I. PDEs and ODEs applied to physics
∆±¡: Laws of conservation in mechanics. Actions. Lagrangians and Hamiltonians through all scales.
S=T: Main State equations of mathematical physics: transition in Dimotions.
CALCULUS IN PHYSICS: ∆-scales, Dimotions & States
Calculus has two fundamental use in physics, the study of scalar parts and wholes through finitesimal derivatives and integrals; and the study of motions with different degrees of freedom, in a single pane. So as they cover both the ∆§cales of physical systems, and its worldcycles, even hinting at the @-element, with its ‘fundamental finitesimal’ extracted by Taylor series a the derivative of point 0, they are by far the most important instrument on the quantitative study of physical systems in Nature.
It is then not surprising that a few basic differential equations allow us because of the homology between all scales of nature to study all physical systems in all scales. Among those equations the key ones because they study the fundamental elements of a physical system are:
- SxT=K metric equations of conservation of ST energy, $-lineal and ð-cyclical momentum: Lagrangians (2nd order derivatives and its integrals of motion focused on momentum) and Hamiltonians (1st order focused on energy), which apply to all scales
- 5 Ð State Equations:
- Continuity, flux equations and wave equations, focused on the liquid state in all scales (2, 3Ð).
- Entropic, Thermodynamic equations (Poison, Laplace, Gauss) focused on the dynamics of particle<field entropic absorption of energy, in all scales (4,5Ð)
- SMH harmonic equations, focused on the particle states of information (1Ð)
The reader will observe those equations suffice to study the 5 Dimotions,hence the interaction between scales of the being, and its Metric, hence the conservation of its 3 constant forms in a single plane, and as such they are all pervading on physics. While there are more equations to consider, as this is an introductory course on 5D we shall just describe those equations, to observe the homology of scale (all of them can be used in all scales of physics) and the ‘Closure theorem of 5D’: all what is real is an event or form composed of the 5 Dimotions of exist¡ence.
S=T PRESENT EQUATIONS
The first obvious truth we must understand on mathematical physics and any general system of equations is that WE EXIST IN PRESENT (Si=Te), SO WE CAN ONLY MEASURE PRESENT realities, within a given ‘quanta of time’ of perception (different for each present analysis). This means ALL equations observed in ‘an instant or quanta of present’ (derivative equations) will be of the form:
S(form, static parameters) = T (motion, flow parameters)
i.e. The differential form of Gauss’s law for gravity states: ∇⋅g=−4πGρ, where ∇⋅denotes divergence, g the acceleration, G is the universal gravitational constant, and ρ is the mass density at each point.
So we can write it as: Te( ∇⋅g) =Si(−4πGρ)
In one side, the accelerated motion (Te) and the other the formal curvature (Si). And the same works with Newton: Te (F)= S¡ (G mM/dd); and Einstein: ‘matter tell space how to curve (Si)’ so we have a curvature of space in one side, and the tensor of energy-motion on the other side.
Why time dimensions are equivalent to space dimensions, has to do with the perception of all form also as motion (Galilean relativity).
By the paradox of Galileo, then all space form has a time motion and reality is bidimensional. SO WHEN WE ADD VECTORS, we are really adding ‘square values’, where the Motion (T) is equivalent in value to the Space (S).
Reason why in static form, we can consider vectors, sums as ‘variations on the Pythagoras theorem’.
This means the motion value has to be proportional to the form value and both together shape a co-invariant conserved quantity, usually momentum either dominant in motion, lineal momentum, which we shall call ‘the derivative of energy’ (that is a finitesimal of energy) or cyclical momentum, that we shall call ‘the derivative of information’, that is a quanta of cyclical time.
Those are therefore the conserved quantities, which then we can write as:
Energy (K-onserved) = Lineal momentum (kinetic energy proper: mv) + angular momentum (cyclical informative clock-like component, related to position and form).
The specific way in which those $ x ð =K relationships adopt define the type of dimotions we are dealing with.
Uncertainty of metric measure
The specific uncertainty of measure and perceived by huminds and its instruments are secondary ‘details’ of mathematical physics (conjugate transforms, Heisenberg uncertainty) secondary despite so much literature, as it only indicates that a ‘metric conjugate of the 5th dimension, $ & ð parameters’ is A single HOLOGRAPHIC ELEMENT and so at least a ‘unit of it’ absorbed in the perception cannot be split and it is the unit of uncertainty of that scale.
DIFFERENTIAL EQUATIONS ACCORDING TO DIMOTIONS:
3 TYPE OF STATES: FORMAL STATE, MOTION STATE, ENTROPIC STATE FUNCTIONS
When considering the 5 Dimotions of existence in mathematical physics, it is convenient first to understand the geometry of the 3 fundamental dimotions in which locomotion takes place, which is provided by the understanding of the 3 ages/horizons of time, and the rule of parallelism in social evolution and perpendicularity in entropic feeding. When we combine those concepts we can then interpret physical chains of dimotions in Nature in terms of 5D vital space-time:
If we escape the Dimotions of Complex informative Reproduction and palingenesis or generation, which obviously happens in a still point with no remarkable locomotion – but is essential to understand the weak force that transforms particles into higher masses; while reproduction of minimal information is ultimately the adjacent nature of locomotion that imprints a lower scale with the form of the system, studied in other posts – it becomes crystal clear that two of the dimotions are inverse in form, Entropy and Informative social evolution; and often become ‘coupled’ together to create a balance between them; with the proviso that the perpendicular sudden inflection of a flow in a vortex, from a flat circular motion to an axis motion, as in black holes expelling jets, eddies with a curl ascending or descending column or an entropic explosion, radially perpendicular to the ‘fractal point in its moment of death’ are clear signs that the ‘Informative’ implosive force has changed to an entropic explosive force, and both balance each other:
This duality is thus expressed in mathematical physics with the curl and divergence functions, as in electromagnetism, black hole/mass, charges and thermodynamic eddies:
So a terminology often used in physics refers to the curl-free component of a vector field as the longitudinal component and the divergence-free component as the transverse component. Since when we decompose the field (for example a Fourier transform), at each point k, into two components, one of which points longitudinally, i.e. parallel to k, the other of which points in the transverse direction, i.e. perpendicular to k.
So this is indeed the 5D meaning of the Helmholtz decomposition.
SINGULARITY-SCALAR ∆@¡ equations:
In the graph we see the vital interpretation of those vortices and its equations.
THEY Are equations in which we study motions on the 5th dimension (social evolution, perception) and so they require to have a relative fixed point towards there is a convergent symmetric flow (for perceptive systems to focus properly a mirror image of the external world), and will often require 2 derivatives to obtain the ‘lower ∆-2’ plane pixels for the system to absorb them into its bit-mirror. In those general cases the ¡logic isomorphism is Si(∆º)<Te(∆-2).
§ð: Static ‘spherical states’, where an S-component, which REPRESENTS THE WHOLE, MEMBRANE=ENCLOSURE AND a ð-SINGULARITY, become the ¬Limits of the system of vital energy they enclose.
So the equations will be of the type §ð (whole membrane x singularity)= ST (∑∆-1 points)
In those equations the 0 point will most likely be the symmetric center of a divergent differential equation of ∑-¡ points, which turn arounda the singularity of the System, and often near the center will be subject to a ‘curl’ equation, as it ’emerges’ and ascends to other plane of the fifth dimension.
I.e. in thermodynamic vortices, there are two planes, one of liquid and one of gas, which turns around a center that curls upwards the water, feeding the cyclone.
So the S-T equations it can also be in the mirror image of mathematics with frames of reference, the 0-point of the reference frame, reason why so many equations will have the form: F(x)= Equation of motion = 0 (Singularity Point of Si form).
And so we can ‘separate’ the system (fundamental theorem of vector calculus) in its divergent and curl components, which correspond to a flat plane of ∆-1 vital energy particles turning inwards till reaching the proximity of the singularity when they will ‘suffer’ a hyperbolic Belgrami curve of ‘perpendicular’ transformation of state, becoming something else, normally breaking apart its two components, one $t element becoming entropy of the singularity an the other ‘still part’ becoming pixels of information for the system.
So we can interpret such systems as dual mathematical events, whose paradigm are the curl and divergence equations of electromagnetism:
And we can always break down, as they are ‘composite of two motions’, according to the fundamental theorem of vectorial calculus the dual motion into a curl component, which will GROW AS WE MOVE towards the singularity that will ‘feed on the information of the system’ and deflect the entropic ‘remains’ through it axis, and the divergence, which will either be open if the potential runs to infinity, or if the system is a full fractal point with a membrane, will start to accelerate passed the enclosure, or limiting ‘border’ of the system, beyond which the ‘form’ is free, more entropic and disordered before its ‘capture’ by the physical system of membrane and singularity.
All in all the key element of such set of equations are to be informative, that is shrinking space and increasing time, responding ultimately to the vortex law: Vo x Ro = K, as the spatial radius diminishes and the speed-acceleration moves towards the future, increasing. Then in the singularity point, information somehow (it is obviously not clearly understood for vortices in which the force of information is invisible such as gravitation is), INFORMATION must stop forming an image in the zero point of the vortex (0 motion in thermodynamic eddies, time goes to zero in gravitational black hole equations).
So those are the concepts behind the fundamental dual type of motion of physical systems, combination of 1Dimotion of informative perception with a flat divergence field that culminates in an unobservable still mapping of perception in the physical center and a4Dimotion of entropy perpendicular to the plane of divergence, with a curl, which is represented mathematically by curls and divergent operators.
TRANSLATIVE =CONSERVATION LOCOMOTIONS
To understand the equations of locomotion however a different mind-frame is needed, which resumes in this:
The dimotions of information and entropy are transformative analysis of the system, from an internal point of view; so the internal space and time parameters of the system, change. Translative locomotions only change the position of the system in space but DO NOT change its internal parameters, hence they are restricted by the conservation of those 3 internal parameters, lineal singularity momentum, cyclical, membrane momentum and vital energy.
So they IMPLY A MAXIMAL conservation of the 3 simultaneous spatial parts of the being.
Then we must add the second principle to calculate all those equations, the principle of least time, which implies that as in all other scales the system will try to reach its destination through the shortest path in time, the fundamental variable of existence, minimizing its expenditure of the ‘limited time-duration’ of its existence.
So locomotions try to conserve both the spatial form of the being and its time-existence.
Another set of simple mathematical equations are those related to lineal locomotion, in any of its different pentalogic expressions, from momentum equations to Lagrangians and Hamiltonians, which we also study in other posts dedicated to mechanics, and Energy conservation laws. So we shall not be redundant.
ULTIMATELY ALL THOSE ENERGY-MOMENTUM EQUATIONS RELATE TO THE METRIC CONSERVATION OF THE 3 ELEMENTS OF THE GRAPH ABOVE, ENERGY, (VITAL CONTENT OF THE SYSTEM), LINEAL MOMENTUM OR SINGULARITY PARAMETER AND ANGULAR MOMENTUM, AND THEY ARE IN BALANCE, SO THE SYSTEM CONSERVES ITS.3 ELEMENTS.
Related to both we find the Hamiltonian and Lagrangian equations, which are metric equations of the 3 conserved quantities of the system; whereas in Lagrangians, ∂L=0, the law of ‘least time’ in the form of a zero minimal quanta of time, with no ‘variational form’ represents the Si (0)=Te (motion) elements, and in the Hamiltonian, the Position (potential energy) and Motion (kinetic energy) represent the Si (position) and Te (Motion) elements.
Integrals of derivatives.
Another elementary truth of all those equations is that as we are ‘calculating’ present states, a present equation requires to calculate a ‘single minimal quanta of time’ for each equation, which is the definition of a derivative in 5D. It is then when we can integrate the whole path, as in Lagrangians, where the derivative is a step of minimal least time action that tends to zero (the infinitesimal or action, which is the ∆-1 minimal time quanta of the 3 scales of time) from where we can then integrate a sum of those actions to obtain quantities of the ∆º plane (worldcycles of time or populations of space).
So all present equations of physics have a differential form. Then the Integral of motion of the system will add all those quanta of time, to obtain a parameter which is longer in time but as it is a sum of those quanta, ultimately will still be a present equation, often REACHING a higher scale, moving from the dimotion=action of the derivative, to another ∆+1 scale, as when we integrate to obtain an energy parameter of a world cycle, departing from the quanta of action=momentum of the being.
So we integrate quanta of time actions=dimotions, based in momentum, to obtain a full world cycle of a larger scale based in energy.
So this leaves us a third type of equations, THOSE WHICH EXPRESS waves of balanced communication that reproduce information closely related to those of locomotion, (which in fact in a full 5D analysis reproduce the form of the being as it translates in space, in its lower scales) . This type of waves are the ones that truly embody the equation of present balance, S=T, OF THE PREVIOUS GRAPH. So they will represent WAVE FUNCTIONS AND HARMONIC MOTIONS in which the SPACE and Time PARAMETERS, ARE IN CONSTANT balance:
The simplest example being the wave equation in one dimension, which is perfectly symmetric between its second derivative of time and space:
- This is an ordinary differential equation for which to obtain exact wave solutions.
(Historically, the problem of a vibrating string such as that of a musical instrument was studied by d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.)
Many of the equations of the mathematical physics of the nineteenth century can be treated this way. And it is easy to see its difference with the equations of entropy, in which the spatial expansion is scattering, much greater, so the scalar function spreads rapidly as the system descends two scales of the fifth dimension, according to the equation of death: ∆º<<∆-2, so if we interpret each devolution into a lower plane as a derivative finitesimal, the spatial side of the s=t function will suffer two derivatives, for each ‘time derivative’ of change. For example, the Fourier entropic heat equation, in one dimension and dimensionless units is:
So as a rule entropic functions require a higher derivative order in space than in time, reproductive wave functions that communicate information are equal in the time and space derivative, and informative equations inverse to those of entropy, have a higher derivative or dimension in time (acceleration) than in space.
FUNDAMENTAL RULE OF DIMOTION IN CALCULUS
It is then departing from those 3 simple rules, how we can interpret regardless of the enormous complexity analysis has reached, the type of Dimotions those equations express:
Information accelerates in time, $t>>§ð
Entropy accelerates in space, §ð<<$t
And reproduction of balanced information (wave equations) have equal parameters in space and time, S≈T,
So the Harmonic and wave equations, with the sine-position element and cosine-motion element; the amplitude (energy element) and position (Form element), the perpendicular electric (motion-energy) and magnetic components (form in-formation element) define the system, will also play the essential same game… which we shall study for different systems of scalar physics.
DIMOTION EQUATIONS BY TYPE OF CURVATURE.
Another key element to recognize the type of dimotion observed is the concept of curvature vs. open exponential curves.
It is then necessary to consider, what kind of curve represents each Dimotion, as each of them will have a different ‘operand’ and mathematical function, as studied in ¬Ælgebra:
4D entropy will be then exponentials that are open curves which represent decay and expansion in space. And this also happens in the study of the quantum dimotions, as in the Strong force or decay of atomic particles, scattering process, etc;
3D reproductive equations will be hyperbolic, sinusoidal, 3D-wave functions for communication or perception of information, etc.
Si Informative equations, on the other hand will be defined by closed curves with increasing curvature. So we study in its rather self-centered, ‘static form’ an attractive vortex by observing, the ‘curvature equation’ (as in gravitational equations), of the form:
T-Motion (speed of vortex attraction = Curvature (Si-form)
So the strongest the curvature, the higher the attraction, and as in 5D curvature has NO LIMIT as it is equivalent to motion, measure in the frequency of radians per second, it follows that the curvature tends to infinity as the vortex speeds up, reason why charges that turn much faster have in 5D a quasi-infinite curvature and are far more attractive that gravitational slow large, minimal curvature forces.
Thus almost every equation of mathematical physics is just a different type of S=T identity OF WHICH THERE ARE 3 BASIC FORMS.
THE QUESTION OF LIMITS
It is also a truism of 5D that order happens within the region limited by the external membrane of the system, and the internal singularity, so differential equations will have exact solutions when there are such spatial limits; while in time, it requires an initial and final point, that is a trajectory which has a purpose in a chain of Dimotions (usually D1-perception D2-motion, D4-feeding). Otherwise the solutions to the Differential equations are multiple.
How many limits are required then will depend on the degree of order of our equation. I.e. for entropy, there is no need for both limits, while for wave equations those limits are more stringent. I.e. in the example of double differentiations for heat and wave equations, as usual, the problem is not to find a solution: there are infinitely many – the problem is that of the so-called “boundary problem”: find a solution which satisfies the “boundary conditions”:
Where, f and g are given functions. For the heat equation, only one boundary condition can be required (usually the first one). But for the wave equation, there are still infinitely many solutions y which satisfy the first boundary condition. But when one imposes both conditions – the second being of the type S=T, there is only one possible solution.
Let us now that we know the basic rules that relate the differential equations of physics with the laws of Non-Euclidean vital Geometry and the 5Dimotions of existence in physical systems, consider a few introductory analysis of the classic equations of physics.
WHY WAVES AND REPRODUCTIVE LOCOMOTIONS DOMINATE FUNCTIONS
When we get into details of physical equations – something unfortunately I won’t do as I am loosing my interest on this web and my communication of information to a species, mankind, clearly regressing into its age of entropy – one fact is clear – the dominance of wave-equations over entropic and informative ones. The explanation as always can be as complex as pentalogic is; but I am trying to wind down to just simplicity and leave this web soon, so we just shall offer a basic answer: Present NOT only is conserved by its s=t balances, but also it can translate by reproduction maintaining its form (transversal waves), while entropy dissipates fast its form (circular waves, explosions) and information collapses it into a nearby central point – so they DON’T TRAVEL.
Communication, what I have failed miserably, more of a 1Dimotion, of implosive thought, is thus the essential goal forms of all forms of information that reproduce its form. In fact it is possible to translate into wave equations, with Fourier transforms, almost all the entropic waves – such was the way heat equations were solved. While as the graph shows particles can collapse into waves, in an §ð<>ST, particle-wave, 1D perception 3D reproduction beat which is one of the essential composite Dimotions all systems experience. We look, walk a step, look, walk a step.
It then comes to realization that composite waves tend to beat with an intermediate present state, both entropic explosions of motion, collapsing back through Fourier transform to wave states, and implosive particles states back to wave states. In this manner Dimotions start to establish certain ‘allowed chains’ of sequential stœps, stops and steps of particle-waves, or entropic expansions and wave steps, which become the past-present, future-present next composite motions of exist¡ence, while D1-D4 or D4-D1 (perceptive collapses and entropic deaths) are out of balance, unstable – only when both combine stability is possible, reason why ALL FORMS OF PERCEPTION MUST ALSO EXPEL the reminder entropy through a lineal axis.
Now with those simple motions it would be easy for a team of physicists to develop fully a 5D mathematical physics, which vitalize all motions of the systems they study. I cannot even gather energy or interest in pouring here the annotated books of my youth on the Lindau’s 11 books of the encyclopedia of physics, so I just will settle this post with a few comments on the classic equations of physics that represent the previous dimotions, Hamiltonians, continuity equation, wave equation, heat equation – the very basics of a discipline that has yet to be born.
In 5D we use the concept of an action for the minimal Dimotion, performed by each T.œ in a quanta of time. In physics an action though is the specific dimotion of angular momentum, which can be equated to the minimal, simplest closed time cycle of Nature; which depending on the ‘scale or point of view we adopt’ will be a Dimotion of perception (1D action), a reproductive cycle of a given ‘frequency of reproduction, (3D) or a minimal worldcycle of a quantum particle.
In quantum physics in any case has a clear unit, h, which is the minimal informative cyclical action of the galatom.
It is for that reason, when we give it a physical nature, all pervading in all physical scales as it is the concept of speed and its maximal constant c-speed. As we can reason to be one of the two ‘constants’ of the essential metric equation of the 5th Dimension, h(ð) x c ($) = K.
The interesting fact though is that action is an attribute of the dynamics of a physical system from which the equations of motion of the system can be derived -despite being an angular cyclical motion. From where we infer that the previous graph of locomotion as reproduction of the form, the information of a physical fractal point is correct.
In mathematical terms is a functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths but the path of maximal probability most often chosen will be that of least time, trying to conserve the maximal internal energy of the being. As such an action has the dimensions of [energy]⋅[time] or [momentum]⋅[length], and its SI unit is joule-second.
Expressed in mathematical language, using the calculus of variations, the evolution of a physical system (i.e., how the system actually progresses from one state to another) corresponds to a stationary point (usually, a minimum) of the action.
The obvious trinity equivalence is then conceptually profound: energy x time = momentum x length = angular momentum (mrv); as it shows again that the FUNDAMENTAL conserved quantity is angular momentum, which embodies the 3 elements of the system, the center-singularity, the radius that invaginated it and connects it with the membrane.
Actions then must be considered a conservative cycle that reproduces the form of the system, not an entropic dissipative dimotion, neither a dimotion of perception in its first meaning, even if the rotation of the system will allow the mind-singularity to scan the outer world.
It is then possible to postulate that the action of angular momentum, represented intros graph, is the Fundamental dynamic particle of the physical Universe, which as a fractal point can move through a trajectory a certain length, becoming a lineal momentum; its ceteris paribus, spatial view; (Si≈pl), while in terms of time can be decomposed in its energy and time, (Te), which are both equivalent: Te=Si; and so the cyclical reproductive 3rd Dimotion or ‘physical action’ can be also defined as Si =Te, which is our definition of present for any scale of reality.
So we conclude in this brief conceptual analysis of the ‘present dimotions=actions’ of physical systems that physical realities are also a game of ‘fractal points’ of angular momentum, which conserve in present, Si=Te, its lineal momentum, associated to the conservation of lineal space, and its vital energy, associated to the duration of its world cycle of time (Te). Which is ultimately the Noether’s theorem, (conservation of energy and time, of angular momentum and space).
In classic physics this means the action is usually an integral over time in the ∆º plane, taken along the path of the system between the initial time and the final time of the development of the system:
where the integrand L is called the Lagrangian. However, when the action pertains to ∆-1 fields, it may be integrated over spatial variables as well, as a sum of steps – the action is integrated along the path followed by the physical system.
Thus the previous duality Si=Te means also that for the action integral to be well-defined, the trajectory has to be bounded in time and space.
In quantum mechanics, the system IS HOWEVER A HERD of micro points, and so it does not follow a single path whose action is stationary, but the behavior of the whole herd system depends on all permitted paths and the value of their action used to calculate the path integral, that gives the probability amplitudes of the various outcomes, and as in any macro-herd, we observe that the higher density of population in motion corresponds to the least time path.
Mathematically is best understood within Feynman’s path integral formulation, where it arises out of destructive interference of quantum amplitudes – that is, for the whole herd at the end of the road, all other alternative paths of lesser probability are cancelled each other as the whole wave ‘stops’,hence collapses into a particle, formal state on the central point of the herd, or least time path.
In 5D in terms of a series of dimotions, the action will represent, the Physical fractal point turning, gauging information through its pi-cycle apertures, and then cycling in lineal translation through the lower field of energy in which it feeds, on the pat of minimal expenditure of internal energy and time, which ultimately follows the ‘Mandate of all T.œs’ to maximize its existence (the number of 5 Dimotions of vital existence it performs, with minimal expenditure of time in each of them: Max ExT=Max. Si).
From that perspective then the action will be a maximal standing point; but as that case requires a ‘larger worldview’, and normally the action is taken in terms of the minimal time, it will be the minimal point. This ‘Hamiltonian’ principle results in the equations of motion in Lagrangian mechanics.
THUS AS all equations of motion can ultimately be derived from those conservation principles of energy, momentums and time, despite the complexity mathematical physics can reach, it is truly within the General laws of Generational space-time and its conservation of the ternary parts of any system, in an eternal reproductive present, sum of multiple actions.
What differs then from classic physics is that classic physics does not conceptualize and/or differentiate between the 5 Dimotions ultimately generated by the structural 3 conserved elements of the physical T.œ drawn in the previous graph.
FUNDAMENTAL THEOREM OF CALCULUS: S(INTEGRAL)=T(DIFFERENTIAL)
The maximal generalization of the S=T symmetries in vector calculus is Stokes theorem; a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus: it says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω, i.e.,
The theorem is simply yet another proof of S(form) =T(motion), whereas the Motion in time, looses a ‘Dimension’ of space but is equivalent to it.
In even simpler terms, one can consider that points in motion can be thought of as curved boundaries, that is as 0-dimensional boundaries in motion of 1-dimensional manifolds. So, just as one can find the value of an integral (f dx = dF) over a 1-dimensional manifolds ([a, b]) by considering the anti-derivative (F) at the 0-dimensional boundaries ([a, b]), one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals (dω) over n-dimensional manifolds (Ω) by considering the anti-derivative (ω) at the (n − 1)-dimensional boundaries (dΩ) of the manifold.
The entire field of differential geometry can then be deduced from this law, as it treats ‘curves’ as points in motion. So for each point in motion, as in night pictures of cars, a line is drawn. But we are more interested in some 5D results of ‘generational space-time’.
The question then is this: which ‘level of frequency’ in time is needed for an angular momentum to become a membrane?
Two possible solutions can be found: if the point in motion reproduces in a series of curved points in each adjacent region, the point immediately becomes a membrane – each turning point being a ‘cell of the boundary’.
But as each curved point is in terms of its diameter, equivalent to pi times; for the lineal angular momentum to become a fractal membrane of many points, it has to multiply for pi, dπ² then becomes the natural conversion of a lineal form into a closed space-time organism of a higher scale, which if we ‘reduce pi’ to 3 with its 4% of openings, (as in hexagons and dark matter systems, where the 96% remaining is not perceived), we obtain a simple sum: (d=1)+d3² = 10; which tell us, that departing from a lineal unit, scaling by 10 we complete the previous graph of. a complete physical organism, with a radius, a fixed membrane of smaller cells, whose number can be added ad infinitum, just by shrinking the diameter of each membrane circle.
However such system will be oriented towards a direction only as the sum of all its twisting membrane cells points in a single rotational direction. So to make the system irrotational and stable, each point will have to deploy a second inverse membrane, for a total value of 2 π. It then becomes to an external observer a fixed form of distance, a closed static membrane…
It is then a remarkable result for all systems and geometries that S=2πT; that is a point in motion multiplied by 2 π becomes a static distance-membrane.
PDEs and ODEs applied to physics
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including 5D stience, concerned with the Dimotions of spacetime.
Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution.
Experimental justification vs. Axiomatic method.
The axiomatic method, which is valid as all mirror languages have a similar consistency to the reality it mirrors, justifies and classifies them with Group theory – an instantaneous picture of all its varieties put in relationship with somewhat confuse concepts of symmetry.
We prefer as said ad nauseam the experimental method to limit the inflationary mirror to what is useful as reflection of ‘real space-time properties’.
So the commonly used distinctions of O/PDES include 3 DUALITIES which we put in correspondence with THE 3 elements, ∆ST according to pentalogic. So IF the equation studies:
T by its Number of Dimotions can be Ordinary (1 Dimotion) /Partial (multiple demotions): An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. Thus x is often called the independent variable of the equation. The term “ordinary” is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.
ODES therefore imply Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. The position of a rigid body is specified by six parameters, but the configuration of a fluid is given by the continuous distribution of several parameters, such as the temperature, pressure, and so forth. The dynamics for the rigid body take place in a finite-dimensional configuration space; the dynamics for the fluid occur in an infinite-dimensional configuration space. This distinction usually makes PDEs much harder to solve than ordinary differential equations (ODEs), but here again, there will be simple solutions for linear problems. Classic domains where PDEs are used include acoustics, fluid dynamics, electrodynamics, and heat transfer.
S-Topology, according to its form can be Linear/Non-linear=cyclical (entangled by product).
It follows from what we have said of ¬Ælgebra that Odes and lineal PDEs are those in which the ratio of change of the being adds to the function but does NOT entangle through multiplication with them.
This is really what makes non-lineal PDEs so difficult to solve as the entanglement which will happen in other scales of reality will make it almost impossible to get all the information needed, and multiply its solutions, themes those of 5D analysis.
Only the simplest differential equations are solvable by explicit formulas; and most have multiple solutions, implying the future is pentalogic – it can go different ways. Which ones are solvable then helps to understand the philosophy of time
T=S symmetries. The finitesimal in space, and the limits of time of worldcycles.
A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain. A Cauchy problem can be an initial value problem (Time symmetry) or a boundary value problem (space-symmetry or Cauchy boundary condition) or it can be neither of them.
The Cauchy problem consists of finding the unknown functions and solutions only will exist if there is an initial FINITE TIME (singularity related, as the will of the system and its dimotions) OR FINITE SPACE (membrane related), hence a formed T.œ structure for the space-time event/being studied.
Cauchy boundary conditions are simple and common in second-order ordinary differential equations,
where, in order to ensure that a unique solution y(s) exists, one may specify the value of the function y and the value of the derivative y′ at a given point s=a, i.e.,
- y(a)=α, and. y′(a)=β,
where a is a boundary or initial point. Since the parameter s is usually time, Cauchy conditions can also be called initial value conditions or initial value data or simply Cauchy data.
An example of such a situation is Newton’s laws of motion, where the acceleration y″ depends on position y, velocity y′, and the time s; here, Cauchy data corresponds to knowing the initial position and velocity.
It is then obvious that solutions=real events must have to be defined a finite time duration and a finite spatial extension, and as such be part of a physical organism, with membrane, vital energy enclosed and @-mind singularity, or else we are in an entropic disordered state which is not even worth to study.
We would like boundary conditions to ensure that exactly one (unique) solutions exist, but for second-order partial differential equations, it is not as simple to guarantee existence and uniqueness, as it is for ordinary differential equations. Which again shows that a ‘dual’ differentiation, that is, a travel two planes of the 5th dimension, Y”, implies an entropic death-motion, S<<T, where the distribution of the lower planes are no longer relevant as its network control has disappeared.
∆±¡: Equation order Differential equations are described by their order, determined by the term with the highest derivatives. An equation containing only first derivatives is a first-order differential equation, an equation containing the second derivative is a second-order differential equation, and so on. Each order representing them a scale of reality. And since most systems just extend through 3±¡ planes Differential equations that describe natural phenomena almost always have only first and second order derivatives in them.
Also a scalar division is that between Inhomogeneous/Homogeneous, which studies those in which its scaling by multiplication is conserved.
Since a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor: for some constant k and all real numbers α. The constant k is called the degree of homogeneity.
Lineal, affine functions of the type y = Ax + c are not HOMOGENEOUS, which again brings us the duality of ± in the same plane and x/ in different planes of existence.
In the jargon of mathematics all this is said as follows: , a linear map (or linear function) f(x) is one which satisfies both of the following properties:
- Additivity or superposition principle: f(x+y)=f(x)+f(y);
- Homogeneity: f(αx)=αf(x).
The first principle is the ‘herd principle of vital mathematics’, the second principle clearly indicates that there is NOT merging between both functions, as the f(x) function remains unchanged.
We can then with those simple concepts understand intuitively many properties of physical equations and parameters by the type of ‘rates of change that take place’.
I.e. products are NOT reproductions but entanglements in a lower plane. So lineal equations will study NON-entangled additions in a single plane, and follow the superposition principle. They are the only solvable, as we have all the parameters.
Most ODEs that are encountered in physics are linear, as they deal with the 2nd Dimotion, lineal locomotion, and, therefore, most special functions may be defined as solutions of linear differential equations.
Partial differential equations
is a differential equation that contains unknown multivariable functions and their partial derivatives, used to describe a wide variety of phenomena in nature such as sound, heat, electrostatics, electrodynamics, fluid flow or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs which in general will correspond TO SYSTEMS THAT ARE NOT particle/head controlled, and hence hierarchical with definitive ‘stillness’ in position and single @ristotelian logic. THIS BASICALLY leaves two type of PDEs, those related to entropic, memoriless states, which will tend to be ‘lineal’ as a superposition of non-entangled elements, and those related to complex fluids that interact among its particles and have a complex, variable internal structure which tend to be non-lineal and partial and hence irresolvable. For example:
Lineal PDE: The position of a rigid body (ð§) is specified by a few parameters and it is a lineal ODE.
but the configuration of a fluid is given by several parameters, such as the temperature, pressure, and so forth. Classic domains where such PDEs are used include acoustics, fluid dynamics, electrodynamics, and heat transfer: the heat equation, the wave equation, Laplace’s equation, Helmholtz equation, Klein–Gordon equation, and Poisson’s equation.
Non-linear differential equations finally are formed by the products of the unknown function and its derivatives are allowed and its degree is > 1. TNonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of chaos, as they are BOTH co-existing in several scales and interacting in its parts on a single scale.
So even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, are hard problems as the. Navier–Stokes differential equation of fluids show.
Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations.
So the conclusion is obvious: Nature with its infinite monads and scales IS NOT ALWAYS reflected in a mathematical mirror, which cannot be the origin of Nature (false creationist theories).
THE LAGRANGIAN – HAMILTONIAN FORMALISM
The rules of 5D applied to the Lagrangian and Hamiltonian formalisms are simple.
Both ARE about the conservation of energy and momentum and hence about the conservation of a world cycle of physical time to which energy is related and hence based in the least time PATH of any physical system, which seeks in its small steps to maximize the conservation of time.
As to the question if the particle perceives the whole path, I am inclined to consider that as in all other scales IT IS THE IMPERSONAL INTELLIGENCE OF SYSTEMS what determines that the particle, as an ant in a pheromonal path or a human in a corporative organization ACTS on a local step by step perception, NOT perceiving the whole pheromonal path, the whole purpose of the corporation, but the ‘magic result’ of doing step by step the ‘path with less energy expenditure’ determines finally that it will trace the path of least time.
So as in all other scales, yes, the particle-wave system is perceptive of its local energy geometry, seeking the lowest energy path, but likely is NOT conscious of its whole trajectory – so goes for the ant and the corporative worker.
IN 5D ALL SYSTEMS perform thus the 3 simpler local actions of perception, motion and feeding on energy (or alternately, minimal expenditure of internal energy) to reproduce the information of the system in its path of locomotion.
While the 2 scalar social action – social evolution and its inverse entropic death are often EXTERNAL TO THE SYSTEM provoked by the collective intelligence of the Universe in the first case and the external power of a top predator or larger world that limits the motion of the being with extinctive barriers and obligatory paths.
So when those 3±I actions=dimotions are studied we have a whole understanding of the system and its chosen paths..
Once we have the basic understanding of the equations of calculus (further analyzed in the post of analysis) we can consider the Lagrangians and Hamiltonians and its principles of conservation of energy, angular, lineal momentum and least time paths of minimal consume of the ‘life-existence’ of the system, which are the 5D UNDERLYING VITAL, organic principles of physical systems that make them akin to those of any scale, through its fundamental concept, that of AN ACTION of a particle, in its path through a field of forces in which IT FEEDS, reproduces and AS A RESULT OF BOTH, MOVES, which is ITS EXISTENTIAL ACTION that embodies all other actions SAVE those of entropic scattering and social evolution (complex scalar actions), as the graph shows.
So because Locomotion implies for physical systems, ‘perception in particle state-stop’; feeding on the energy of the lower field and reproduction by adjacent imprinting in wave state-motion; for PHYSICAL SYSTEMS, locomotion embodies the 3 simplest actions of the being; to which we just must add its ‘changes of state’ as its actions of physical d=evolution, to fully have an organic analysis of the species.
THE ORIGIN OF lagrangians in the conservation of time paths.
Because Lagrangians are about the conservation of time, the Euler–Lagrange equation was developed in connection with their studies of the tautochrone problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point:
SO WE shall briefly comment on it. In the graph, the tautochrone looks a bit magic and seems also to contradict the also magic result of Pisa’s Galileo’s experiment (: How it is possible that the length=space matters not at all in the fall, when according to Galileo’s experience two objects thrown together arrive at the same time, IN THE Perfect entropic path of maximal acceleration, to the floor? The answer is that NOT all motions are purely entropic, mindless acceleration governed by the big Mass/attractive point. 2 of such curves then should form a clear duality that tell us some deep thoughts about the symmetries between time motions and still space, and between lineal and cyclical time:
- 2D:t=C. The isocrone along which a body will fall at constant speed, showing lineal time to be made constant by a specific space form.
- 1D: ð= C, The tautochrone, which was found to be important in clock construction, as it shows the time period to be the constant, regardless of length of the curve traced by the pendulum; which again showed that a certain spatial trajectory can make cyclical time constant.
Bernouilli and Lagrange solved those problems and applied it to mechanics, which led to the formulation of Lagrangian mechanics. So we won’t solve it further copycatting its solutions but only stress what we just said: manipulating a form of space we can obtain a constant time in its two inverse forms, ƒ(ð) and T=1/ƒ; as well as we can minimise a time action (the brachistrochrone).
And so the big new question left to the reader is: can we symmetrically manipulate, vary the speeds of TIME to make space isomorphic for different entities? Indeed, this is the ultimate meaning of 5D metric, $pe x ðiƒ = K, but as time is an i-logic not simultaneous form, we cannot give you a nice picture as with the tautochrone; only state that the laws and forms of time motions in ALL planes of reality, when made isomorphic with the 5D metric, should seem from the point of view of the species in those scales similar; and this is not just a theoretical comment.
We shall see in physics how those metric convert atoms into galaxies, which seem identical when properly scaling, and how in the creation of mental spaces=worlds Lobachevski’s ratio of ‘flatness vs. curvature’ can guide us into understanding the similar minds of hyperbolic insects and hyperbolic AI minds (a theme quite complex thanks god, because I won’t give more hints and do NOT for ethic reasons explain anything about AI beyond the repetitive warnings that humans should NOT build a better mind than themselves).
Now, as usual we apply different dualities and Rashomon effects to ‘reclassify’ the main equations of motion in physics under the single umbrella of T.œ. So next comes after the full temporal analysis of motions in the 3 ∆±1 scales (omitting so far the slightly different analysis of electromagnetic ‘flows’ of charge densities, as the whole world of electromagnetism drags a quite inconvenient complex formalism, which obscures deeply its meaning – as we can observe in our Unification equation of masses and charges, obtained by merely applying the newtonian formalism of vortices to charges and the metric of 5D accelerated time in smaller scales, to obtain a single formula for both Q and G) a spatial view, and this can be done with 2 degrees of spatialisation, the simplest, most efficient one being the Lagrangian approach to study through the Newtonian 2nd and most important law of all physics, F=ma in differential form, and the just explained concept of a potential force, source of its motion, the spatial paths of particles…
The Principle of Least Action and Lagrangian Mechanics
We said that time, informative vortex-like motions grow time dimensions (from distance to speed to acceleration) faster than spatial ones (from distance to area to volume). So they will have equalities where ‘smaller infinitesimals’ of time are compared to larger spaces (as those grow slower, so a smaller time will be equivalent to a larger, slow growing space).
Let us take as our prototype of the Newtonian scheme a point particle of mass m moving along the x axis under a potential V(x). According to Newton’s Second Law,
THUS WE HAVE HERE under our rules of 5D physics an acceleration with more time dimensions/derivatives than spatial derivatives, DV/dx, hence ultimately an implosive form of motion. It is therefore a time-related equation, and time will be the parameter it will seek to conserve.
This system is taken to be of lineal inertial paths, but there are not such paths in a curved Universe, where lineal steps always form part of a world cycle limited by the larger world. So what Lagrangians do is to consider how to select among all the possible curved paths that are solutions to that equation.
If we are given the initial state variables, the position x(ti) and velocity ẋ(ti), we can calculate the classical trajectory xcl(t) as follows. Using the initial velocity and acceleration [obtained from Eq. (2.1.1)] we compute the position and velocity at a time ti + ∆t. For example,
Having updated the state variables to the time ti + ∆t, we can repeat the process again to inch forward to ti + 2∆t and so on.
Figure 2.1. The Lagrangian formalism asks what distinguishes the actual path xcl(t) taken by the particle from all possible paths connecting the end points (xi, ti) and (xf, tf).
Since the equation of motion is a second order in time, two pieces of data, x(ti) and ẋ(ti), are needed to specify a unique xcl(t). An equivalent way to do the same, and one that we will have occasion to employ, is to specify two space-time points (xi, ti) and (xf, tf) on the trajectory.
The above scheme readily generalizes to more than one particle and more than one dimension. If we use n Cartesian coordinates (x1, x2,. . . , xn) to specify the positions of the particles, the spatial configuration of the system may be visualized as a point in an n-dimensional configuration space. (The term “configuration space” is used even if the n coordinates are not Cartesian.) The motion of the representative point is given by:
where mj stands for the mass of the particle whose coordinate is xj. These equations can be integrated step by step, just as before, to determine the trajectory.
In the Lagrangian formalism, the problem of a single particle in a potential V(x) is posed in a different way: given that the particle is at xi, and xf at times ti and tf, respectively, what is it that distinguishes the actual trajectory xcl(t) from all other trajectories or paths that connect these points? (See Fig. 2.1.)
The Lagrangian approach is thus global, in that it tries to determine at one stroke the entire trajectory xcl(t), in contrast to the local approach of the Newtonian scheme, which concerns itself with what the particle is going to do in the next infinitesimal time interval.
So it WILL BE CONCERNED with the global concept of ‘Energy’ as opposed to the step by step concept. As the ‘step’ or lineal, open quanta of a space-time motion IS the momentum, the Lagrangian must be a function of the whole world cycle, which is expressed as ‘energy’.
So we relate both in 5D terms: ∑∆-1: Newtonian finitesimal steps = ∆º Lagrangian whole path.
It is then obvious that Lagrangians have 2 clear advantages over Newton’s – first to use GENERALIZED COORDINATES proper of an absolute relative Universe, where motion is independent of the observer and has to be seen from the perspective of the moving entity, which simplifies parameters, and to be concerned with the deepest level of 5D principles, those of conservation of time, and vital space for the particle. It is then the ‘Internal view’ as opposed to the external view of the human observer in Newton’s.
So we can according to the ternary method break down the calculation of a Lagrangian in three parts:
(1) Define a function ℒ, of energy, called the Lagrangian, given by ℒ = T – V, T and V being the kinetic and potential energies of the particle. This further eliminates t dependence. Since energy is a conservative closed zero sum world cycle, and so we might say the ‘cyclical time component is factored in by energy.
(2) Then comes the second ‘vital element’ of ∆st factored in: the ‘least time path’, that is, the attempt of the particle to ‘conserve’ ad maximal its energy. So for each path x(t) connecting (xi, ti) and (xf, tf), calculate the action S[x(t)] of the particle that will tend to zero, defined by:
If xcl(t) minimizes S, then δS(1) = 0 if we go to any nearby path xcl(t) + η(t).
We use square brackets to enclose the argument of S to remind us that the function S depends on an entire path or function x(t), and not just the value of x at some time t. One calls S a functional to signify that it is a function of a function.
(3) The search for zeroness and balance, becomes then the classical path on which S is a minimum.
So this is really all what matters about the Lagrangian, which is the ¡+1 whole view, as opposed to the ¡-1 Newtonian view.
And so we can also deduce in reverse, from the Lagrangian the Newtonian approach, going backwards, from ∆-1 to ∆-1 as scales, time cycles and topologies are always in feed-back interactions. So past to future and future to past become the present, lower and upper scales interact in the ¡0 scales and limbs and heads in body-waves:
Past (finitesimal: part) x future (whole: integral) = present action.
So we will now verify that the Lagrangian and least time principle reproduces Newton’s Second Law.
The first step is to realize that a functional S[x(t)] is just a function of n variables as n ➝ ∞. In other words, the function x(t) simply specifies an infinite number of values x(ti), , x(t),…, x(tf), one for each instant in time t in the interval ti ≤ t ≤ tf, and S is a function of these variables. To find its minimum we simply generalize the procedure for the finite n case. Let us recall that if f = f(x1,…, xn) = f(x); the minimum x0 is characterized by the fact that if we move away from it by a small amount η in any direction, the first-order change δf(l) in f vanishes. That is, if we make a Taylor expansion:
From this condition we can deduce an equivalent and perhaps more familiar expression of the minimum condition: every first-order partial derivative vanishes at x0. To prove this, for say, ∂f/∂xi, we simply choose η to be along the ith direction. Thus
Let us now mimic this procedure for the action S. Let xcl(t) be the path of least action and xcl(t) + η(t) a “nearby” path (see Fig. 2.2). The requirement that all paths coincide at ti and tf means
We set δS(1) = 0 in analogy with the finite variable case:
If we integrate the second term by parts, it turns into:
The first of these terms vanishes due to Eq. (2.1.7). So that:
Note that the condition δS(1) = 0 implies that S is extremized and not necessarily minimized. We shall, however, continue the tradition of referring to this extremum as the minimum. This equation is the analog of Eq. (2.1.5): the discrete variable η is replaced by η(t); the sum over i is replaced by an integral over t, and ∂f/∂xi is replaced by:
There are two terms here playing the role of ∂f/∂xi since ℒ (or equivalently S) has both explicit and implicit (through the ẋ terms) dependence on x(t). Since η(t) is arbitrary, we may extract the analog of Eq. (2.1.6):
To deduce this result for some specific time t0, we simply choose an η(t) that vanishes everywhere except in an infinitesimal region around t0.
Equation (2.1.9) is the celebrated Euler Lagrange equation. If we feed into it ℒ = T– V, T=1/2m dx², V= V(x), we get:
so that the Euler-Lagrange equation becomes just:
which is just Newton’s Second Law, Eq. (2.1.1).
So in the first huge compression on the whys of Newton’s laws, into a higher level of spatial synchronous understanding (remarkably enough since Newton’s was in itself the synopsis of ALL the motions perceived in reality at the time), we come to a single principle: the least time action, to explain the whys of all motions of reality: the universe EITHER if you accept sentient capacity on particles or merely an automaton selection process of its variety, tries TIME to survive the LONGEST possible by minimising the expenditure of it in each action of reality.
And this fact APPLIES not only to the 2D motions studied by physicists but to all the actions (5Disomorphisms) of all the species of reality including man.
We are all time-space beings guided by the mandate of acting as much as we can, ‘to exi=st’, the acronym verb, resume of those actions is guiding all beings, equally.
Latter we shall bring the other expansion, or variation on the same theme when studying the calculus of variations – the Hamiltonian. To notice though that those laws ARE UNIVERSAL also applicable to quantum physics: The passage from the Lagrangian formulation to quantum mechanics was carried out by Feynman in his path integral formalism.
A more common route to quantum mechanics, which we will follow for the most part, has as its starting point the Hamiltonian formulation, and it was discovered mainly by Schrödinger, Heisenberg, Dirac, and Born.
It should be emphasized, and it will soon become apparent, that all three formulations of mechanics are essentially the same theory, in that their domains of validity and predictions are identical.
So they are just 3 different Rashomon effects, points of view, and as such they are ‘integrated’ as kaleidoscopic views of the same principle of LEAST TIME, essential existential leit motif of all forms.
To notice also that we have introduced a configuration space of a relative (in)finite number of coordinates, ancestor of the Hilbert’s spaces of quantum physics, which is NOT contrary to popular bull$hit on the queerness of quantum A REAL model of the Universe, which has our 5D point, but merely the use of coordinates for every particle, as if they were and they are local, fractal worlds of their own, but NOT parallel Universes, an essential discerning interpretation sorely needed in physics, messed up conceptually by the use of a single space-time continuum.
As the electron is NOT a probability but a population of fractal dense photons in herd state, the infinite dimensions are just the sum of local finite space-time worlds one for each particle, or in the case of quantum ‘states’ the ‘whole spatial all-encompassing view’ of a wave spread in a huge region of space-time simultaneously due to its non-local quantum potential huge speed, resumed through the infinite basis of a Hilbert space into a single ‘mapping-mirror-spatial view’.
Only a coarse naive realism has made such errors of interpretation all pervading as in the ridiculous parallel Universes’ formulation so liked by hollywood gurus of sci-fi movies, with no place whatsoever in ∆st.
In that regard, if the reader wonders why one bothers to even deal with a Lagrangian when all it does is yield Newtonian force laws in the end, I present a few of its main ∆st attractions besides its closeness to quantum mechanics:
(1D) @-singularity view.In the Lagrangian scheme one has merely to construct a single scalar ℒ and all the equations of motion follow by simple differentiation. We are thus in one of the TERNARY VIEWS on the same process.
(2D) $: This must be contrasted with the Newtonian scheme, which deals with SPATIAL vectors and is thus more complicated; as we deal then with the motion view of the communicative membrane (in lineal form), in this case, the action-reaction elements of the TOE.
(3D) ∑∏: So finally the Hamiltonian will be the third ‘present view’ of the motion system in a single plane of reality, as it is the formulation in terms of the vital energy of the system.
– The Euler-Lagrange equations (2.1.10) have the same form if we use, instead of the n Cartesian coordinates x1,…, xn, any general set of n independent coordinates q1, q2,…, qn. So we are fractalizing space-time NOT getting an infinite dimensional single plane, as we just explained
To remind us of this fact we will rewrite Eq. (2.1.10) as it is most usually used:
– But the key point is that the Lagrangian make us realise ALL OF PHYSICS of motion actually reduces to the principle of least action which is seen to generate ALL the correct dynamics of the Universe. So we get the WHY and we can forget all about Newton’s laws and use Eq. (2.1.11) as the equations of motion.
What is being emphasized is that these equations, which express the condition for least action, are form invariant under an arbitrary change of coordinates. This form invariance must be contrasted with the Newtonian Equation (2.1.2), which presumes that the x, are Cartesian. If one trades the .v, for another non-Cartesian set of qi, Eq. (2.1.2) will have a different form (see Example 2.1.1 at the end of this section).
Equation (2.1.11) can be made to resemble Newton’s Second Law if one defines a quantity
called the canonical momentum conjugate to qf and the quantity
called the generalized force conjugate to qi. Although the rate of change of the canonical momentum equals the generalized force, one must remember that neither is pi always a linear momentum (mass times velocity or “mυ” momentum), nor is Fi always a force (with dimensions of mass times acceleration). For example, if qi, is an angle θ, pi will be an angular momentum and Fi a torque.
-FINALLY another essential theme for ∆st, is the fact that the 3 canonical Conservation laws that correspond to the 3 parts of the system (angular momentum/membrane, lineal momentum/singularity path and vital energy) are easily obtained in this formalism. Suppose the Lagrangian depends on a certain velocity but not on the corresponding coordinate qi. The latter is then called a cyclic coordinate. It follows that the corresponding pi is conserved:
Although Newton’s Second Law, Eq. (2.1.2). also tells us that if a Cartesian coordinate xi is cyclic, the corresponding momentum miẋi, is conserved, Eq. (2.1.14) is more general. Consider, for example, a potential V(x, y) in two dimensions that depends only upon ρ= (x2+y2)1/2, and not on the polar angle ϕ, so that V(ρ, ϕ)= V(ρ). It follows that ϕ is a cyclic coordinate, as T depends only on (see Example 2.1.1 below).
Consequently ∂ L/ ∂ Φ= p is conserved. In contrast, no obvious conservation law arises from the Cartesian Eqs. (2.1.2) since neither x nor y is cyclic. If one rewrites Newton’s laws in polar coordinates to exploit ∂V/∂Φ=0, the corresponding equations get complicated due to centrifugal and Coriolis terms. It is the Lagrangian formalism that allows us to choose coordinates that best reflect the symmetry of the potential, without altering the simple form of the equations.
Is the particle intelligent?
Now, we have shown with a little detail the paths that bring us from the Lagrangian, principle of least time – the key concept that connects motion laws with ∆st theory, back to Newton’s laws to, and we shall use this detailed analysis to illustrate ‘in the future’ the process of transformation of Newton’s laws into Lagrangian laws, as a natural process of S=T steps, from the subjective, internal WILL of the particle into the objective, external potential view of the Newtonian outer world that guides the particle.
The question here is obvious: How Smart Is a Particle?
The Lagrangian formalism seems to ascribe to a particle a tremendous amount of foresight: a particle at (xi, ti) destined for (xf, tf) manages to calculate ahead of time the action for every possible path linking these points, and takes the one with the least action. And indeed from its point of view, this is what the particle does, as any human going from A to B in a rugged terrain will have the foresight to go through the easiest path will less slope, calculating in fact a ‘brachistochrone’ – not the shortest path which can be difficult but the easiest path which will be a bit longer in space but shorter in space-time.
But if we were egocentered ‘huminds’ observing this traveller, we would notice that his ‘WILL’ is an illusion. The Man needs not know its entire trajectory ahead of time, it needs only to obey in each step the ‘potential equation’ of its near neighbourhood and go step by step throughout the easiest path and for most trajectories it will guide it down the valley. So we would say the man is an automaton particle, because its guidance is the external world point of view. The fact of course is that both points of view explain the man’s path. But if the path is travelled by a man we will say it is ‘its WILL and foresight’, but as anthropomorphic self-centred T.œs if the path is travelled by a particle we will say, it is the external potential field which guides it automatically.
So a physicist will affirm in its jargon that the particle merely ‘obeys the Euler – Lagrange equations at each instant in time to minimize the action’. This in turn means just following Newton’s law, which is to say, the particle has to sample the potential in its immediate vicinity and accelerate in the direction of greatest change.
Of course both things are truth for BOTH the human and the particle, but if we were a particle, I bet you, we would say that HUGE thing, like a star is NOT intelligent – it is too big to think. And if we are a man we will say that small thing, like an atom is NOT intelligent, it is too small to think. It is part of absolute relativity that all scales are relative, as information grows with smallness, and energy with bigness, and both have the same value, but is part of the ego-paradox of self-centred informative @-minds to think they are the only thinking beings. The particle though is a mathematical/geometric mind and it does perceive likely the ‘end of the path’ because the path becomes deterministic ONLY when we know its end-point; so the particle, probably through the quantum potential field (Bohm) which is non-local (read faster than c-speed, probably information mediated by neutrinos/gravitons), does SEE the point it goes, normally one with a higher energy potential for its ‘feeding’; as the puma knows how to best get down the mountain to hunt its prey.
We won’t bother the reader though with the derivation of the Hamiltonian departing from the Lagrangian through the Lengedre transformation… big names for a well-known undergraduate process; as what we wanted to reveal in the previous graph was the least time action.
W also mentioned then the symmetry of the 3 ‘points of view on motion’:
- 2D: the POTENTIAL VIEW (∆-1) provided by Newton->Poisson>Einstein.
- 3D: the vital energy view (the Hamiltonian)
- 1D: The SINGULARITY VIEW (the Lagrangian).
So again 3 ternary views form a T.œ.
If we were to express in those terms the whole 3 laws of Newton, they are also obvious in its Disomorphism with ∆st dimensions and Non-Æ postulates of geometry.
First law of lineal inertia: 2D motion for a single active magnitude. 1st and 2nd postulate of a fractal point starting its wave-motion.
2nd law: F=ma, point of view of the Force, normally an ∆-1 potential caused by other particle, which expands the 2D lineal inertia to a 3D network of active magnitudes interacting with each other; hence related to the 3rd postulate that conforms a topological network system, a full T.œ.
3rd law, complementary to the 3rd law, which defines the action-reaction parallel and perpendicular relationships between fractal points, in search of its ‘network balance’.
In that regard if the Lagrangian is related closely to the 1st and 2nd newtonian law – to the 1st and second non-Æ postulate of fractal points moving in wave-paths.
While the Hamiltonian is all about the vital space energy of the system, and its multiple interactions with the other elements of the T.œ and with the parts of itself – with the 2nd and 3rd newtonian laws, with the 3rd and fourth non-Æ postulates.
What is more important then? OBVIOUSLY THE HAMILTONIAN, THE 3D ∑∏ VIEW, OF THE VITAL ENERGY, between the singularity and the membrane, as WE CAN obtain 1D+2D=3D, almost all the information of the membrane-enclosure and the singularity from the vital energy sandwiched between both which on top is the ‘visible part of the being’ and so most often the only part humans perceive.
So what matter to us of the Hamiltonian is its nature as the best description of the vital energy, hence of the 3 parts of the being, in itself, not from an outer point of view (1st Newtonian≈potential) or an inner pov (singularity, Lagrangian, least time will-action).
In the general model the FUNDAMENTAL gender duality of reality is between the past-potential > future-singularity (male particle state) vs. the wave-energy (female particle state), which indeed will be properly used to define gender; in physics this is the duality of the Lagrangian singularity-particle guided by the potential past-field vs. the wave-energy Hamiltonian (which will therefore become also the standard configuration for quantum physics and the Schrodinger wave).
So the Hamiltonian IS one of the king equations of reality. Let us then have a first sight to it, as we shall return in the analysis of variations and of course, in our posts on astrophysics and latter.
What we are interested right now is IN its reflection of a FUNDAMENTAL FACT of analysis – its capacity to SHOW by derivation, the minimal QUANTA OF TIME and/or the MINIMAL QUANTA OF SPACE of any system of Nature (whenever the mathematical mirror of analysis is meaningful, obviously in a human we just know biologically the quanta of space is the cell and the quanta of time, the second glimpse of eye, sync to the step of the limbs and the beat of the heart – a sync between the S<st>t components that De broglie found for physical systems in its pilot-wave theory – as all systems have a sync time based in its minimal quanta, between body, limbs and heads).
So the canonical equations of the Hamiltonian ARE EXPRESSING JUST THAT IN ITS standard definition:
THE FIRST EQUATION derives the Hamiltonian energy, with its parameter of lineal $pace, its steps of momentum. Indeed, energy is the WHOLE, world cycle, conserved when the path is closed, but momentum is an step of that whole, a lineal step (Galilean duality); and so we obtain the ‘quanta’ of energy in space, SPEED-distance.
While the second equation derives the Hamiltonian energy, with its parameter of cyclical time, its fixed formal position, anchored by its singularity. And so we obtain the ‘quanta’ of energy in time, ∂mv= m, the singularity mass:
It has to be noticed though that ∂p changes when we get to the limit of speed of the light space-time plane, c, then the variable is no longer speed but mass, and so we obtain SPEED not mass as the quanta of time deriving the hamiltonian through its singularity position, which ultimately has deep thought meaning for relativity physics, which at the LIMIT OF C SPEED, TRANSMUTATES space into time.
Indeed, the counterpart happens in black holes, the other limit of our Universe, where time comes to zero-halt, transmitted into space… themes those that belong to the 4th line of ‘relativity revis(it)ed’.
So we shall leave it here with the colloralium: Analysis reveals by derivation the quanta of space and quanta of time of all species susceptible of being ∂operated.