When you release a toy gyroscope on its pedestal, it
initially has no precessional rotation around that pedestal. A
moment later, it is merrily precessing around the
pedestal! **
Both Kinetic Energy and Angular Momentum now exist where they did not
exist a moment earlier. ** We
will discuss below that the Kinetic Energy is easy to explain, as appearing due
to the body of the gyroscope dropping a tiny fraction of a millimeter in the
Earth's gravitational field, giving up exactly the correct amount of Potential
Energy. So Conservation of Energy definitely does apply as we have always known.
** But the Angular
Momentum of the Precession which just appeared does not come from anywhere!
This is therefore a Violation of the
Conservation of Angular Momentum, which still appears to be universally true
for all other situations than gyroscopic-type precessional motions.**

** The Earth is an even more impressive example of this same effect. ** In
school, you were taught that the Earth SLOWLY "wobbles" ( precesses),
somewhat resembling a child's wobbling top, in about 26,000 years. ** Even your
Teacher did not know that on every March 21 and September 21, that Precession
absolutely STOPS COMPLETELY.** (This
statement is technically only true for the SOLAR portion of the Earth's
Precession, and a similar statement can be made about twice every month
regarding the Moon's portion of it . We are simplifying
a little here in just considering the Sun's for now! ) **
And then it increases to become about twice the
average rate around June 21 and December 21. During
this complex dance, the Earth's Precessional motion CREATES massive amounts of
Angular Momentum, between Mar 21 and Jun 21, and then gives it back up, between
Jun 21 and Sep 21.**

There is NO SOURCE FOR THAT ENORMOUS AMOUNT OF ANGULAR MOMENTUM TO HAVE COME FROM, OR TO HAVE GONE BACK TO! (It is related to the gravitational field of the Sun or Earth, but in a peculiar way. ) I have a different web-page that discusses what acts as a VIOLATION OF THE CONSERVATION OF ANGULAR MOMENTUM, in that precession related angular momentum appearing and disappearing on a regular basis.

** Energy is what is called a scalar quantity, a number
without any direction, and energy can be converted back and forth between being
Potential Energy and Kinetic Energy or other forms.
**Angular Momentum is very different, as it is a Vector quantity. It has a
direction! In both the toy gyroscope and the Earth,
the axis if the Precession is what can be referred to as vertical
. It cannot be converted or changed into anything else, even
any vector into any other direction, and it could not have come from anything
else or even any angular momentum vector along any other axis.
A true problem!

Down below, we will show the calculations of these effects. **
As to Kinetic Energy of the Earth's Solar Precession,
that amount of energy is zero on March 21 and it increases to the Earth having
10,000 trillion joules of Precessional Kinetic Energy just three months later
. **This is a significant quantity of energy being transferred into the Precessional
motion, which then has to all be returned to its source during the next three
months to again become zero. **
The RATE of energy transfer is rather impressive! **The
fact that it is spread out over a three-month period means that **
the average rate of energy transfer (power) into and
out of Precession is about 1,250,000 kiloWatts**, but which constantly
varies itself.

** For the record, if we could ever figure out a way of
capturing that energy and making it into electricity, that's more than a
nuclear power plant could provide!**

In fact, the similar Precession caused by the Moon on the Earth is actually both
larger and faster changing. Roughly 40,000 trillion
joules is transferred into and out of the Earth's Lunar Precession over
approximately a week interval. **
This results in a RATE of energy transfer being far
greater than that due to the Sun, being around 63,000,000 kiloWatts as an
average. **This
SEEMS potentially interesting to us humans as that AVERAGE amount of power from
just Lunar Precession is actually comparable to the ENTIRE electricity
production of ALL the 104 nuclear power plants in the United States
! Unfortunately, mankind does not seem to know of any way of
CAPTURING any of that power! (All these calculations are included below).

Actually, the two examples of the toy gyroscope's precession and the Earth's Precession are OPPOSITE each other in one way. The effect of the Earth's gravity on a toy gyroscope acts to try to INCREASE THE TILT of the spin axis of the gyroscope, which then causes the Precessional motion (per Euler's Differential Equations) to tend to fall over. In contrast, the effect of the Sun's gravity on the tilted Earth acts to try to STAND UP the Earth's spin axis, which then causes the opposite sort of Precessional motion.

We provide another web-page which is specifically on Precession which provides the math to support both of these situations.

The work of Euler in developing the three-dimensional equations of Newton's one- and two-dimensional Mechanics (which are generally called the set of three Euler Differential Equations) has rarely been fully comprehended or used.

LaPlace, LaGrange, Leverrier and other brilliant early astronomers all made an
ASSUMPTION that now appears to have been slightly wrong! They
correctly recognized that Conservation of Energy must apply to planets orbiting
the Sun. ** They also accepted that Angular Momentum is always Conserved** which turns out
not to always be true! **
Given those two conditions, it is rather easy to mathematically show that
two planets might perturb each other in a variety of ways, but NEVER by ever
being able to change the radius of each other's orbits.
**And so they all agreed on
an absolute statement to that effect, that planets could never alter the
semi-major radius of each other's orbits. **
THAT statement is now seen to be clearly incorrect,
but ONLY when gyroscopic-types of effects apply.**

Their reasoning and their resulting conclusion is generally true. **
However, this presentation shows that there IS one
process which violates the Conservation of Angular Momentum, which then PERMITS
planets to (very, very slowly) alter each other's orbital radii. **It is
the gyroscopic precessional processes by which planets can affect each other.

Those brilliant scientists and mathematicians and others must certainly have seen toy gyroscopes that APPEAR to have very constant Precession, and made that incorrect assumption. They even developed the calculations to show that such a toy gyroscope gradually precesses FASTER as friction gradually slows the rotor spinning down, but that turns out to be a rather different effect.

** They had overlooked an obvious fact that when a toy
gyroscope is FIRST released, its Precessional motion MUST ACCELERATE from the
initial non- precessing situation. Angular
Momentum around the precession axis simply BEGINS out of nothing! **ZERO
Angular Momentum around the precession axis rather quickly becomes some
non-zero number, which then remains essentially constant thereafter. ** The so-called Law of
Conservation of Angular Momentum says that cannot happen!
**This is the simplest and
most obvious example of a clear Violation of that Law of Conservation of
Angular Momentum.

** As long as gyroscopic precession effects do not occur,
then it IS true that Angular Momentum IS Conserved, and the mathematics and the
conclusions of those brilliant scientists are true. It
is ONLY when precession occurs that this Violation can and does occur.**

Among the solar system, IF a perturbing object (planet, moon or sun) happens to be on the equator of a spinning other object, that second object is NOT perturbed in this gyroscopic manner. However, most of the time, the perturbing object is either ABOVE or BELOW that equatorial plane, and a precessional effect DOES occur. Kinetic Energy is then transferred from one to the other, an effect which has long been known, and which results in effects such as Regression of the Nodes of the orbit of the target object. It has always been ASSUMED that Angular Momentum was also Conserved, but that is NOT precisely true, due to this Violation of the Conservation of Angular Momentum.

However, in practical terms, the effect of this Violation is quite small, and it has apparently always been neglected. The fact that this is such a small effect for objects like the planets and moons of the solar system, makes the effects VERY small and therefore slow to cause changes great enough for detection.

It also turns out that the common belief that planetary Precession is constant is not remotely true! In fact, for the Earth's Precession, which school children learn takes around 26,000 years to complete one wobble, the actual motion is extremely complex. EVERY March 21 (or more precisely, at the instant of the Spring Equinox), Precession due to the Sun's effect (Solar Precession) briefly vanishes! At that instant, there is ZERO Solar Precession! The effect then rapidly increases, up to a maximum Precession rate around June 21 (Summer Solstice), when it has become roughly TWICE what the average annual Precession rate is measured to be. Then it decreases again to again become zero at the Autumnal Equinox (around Sept 21) and again increases to a (positive) maximum at the Winter Solstice (around Dec 21).

The math behind these statements are fairly simple and they are provided in a linked page about Precession.

The huge amounts of Kinetic Energy involved in these rather rapid and enormous changes of speed of the Earth's Precession, actually do NOT violate Conservation of Energy! The energy "which appears" actually came from an identical amount of Kinetic and Potential energy of the Sun-Earth system, and those amounts get returned to the Sun-Earth system during the following three months! But as to Angular Momentum (of the Earth-Sun system), it is NOT Conserved, as neither the Sun nor Earth loses any Z-axis Angular Momentum in those three months while the Earth's Precessional motion is accelerating (Mar 21-Jun 21 or Sep 21-Dec 21).

This all turns out to be immensely important ! The usual Euler Equations are used, but they must be Integrated, in order to analyze the VARIATIONS in the rates of Precession and the consequences of that IN A DIFFERENT AXIS (direction). THIS relatively simple mathematical procedure can then explain WHY AND HOW a toy gyroscope has a precessional motion "appear" where it had not previously existed. At least, it explains where the Energy came from!

** This then results in a VIOLATION of an universal
concept in science, the Conservation of Angular Momentum!
**(The results of
this effect are rather small and for planets are only of major effect after
many thousands of orbits, thousands of years for planets and moons.)

And THAT then allows all sorts of incredible consequences!
LaPlace and the others ASSUMED that when planets perturbed
each other, that they COULD NOT alter the overall orbital radius (because that
would be impossible if both Energy and Angular Momentum are conserved)
. But that is now seen to be wrong, which then provides
explanations for many astronomical phenomena which have never been properly
explained before. Much of Astrophysics probably needs
to be re-examined and re-written to be more correct. The
differences are very slow in effect. For the four
Galilean moons of Jupiter to have gotten themselves into a fascinating mutual
pattern, certainly took many millions of their orbits to create
. And IF these effects would ever cause an EXACT
commensurability of two moons or planets, the same effects would destabilize
the situation. So the four Galilean moons are not
QUITE in perfect commensurability, but tantalizingly close! Even
the amount of this meta-stability can be calculated, with results that are very
close to observed differences of those moons, the relationship between Jupiter
and Saturn, aspects of Saturn's rings and moons, the Kirkwood Gaps in the
asteroid belt, and much more. **
Such patterns are not just coincidences or accidental
as some scientists claim! They are not only logically
sound, but the math confirms that fact!**

** The AVERAGE Precession rate of the Earth's wobble is
as we all learned in school, but it ACTUALLY is constantly rather wildly
CHANGING, constantly accelerating and decelerating.**

** All those CHANGES in the rates of precession have
effects similar to those seen when releasing a gyroscope.
**There are EFFECTS
which cause the planets to constantly be increasing and decreasing in their
precessional rates. They are called Perturbations, and
they are forever causing slight changes in the shape and orientation of each
planet's orbit, and in the rotation and orientation of each planet's spinning
. The REALITY in the solar system is as though we would see
a toy gyroscope's precession keep starting and stopping in a herky-jerky
motion!

Precession between planets can act on planets in two very different ways. There is the spinning precession like that we are all familiar with, where the effects of changes in the rate of precession only has the effect of tiny wobbles in the precise orientation of the spin axis. The Earth shows such wobbling of a small fraction of a mile, where the North Pole axis is actually never quite where maps say the North Pole is! It is actually about 900 feet away from that map location, gradually circling around the map location of the Pole, in a very complex dance.

But what is of more interest here is that the second way that precessional effects can occur is by ORBITAL changes such as what is called Regression of the Nodes.

** IF the previous assumption had been true, then the
cumulative effect of many years of Regression of the Nodes would always have
resulted in no net change in the orbital radius. But since
Angular Momentum is NOT Conserved in this specific situation, it turns out
that, very slowly and very gradually, planets CAN cause modifications to the
orbital radius of each other.
**They DO still comply with the Law of
Conservation of Energy!

This all means that in SHORT-TERM viewing, there does not SEEM to be any observed effects of semi-major radii being altered due to precession effects, but that over LONG-TERM periods, such changes can and do occur.

After many thousands or millions of orbits, the two planets (or moons) therefore wind up having orbits which have changed and that are now CLOSE to having some simple fractional proportion regarding orbital period. They CANNOT have stable orbits which are EXACTLY commensurable, as there are severe effects of instability then. But the result is a meta-stable pair of orbits which are quite close to seeming commensurable, and which can maintain themselves in that relationship for very long periods of time.

I recommend that mathematical Physicists set up such a problem in a computer, of any two planets orbiting a much more massive primary, and given RANDOM orbital periods to start. This process also works regarding differences in orbital inclination and regarding the eccentricity of both orbits, but those processes seem to take longer to occur. A computer experiment where two co-planar planets of minimal orbital eccentricitys, shows these changes in the semi-major axis dimension of both, where during a million orbits, significant changes will have occurred.

This then provides an explanation for WHY the four large Galilean Moons of Jupiter have orbits which are very close to being in the ratio of 1 : 2 : 4 : 8 regarding their orbital periods, BUT they are not EXACT in those proportions but are necessarily slightly different for the meta-stability to be possible.

It also explains the Long Inequality of Jupiter and Saturn, the Kirkwood Gaps in the Asteroids due to Jupiter, very distinct gaps in the rings of Saturn, and many other orbital relationships in the Solar System, possibly even including Titius-Bode's Law regarding the orbital periods of all the planets!

**There are astounding implications for
Nuclear Physics as well. **In
the 1930s, Physicists saw that electrons only appear to revolve around nuclei
in very specific orbits. THAT was the basis for the
development of Quantum Dynamics, to provide an explanation for that apparent
"graininess" of many small phenomena. But
this new concept indicates that if an electron is thrown into an atom with ANY
initial orbit, within a few million orbits (maybe a trillionth of a second, too
short for us to ever detect or be aware of), the electrons could MUTUALLY
PERTURB each other into those orbitals that Nuclear
Physicists always see! (I refer to this as that we
have "slow eyes" to not be able to see the gradual perturbations of
the orbits).

The point here is that, possibly, there is a far better explanation for countless nuclear processes, of actually using Classical Mechanics (but with far smaller time graininess than we could ever detect) instead of using the assumptions on which Quantum Dynamics is based. There seems a valid chance that much of Nuclear Physics may need to be re-examined to become more correct.

Newton, Joule and others established several Laws of Nature, including Newton's Laws of Motion, Conservation of Energy, Conservation of Momentum and Conservation of Angular Momentum. These laws have been used extensively to develop many later aspects of science, particularly in Astronomy and Astrophysics.

Specifically, regarding Perturbations of planets or other objects by other planets, it has
always been assumed that the orbital radius (called the semi-major axis) cannot
be affected by perturbations of other planets. The
reasoning always seemed sound. If both Conservation of
Energy and Conservation of Angular Momentum apply, then the semi-major axes
could not change. If the TOTAL (kinetic) energy of the
two objects remained the same (one becoming greater and the other less, in the
exact same amount), then the TOTAL Angular Momentum of the two could not remain
the same, if their orbital radii had been altered. The
reason is that the kinetic energy is proportional to the SQUARE of the
velocities in orbit, while the angular momentum is proportional to the
velocities themselves. Kepler's
work showed us that the velocity has to change with the distance from the
central body, which always seemed to mean that Perturbations might affect other
Orbital Parameters but could never affect the actual average distance from the
central body. ** This is a universally accepted conclusion among Astrophysicists today.**

** It is incorrect!**

But only in a very specific and very peculiar way and the effects could only arise very slowly, over very long time periods, longer than is ever considered by any existing perturbation theories. In practical situations, this effect is never seen, as Conservation of Angular Momentum is seen as being valid to within measurable amounts.

Those statements ARE true, if only one plane of motion is considered . However, all of those brilliant people ASSUMED a situation which neglected a central result of the process of gyroscopic precession.

The most obvious way of first presenting this is with a high-quality child's
gyroscope. Consider one where all the support bearings
are perfect, that is, there is no friction whatever, and it is operated in a
total vacuum, where there is no air friction, such that the gyroscope rotor
will spin forever and never slow down. Placed on the
usual pedestal in a axle-horizontal position, we all know that the gyroscope
will do two unusual things: it "hangs there", apparently defying
gravity, and it also precesses (slowly revolves) around the pedestal. ** But
we note here the important fact that the gyroscope does not START OUT precessing!
**The current question is now related to the issue that,
"** when the gyroscope is released, it
necessarily ACCELERATES up to the final precessional rate.
**So what is the source of
that energy that is used up in that acceleration?"

** According to the conventional description, this is a
clear violation of the Conservation of Angular Momentum! **The rotor did not
slow down, so that was not the source of any angular momentum.

These are the Euler Equations, the expression of Newton's Laws of Motion as Differential Equations for motion in three dimensions. As usually interpreted for a child's gyroscope, the first Equation considers the motion about the gyro spin axis (which is horizontal in the simplest case), in other words, bearing friction and air resistance and any motors that might affect the rate of spin of the gyro rotor. In this case, there are no changes and this equation is 0 = 0.

The second Equation considers the motion about the "2" axis (which is also horizontal but normal to the 1 axis), in this case the effect of the gyro falling due to gravity, and therefore attempting for the gyroscope body to rotate around the support point at the top of the pedestal. The third Equation considers the motion about the "3" (vertical) axis, that is the precessing motion of the whole gyroscope body about the vertical axis (also around the support point at the top of the pedestal).

We need to now look carefully at the second and third Equations, which will be
seen to be inter-related. The third Euler Equation,
for this horizontal gyroscope, is:

M_{3} = I_{3} * (d** ω**_{3}/dt)
+ (I_{2} - I_{1}) * ** ω**_{2}
* ** ω**_{1}

We can first look at the situation AFTER the precessional motion has fully
developed. This is the equation that describes the
dynamics of the motion around axis 3, the precession. There
is no external Moment applied (around the 3 axis), so M_{3} = 0
. The other two terms must therefore always add to zero
. In other words, once the precession is at its correct
rate, this equation is 0 = 0.

Now we can look at the situation as the gyroscope is first released, where there is
initially zero precessional velocity. **
A precessional angular acceleration is therefore
required. **The
M_{3} term on the left is the EXTERNALLY APPLIED Moment (torque) which
is zero for this situation, which is still always zero. The
first term on the right involves the angular acceleration of the precession (d**
ω**_{3}/dt) which is what we
need to determine. The second term includes three
terms that cannot change and one which could (** ω**_{2})
. Both of these potentially variable terms therefore become
non-zero for a brief period, immediately after the gyroscope is released
. As the precession accelerates (around the "3"
axis), the gyroscope slightly lowers (around the "2" axis)
. In the case of a toy gyroscope, this all usually occurs in
a fraction of a second, and the distance the body of the gyroscope drops is
extremely small.

The SOLUTION to the long-standing error of assumption is seen if we use the set of
Euler Equations but Integrate them. The directions of
the (acceleration) vectors are similar, defined by the standard Vector Calculus
procedures. ** We can
then see that a (downward, gravitational) ACCELERATION of the axis-2
"dropping" of the gyroscope body (an acceleration vector along the 2-axis)
causes an ACCELERATION in the axis-3 precessional motion.
**Once it has given the
appropriate precessional velocity, the effect then works in the opposite
direction to STOP accelerating the precessional speed and also stops the
downward acceleration of the body of the gyroscope.

The usual Right-Hand-Rule applies, which establishes which way the precessional motion will accelerate, and therefore which direction the gyroscope will precess.

This is NOT instantaneous, but both these accelerations proceed in a sine-wave curve . This insight now allows calculating HOW LONG it takes to have the precession accelerate up to its final speed, and also how much of a downward angle the body of the gyroscope "falls" during that time interval. In a related presentation on Precession, linked below, those calculations are done for a representative toy gyroscope . We show there that less than one one-millionth of a joule of energy is transferred from the one axis to the other, and that amount of energy is provided as the body of the gyroscope falls around one four-thousandth of a millimeter vertically. We also calculate there that the entire process for a toy gyroscope occurs in around one ten-thousandth of a second.

The entire process of the precessional speed RISING from zero to its expected rate, in a smooth process, as well as the lowering of the body of the gyroscope (also in a smooth process), is therefore calculable, where the entire process is unambiguously described by the mathematical differential and, more specifically, second-differential equations.

This effect has apparently been overlooked because all practical-sized gyroscopes seem to achieve their proper precessional speed extremely rapidly, and no one seems to have realized the incredible importance of this effect ! (A toy gyroscope gets up to its proper precessional speed in around 0.0001 second and it drops less than 0.001 millimeter, which makes it seem to be essentially instantaneous and of no noticeable effect other than the new precession!)

We can Integrate both variable terms in either the
second or third Euler Equation over the entire time interval of the precession
acceleration, and we wind up with terms which represent **
ω**_{3} (the actual final
precessional rate) and ** M**_{2}
(a change of angle of the tilt of the gyro axis).

The Precession page, linked below, provides the calculations for an actual toy
gyroscope, and the results indicate that the gyroscope physically drops down a
tiny fraction of a degree while the precession accelerates up to speed
. (This commonly represents a lowering of the body of the
gyroscope by around 0.00026 millimeter, a distance that would be hard to notice
and is also even hard to detect! I have confirmed this
experimentally.) The precessional kinetic energy which
appears in our toy gyroscope is just under one one-millionth of a joule (or
newton-meter), which is EXACTLY the same as the amount of
potential energy that was released as the gyroscope dropped that tiny fraction
of a millimeter, ** which properly shows the
Conservation of Energy**.

**The result is that there is an angular acceleration of
the precessional motion (around the 3-axis), which is due to (vertical,
dropping) motion in a different plane (around the 2-axis)! The
support angle of the gyro body is very slightly lowered, which gives up some
gravitational potential energy, which is then converted into the kinetic energy
of the precessional motion. Conservation of Energy is
actually exactly maintained. It would not appear to be
Conserved if just the precessional motion was examined in just the horizontal
plane (or along the "3" axis).
**There was initially
zero kinetic energy of the precessional motion and some kinetic energy would
seem to just "appear"!

**The significant fact is that this demonstrates a
transfer of (potential) Energy from one plane ("2") to another (as
kinetic energy) which seems to give the appearance of NOT conserving Energy in
the process! **It
actually DOES Conserve Energy, but it cannot and does not also Conserve Angular
Momentum in the process! Before being released, only
the rotor is moving, spinning (in the 1-axis), so there is no Angular Momentum
along axes "2" or "3". Once
released, the Angular Momentum of the rotor is not changed, and after the
precession has gotten up to proper speed, there is again no Angular Momentum
along the "2" axis, but now there IS Angular Momentum along the
"3" axis, in the form of the Angular Momentum associated with the
precessional motion.

However, ** Conservation of Angular
Momentum is violated**, where it is always otherwise true
. As the precessional motion begins, angular momentum
"appears" (along the "3" axis) where it had not existed
before. ** This is in
disagreement with the universal acceptance of Conservation of Angular
Momentum in the field of Physics!**

Technically, this is NOT really a Violation because there is an external force acting on the gyroscope, which is the Earth's gravitation . In the Solar System, there is also an external force acting on the Perturbation of planets with each other, the Sun's gravitation

This indicates that the long-held assumption that Angular Momentum is always conserved is not really necessarily true when more than one plane of motion is considered, and gyroscopic precession certainly shows that flaw of that reasoning.

** It has been assumed by all astronomers and Physicists
that planets can perturb several parameters of the orbits of each other, BUT
that they could never alter the semi-major radii of each others' orbits
. **That
conclusion WOULD be true IF all the objects in the Solar System orbited in
exactly the same Plane. But they certainly do not.

The Solar System objects move in various planes. This fact
results in effects that are similar to the non-Conservation of Angular Momentum
of the toy gyroscope. Examples are the Earth's
Precession, the Regression of the Nodes of the Moon's Orbit (and all other
orbits), and any other perturbations where the Z-axis is involved
. Planets ARE causing precessional effects in each other
. Now that the precessions are all established, no
significant violations of Conservation of Angular Momentum SEEM to occur, but **
whenever each of those precessions CHANGES, that is,
they ACCELERATE, they certainly represented clear violations!**

For example, the earth has an equatorial bulge that is rotating in a plane where
each of the Sun and Moon nearly always are acting to gravitationally try to
tilt that plane (trying to stand the Earth more upright), which causes the
Precession that the Earth experiences. There seems to
be a common misconception that this Precession of the Earth is constant, and we
all learned of the 26,000 year time period of the (wobble) Precessional motion
in Elementary School. However, that is not even close
to being true! ** TWICE
each year, the precessional effect of the Sun on the Earth entirely vanishes**,
at the instant when the Earth's orbital motion causes the Sun to appear to
exactly cross the celestial Equator (around March 21 and September 21 each
year). ** The
Precession of the Earth due to the Sun ENTIRELY STOPS on those two days each
year! **After
that, the precessional speed ACCELERATES during the following three months, up
to a point where the precessional SPEED is greatest around June 21 and December
21 each year. After that, there is a DECELERATION of
the precessional speed during the next three months, to get back down to the
zero precessional speed.

Also, consider a "new earth" exactly like ours but not precessing
at all. ** It would
(somehow) START to precess, in other words, the Precessional motion of the
earth would ACCELERATE up to the rate it is now at.
**A motion which takes
26,000 years to occur might seem to not involve very much energy, and so it
seems that it has always been neglected by Researchers regarding energy
considerations. But the Earth is quite massive and
since that Precessional motion starts and stops twice every year, there is
actually an enormous amount of energy involved.

This represents a good deal of kinetic energy of the Precessional motion, and the
Conservation of Energy insists that a source for that energy have provided it
. The energy that would supply that motion comes from slight
variations in the tilt of the Earth's rotation axis, so Kinetic Energy would be
conserved, even with the "precessional acceleration up to the new
precession rate". **
However, Angular Momentum in the Plane of the Ecliptic would NOT be
conserved! **New
Angular Momentum would constantly arise and disappear in that Plane
. The AVERAGE of this is the actual observed velocity of the
Precessional motion.

In fact, a "new Earth" would be no different than our current Earth, since our precessional motion (due to just the Sun) entirely STOPS twice every year! That significant amount of Kinetic Energy involved in the Earth's precessional motion is CREATED and then CANCELLED OUT twice every year! The processed being discussed therefore involve significant energy transfers! These energy transfers occur because of the gravitational field of the Sun (and Moon).

In fact, since the precessional effects of various solar system bodies on each other are constantly CAUSING ACCELERATIONS AND DECELERATIONS in the precessional speeds, this necessarily indicates that the Earth and other planets are also doing a very slight tilt-axis dance that has always simply been considered a part of Solar Nutation! It is quite a small effect, but experimentally measurable!

The same effect occurs as planets perturb the orbits
of other planets and satellites, sometimes also referred to as precession but
more commonly called Regression of the Nodes. These
effects are commonly presented as though they are constant effects, but they
are NOT constant at all! As a perturbing planet varies
from being above or below the orbital plane of a perturbed planet, it crosses
that orbital plane twice in each synodic period
. This causes the precessive
effect to constantly be oscillating, from zero effect to a maximum, during the
synodic period of the two bodies. This
effect occurs for BOTH of the orbital motion and the rotational motion, and
affects both bodies involved. **
This indicates that there must certainly CONSTANTLY be
MANY small violations of Conservation of Angular Momentum occurring.**

The effect described here is fairly small, and the cumulative effects are very slow . In all practical situations, Conservation of Angular Momentum will be seen to appear true. It is only where Euler's equations transfer energy from one plane to another that any variances with that Conservation can occur. Conservation of Energy appears to still always be true.

If we consider that on March 21 of any year, the Earth
has NO Precessive effect due to the Sun, we can
easily calculate some things here. We first calculate
how much kinetic energy there is in our AVERAGE precession. It
is 1/2 * I * ** ω**^{2}
. We know that the rotational inertia (I) of the earth is
8.07 * 10^{37} kg-meters^{2}. We know
that ** ω** is one precessional
revolution in 25,800 years or one radian in 1.296 * 10^{11} seconds
. Therefore, the AVERAGE kinetic energy the Earth has in
precessing is around 2.4 * 10^{15} joules
. In planetary dynamics, that is not very much, but it still
is kinetic energy that did not used to exist!

The AVERAGE kinetic energy of the precessional motion is that amount
. However, we know that around Mar 21 and Sep 21 each year
(considering only the Sun's contribution) that amount is briefly zero, and
arould Jun 21 and Dec 21, it is much greater than that
average amount. Actually, since the Precessional
MOTION is twice as fast at those instants, there was FOUR TIMES AS MUCH kinetic
energy transferred. ** Note
that this means that around 10 quadrillion joules of energy is ADDED to the
Earth's (Solar driven) Precessional motion in a three-month interval, and then
the same amount is REMOVED from the Earth's precessional motion in the
following three months! This is a significant transfer
of energy into and out of that motion, on a very regular basis! **The
Moment (torque) of this constantly fluctuating amount of Precessional Kinetic
Energy is related to a slight axis tilt change of the Earth's spin axis
. The double Integration of the Euler Equations shows that
the energy involved is always Conserved, but that it
is simply transferred back and forth between a slight fluctuating tilt of the
Earth's rotation axis and the varying Precessional speeds.

Ten quadrillion joules might sound like a lot, but since it is spread out over a
three-month interval, that is about an ** average
of 1,250,000 kiloWatts** (because a watt is a joule/second)
. That might not be worth the bother regarding trying to
build any equipment to try to capture it! But it gets
WAY better!

** In addition, similar calculations show that around 40
quadrillion joules of energy is ADDED TO the Earth's (Lunar driven) precession
in about a WEEK,** and then the same amount is removed during
the following week! Many people have noticed and
measured the very small-scale wobbling that the Earth does (collectively called
Nutation) but I have never seen that anyone has realized that it was actually
(primarily) due to a side effect of the constantly varying Precessional effect.

** It might be noted that the energy transfer due to the
Moon's effect here is relatively significant from a human perspective! **There
is about 40 quadrillion joules transferred in a period of about one week (or
637,000 seconds) which means that an ** average
power transfer of that quotient is occurring, or about 63 billion joules/second
or 63 billion watts or 63 million KiloWatts! That
63,000 Megawatts is comparable to the entire output of electric power from ALL
US nuclear generating plants!
**But I do not see how it could ever be captured by
anything that we humans could ever do! Maybe some
human far smarter than me can see some way to capture that energy, and we would
then have an enormous supply of power, essentially forever!

The puny little Moon causes this effect around 50 times greater than the enormously massive Sun does! Interesting!

More significantly, each time when the Precession
Effect is non-zero, there is Angular Momentum that did not used to exist
! ** This fact means that one
of the two pre-conditions that Laplace, LaGrange and
everyone else have always applied is often (slightly) invalid.
**In short-term motions or
perturbations, these effects are not seen, as they are comparatively small
amounts of energy and angular momentum involved, and Angular Momentum appears
to be conserved. However, over very long periods of
time, these effects of continuously modifying small amounts of "new"
angular momentum ALLOWS planets to mutually alter their semi-major axes
! (Which is currently assumed impossible.)
** This then allows some very slow
perturbation effects that are so small that they have not been yet detected**
(or understood). However, they certainly occur,
because there is extensive evidence of near-commensurability in orbits of
planets, satellites, asteroids, ring particles, and more. These
are therefore not mere coincidences, but the very long-term effects of this new
category of mutual perturbations where the semi-major axes are altered
. Again, the Hamiltonian remains true, and Energy is
Conserved, but slight changes in Angular Momentum certainly occur.

We have known for thousands of years that the Moon causes and creates spectacular amounts of energy that exist in the Oceans' Tides (every day). We know that this is due to some gravitational effects of the Moon on the Earth and on the waters of the oceans . Here, we are discussing a DIFFERENT gravitational effect of the Moon on the Earth, which is actually due to the Earth having an Equatorial Bulge so that a Precessional motion can be caused on the Earth by the Moon. In other words, IF we could just figure out HOW to collect some of the precessional effects of the Moon on the Earth, we might easily collect enough energy to power much of modern civilization ! This presentation is a discussion of one (of many) possible approaches we might try. If done effectively and efficiently, we probably could harvest spectacular amounts of energy which exists due to the Moon's existence orbiting our planet.

This presentation was first placed on the Internet in Sept 2006.

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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago