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This page is dedicated to introduce "Energy as a Function of Time". New "concepts" and "terminology" are needed. Methods of closing a system to motion as a "function of position", may lead to a mistake of generalizing a special rule.
Hamilton teaches that energy is not preserved, when motion is a function of time and LaGrange equations do not stand. Our Relative Motion as a "Function Of Time" solves Hamilton statement, by equating time as a form of energy exchange, where energy preservation is re-established, but with time also becomes a source of energy.
Our Relative Motion as a "function of time", teaches that a net force, is not enough to calculate energy of an object in motion, simply because vacuum does not exist in nature, where every motion is part of at least one field ( yet a field role can be ignored when motion calculated energy as a function of Time = motion Energy as a Function of Position ).
As a "Function of Time", we may have more than one dimension or one field of acceleration. that shall not be reduced to the sum of acceleration vectors, nor the object's motion shall be reduced to a particle subjected to a central net force.
To simplify the concept, we may suggest that a similar net force and a similar object would act differently, if the field of motion was different, like a simple field of gravity, fluid or electromagnetic, with different levels of pressure, or combined with a field of relative motion, like a surface and tidal waves acting on a ship, the acceleration vectors can be added to find out its position, however such sum is not helpful to calculate energy needed for the trip.
No matter how complicated such a field can be, from an energy prospective; acceleration can only be of the power square, which shall not be reduced to the sum of two vectors, nor to classical vector multiplication, as we see incorrectly practiced in classical mechanics and in special relativity, whenever motion was a Function of Time.
While acceleration as a Function of position can be measured as a vector of certain magnitude at certain environment, The acceleration of the power square, shall be treated as a motion derivative that governs the relation between virtual and physical distances, and measures the relation between (the field, the momentum and available potential energy).
Cartesian geometry of energy in our Relative-Motion method, is a result of a volume created from area (calculated from the acceleration squared) multiplied by time lapse of acceleration as an elevation on the X-coordinate, which opens any observed system to the equivalency of energy and time, rather than having energy calculated as a scalar.
When motion is a function of time, second acceleration makes a continuous force-equilibrium relation between motion kinetics, and between its field, which is mathematically handled by integration.
Pascal law in its most simple expressions is calculated as equivalency between work input and work output, where for each side (W= pressure * Surface *stroke Distance =pressure *Volume). In the case of our Relative Motion cylinder, our input side [W=pressure* (internal volume of cylinder MINUS space void by floating piston) * stroke distance] where Work output is bigger than work input. As a function of position, this would be considered as a case of non-preservation, however this controversy can be solved if time is considered a form and a source of such energy difference, where complex numbers is needed to calculate Work energy gain ( W =space void *pressure * stroke distance ) or in another expression, energy gain from time = Negative mass * distance. Not always we have the means to calculate space void available, as a negative mass, and another method of equivalency between input and output must be established for motion as a function of time, where we replace physical distance by virtual distance, and where Virtual distance is measured by time (sec) under universal acceleration, like saying virtual distance equals one second, when an object average speed is 4.9 meter/sec
Relativity as a function of time, is method of calculating energy as a function of time, where acceleration is mathematically treated by equation of the second power, and by giving potential energy a Cartesian geometric volume rather than a scaler.
The conservation relation between potential energy and time, is treated independent of the conservation law that governs potential and kinetic energies, where motion is a Function of position.
With these considerations in mind, we can advance to a very important example in applied physics by saying:
Time accelerates potential energy as force accelerates a mass.
E = 1/2 *Mf *g2 *t
where t is a measure of virtual distance under universal acceleration.
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