angle
.
Proof. Using part (d) of Lemma 9, we have
so the result follows.
Thereafter there is alternation of interactions: pointM-pointm, pointm-wall, etc.
Theorem 16 The velocity after
1) 2n interactions (with a last pointm-wall interaction) is
2) 2n + 1 interactions (with a last pointM-pointm interaction) is
Proof. The process ends after 2n or 2n+ 1 interactions.
1) For 2n interactions, we use part (a) of Lemma 9 to get
2) For 2n + 1 interactions, one more pointM-pointm interaction is needed, so
Then from part (c) of Lemma 9, we get
Starting with a pointM-pointm interaction, considering the preceding expressions for the velocity, and applying the stopping criterion, we conclude that the process will end after
1) 2n interactions if the last interaction is a pointm-wall interaction, so
, of angle
, is on the hyperbolic arc of angle in
;
2) 2n + 1 interactions if the last interaction is a pointM-pointm interaction, so
, of angle
, is on the hyperbolic arc of angle in
.
In both cases, if K is the number of interactions we obtain
or
Note that this result also holds for
because
.
7.3. Point Mass and Wall Interaction First
For the sequence of interactions starting with a pointm-wall interaction followed by a pointM-pointm interaction, we must have
with
, so
with
. It also means that
.
Theorem 17 A sequence of two interactions, a pointm-wall interaction followed by a pointM-pointm interaction, is a rotation of angle
.
Proof. Using part (d) of Lemma 9, we get
so the result follows.
Thereafter there is alternation of interactions: pointm-wall, pointM-pointm, etc.
Theorem 18 The velocity after
1) 2n interactions (with a last pointM-pointm interaction) is
2) 2n + 1 interactions (with a last pointm-wall interaction) is
Proof. The process ends after 2n or 2n+ 1 interactions.
1) For 2n interactions, we use part (a) of Lemma 9 to get
2) For 2n + 1 interactions, one more pointm-wall interaction is needed, so
Then from part (c) of Lemma 9, we get
Starting with a pointm-wall interaction, considering the preceding expressions for the velocity, and applying the stopping criterion, we conclude that the process will end after
1) 2n + 1 interactions if the last interaction is a pointm-wall interaction, so
, of angle
, is on the hyperbolic arc of angle in
;
2) 2n interactions if the last interaction is a pointM-pointm interaction, so
, of angle
, is on the hyperbolic arc of angle in
.
In both case, if K is the number of interactions, we have
or
7.4. Summary of Results
Table 1 presents a summary of our results on the values of K depending on the values of
. Let us use the notation
for K associated to
. So let us observe that for
we have
8. Digits of the Logarithmic Constant ln(2)
Let p be a prime number and consider the logarithmic constant
. Since
we can apply the preceding result with
to find the first digits of
. We will consider
to illustrate the process, so our initial point on the hyperbola will be
, with
, and
[3].
8.1. Observations
In any integer base
of a number system, the integer part of
, noted
in base b, add, to the integer part of
, the first N digits of the fractional part of
in base b. So let us consider an angle
and look at the value of the number K of interactions.
To get
we can consider the following two cases:
A)
, so
, which means that
;
B)
, so
, which means that
. It remains to verify that, if K is the number of interactions, the following conjecture is true.
Conjecture For
(i.e.
), or
(i.e.
), the total number K of interactions which is given by its representation in base b by
consists of the digits of the integer part
and the first N digits of the fractional part of
in base b, so
.
In the sequel, we will use the following representations in base b
and
8.1.1. Case (A)
The Taylor expansion of
is
for
. Also
for
. It can be shown that
for
. Multiplying by
and take
, we have
Since
is not an integer
Using the representations in base b, since
under the condition that there exists
such that
, we get
So
consequently
is not an integer and
8.1.2. Case (B)
The Taylor expansion of
,
for
. Also
for
. It can be shown that
for
. Multiplying by
and take
, then
Since
is never an integer, so
Using the representations in base b, since
under the condition that there exists
such that
, we get
Hence
consequently
is not an integer and
8.2. Consequences
There are consequences of the preceding results. For
Case (A): if
, then
Case (B): if
, then
for
. So the result holds for the powers of b from
up to N.
This last observation suggests a Cauchy induction like method [5]. With an algorithm which can find a digit of
at a precise position N without calculating all digits in positions less than N, see for example [3] [6], we could deduce the result for a number of lower positions. We proceed in the following way. Suppose the property true for
. Then look for the smallest
such that in case (A)
or in case (B)
, then the result holds for
.
8.3. Conjecture Almost Proved
It should be verified, with modern computational facilities, that up to very large values of N, no sequences such that
Case (A):
for
,
Case (B):
for
,
are present in the expansion of
. So the conjecture would be verified for up to very large values of N.
8.4. Final Remark
There exists in fact infinitely many angles
for which we get the result
. Indeed, if we use
with
for
, we also get the result. We observe that
where
and
are the masses for case (A) and case (B). Also
, and
Since the result holds for
and
, it also holds for
for any
.
Acknowledgements
This work has been financially supported by an individual discovery grant from the Natural Sciences and Engineering Research Council of Canada.