text:
User:Super Quantum immortal/Abiogenesis

(fix the formating latter)

2.0 Hackolution

CLIC ME CLIC ME the UNIVERSE is bigger then big, and we are smaller then small Absolutly huuuge stars... and us

Lkjhbopfg homrthmj qldkjhlijh ilthilt jrlirklt nhfgg fdgh foigu rpoue. monkey at a typewriter

At the beginning, theres the big bang, mater condenses in to galaxies and stars. After several generations of stars our sun is formed and earth around it. Nothing but a grain of dust in the hugeness of the universe, that just happened to have conditions just right for us. Other wise the universe is a pretty damn barren place Then life appears, there isn't an official guess for this; there are a zillion conjectures, but these are mainly superstitions and heresy. As it stands, the most advanced, plausible, confirmed and official theory, parallels a bunch of monkeys typing on typewriters, and 1 of them writing once, 1 of Shakespeare's plays. Every 1 can say anything and its contrary, without being accused of crankdum. I'll do that too, i'll just take a little bit from here and there that i like, and add a little bit of sauce of my own recipe.

2.1 Bootstrapping Life seems to start very fast after earth has cooled down. A cool theory, is that mars being smaller, cooled faster, hence being capable to harbor life, while earth was still a molten hell. Life actually started on mars, and got transfered immediately on earth, through meteor impact(as bacteria), as soon as she was cool enough. Mars continued to cool fast to death, while earth, being bigger, has still quite some time before she reaches that state.

surfactants bricabrac pop out and full size your browser does not support the video tag, install

Regardless, if we are really earthlings or Martians, at the beginning, various simple molecules get formed spontaneously due to physical processes (lightnings, UV, natural catalysts, extraterrestrials, Santa Claus, and whatever) and accumulate over millions of years(primordial soup theory). We will concentrate on surfactants; probably, mainly fatty acids because they are the simplest, but other types of molecules are not excluded. There is a shitload of possibilities; see for example the 1 with all the flour things. The examples are artificial, or created by living beings, only the 1s that can be created spontaneously by natural processes will be available(organic/inorganic hybrids possible). Surfactants have a head and a tail thing; the head is polarized, and the tail is not polarized. The polarized head tends to stick with H2O molecules because they are also polarized; but around the non polarized tail, H2O molecules tend to stick with themselves only; in other words they try to avoid each other. When natural processes concentrate locally these surfactants molecules, they will collectively try to hide there tails from the H2O by only showing to it there heads, initially forming tiny droplets, all heads out side, all tails inside. If certain conditions are met, the droplets merge in bilayer sheets of molecules, and lastly folding on them selves, into the 3D form of a vesicle. The vesicle's surface has some permeability to any small random free floating molecules(organic/inorganic), but not to big 1s. The small molecules being in a dynamic equilibrium, molecules constantly get in and out of the vesicle, but on average their concentrations in and out stay the same. If for some reason they get polymerized inside the vesicle, then they will be too big to leave. These polymers, take up space, and H2O molecules have a tendency to stick on them, thus reducing the concentration of free H2O molecules inside of the vesicle. In the outside H2O is more pure, and the concentration of free H2O molecules near the membrane is higher. Now, it becomes more probable that a H2O molecule will collide with the membrane at the outside face rather then the inside, inside some H2O collisions are replaced with polymer collisions, thus more get in rather then out. A net flow of H2O will rush inside, swelling the vesicle, until the concentrations of free H2O molecules become identical again(osmosis). This swelling will stretch the membrane of the vesicle and expose the tails of the surfactants. Vesicles look macroscopically inert, but at the microscopic level; individual surfactants molecules are very dynamic; 2D-floating around in their sheet; flipping sheets, leaving the vesicle all together to float on there own, and reincorporate in to a nearby vesicle. The stretching of the membrane will disrupt the dynamic equilibrium between the rates of gained and lost surfactant molecules. The stretched membrane will recruit extra free floating surfactants molecules to beater hide the tails. Concentration of free floating surfactants will diminish, resulting in a net loss of surfactants molecules from vesicles with lower concentration of polymers inside them. In final analysis, vesicles with the most junk inside, will cannibalize it's neighbors and grow at there expense. ("The emergence of competition between model protocells." Science 3 September 2004) A vesicle will not grow indefinitely; when it grew too big, it become instable and deviates greatly from the perfect sphere, mechanical stresses will have a increasing likely hood to split it into smaller vesicles(lipid world theory? 1994?). With it's breakfast randomly distributed among the daughter vesicles but without losing too much of it in the process. Some of this content however, is less junk than regular junk because it has catalytic properties that polymerize simple molecules into regular junk. The more polymerizing catalysts, a vesicle has, the faster it will grow. However, when the vesicle splits, it would confer an advantage to its descendants only if the catalysts can somehow be build at a efficient enough rate. Regular organic polymers can of course have catalytic properties, but it's very hard to get assembled spontaneously. Any organic polymer that can catalyze polymerization, it would be nice for the vesicle, but they arise at random, there no way to built those efficiently enough, they come and go at random.

you're hired monkey at a typewriter

left: clay crystals ready to disintegrate, right: clay theory, (pirated original diagram :)) kaolinite platelates, demonstrate crystal fertility 	crystaline reproduction There is something that can have catalytic properties and still somehow get produced at efficient rates(not merely at random); it is certain types of inorganic crystals(clay theory 1966). Organic molecules are too good at what they do, they are too flexible, spontaneously they assemble in completely random stuff, but if they get it right, they are very efficient. These crystals are indeed less flexible and less efficient, (assuming that the precursor must have been organic because of its efficiency, is silly; this is like arguing that Neanderthals had titanium spears because titanium is light and strong and stones are crap) but they are forced to crystallize in to something specific, they are forced in to whatever shape the previous crystallized layer looks like. Copying across defects, and what interest us here, surface shape. These crystals grow axially, copying across the grooves and other defects of it's surface, that has the catalytic property that interest us, when they become too big, mechanical stress cause them to break, essentially duplicating a primitive form of genetic material, in combination with the split of the vesicles, it would allow natural selection to start bootstrapping the protocell in to heir levels of complexity. Here we have genetic information, stored in the circumference, of 1 tiny crystal, 1 tiny gene, order of magnitude 1KB, fitting in a small text file; as a comparison, human DNA is 2m long, 850MB with 20.000 genes of varying complexity; a gigantic difference in complexity. So we have a bunch of monkeys, typing at random on there typewriters, but success is measured by how many pages they type.

2.2 Variability

At this point a behavior of the vesicles become relevant, vesicles can sometimes merge and mix there content, this isn't energetically very favorable, so it doesn't happen often. If there isn't some type of genetic material and natural selection acting on it, this would be pretty uninteresting. Usually(not always) this confers a selective advantage to the population by increasing variability. Instead of having a pointy Gaussian of genotypes around some average, it spreads the bell to it's tails at the expense of the area around the average. Increasing variability, has short term negative effects, but long term positive effects. Individuals with perfect fitness are less numerous, since there genes get polluted by less then perfect individuals, but, when the environment changes, or when the population colonizes new environments, there are beater chances that some individuals will survive. Sex works so well, for a lot of beings, that natural selection will come up with very ingenious ways to preserve it and enhance it, as the complexity of the cell develops,and meare merger,will not be permitted. Of course its all about striking the right balance in variability(cloning/recombination), if the population is too variable present fitness is excessively sacrificed with no corresponding long term benefits. The optimum levels of variability depend in complicated ways with various factors(size of population, variability of environment, reproduction speed...), in addition variability can be fined tuned with various tricks(sex, cloning, hermaphroditism, serial hermaphroditism, genders....). The problem with overperformance, is an overlooked aspect of competition, staying valid for all forms of competition, not just in biology. And i take the chance to point out, that because of this, eugenics, is as anti-evolutionist as creationism.

Overspecialize, and you breed in weakness. It's slow death. (Motoko Kusanagi)

Some examples: For the gazillions of bacteria that clone them selves every 20 minutes, mutations give them all the variability they need. In a extremely harsh desert, a species of lizards, gave up sexual reproduction all together, they just clone them selves. Incest can be seen as a poor mans cloning technology, once less then perfect genes where all breaded out, the population is HEALTHY, with NORMAL fertility, but very high HOMOGENEITY. The incest taboo(in most animals) is there to prevent this very high homogeneity of occurring, not to prevent the birth of weak offspring. Prehistorically humans were a couple of hundred thousands on the entire planet and need(ed) around 20 years to start reproducing, cloning was absolutely out of the question. Today we are billions, the most successful complex species ever, some cloning would actually be beneficial, for the far future, if we colonize the solar system with hundreds of billions, or if we setup a galactic empire of quadrillions, the level of cloning that becomes beneficial increases proportionally(not 100% of course).

2.3 Cell 0.01

2.3.1 "copying"

left: Drexler-Merkle Diferential Gear, right: MarkIII(k) Planetary Gear a molecular gear 	an other molecular gear

"When one tries continuously, one ends up succeeding. Thus, the more one fails, the more one has chances that it works."

The first generation of genome/crystals, are only polymerizing at random, but they get progressively refined in what they do, in some unspecified way, anything, no mater how small, it will get picked up by natural selection. The main goal it's to increase absorption of H2O, the most efficiently with the small molecules that enter the vesicle, but without blowing up. The least efficient way to create H2O absorbing molecules is to clog all the little molecules in to a dense and impenetrable perfectly spherical ball. Realistically a light sponge-like structure is the most efficient way. Tendency to polymerize in filaments(crystal surface with groves? comb-like? tunnels? spikes?) with not too much branching, is a first amelioration(avoid compacting), hence naturally selected. Fossilized evidence of this is modern proteins, because of the weight of the past, they retained a sponge-like structure, rather open to H2O and not say compact and gear-like. An alternative mechanism, is the production of second order polymers, group of crystals(or just 1) are 1 purpose, hard build, to generate just one polymer, that has the actual catalytic ability, because of the higher quality of polymers over crystals. We can imagine an evolved hard-built multilevel system of crystals and polymers producing just one catalytically useful polymer. To simplify the explanations we consider here that the crystals are indirectly catalyzing. Filaments are beater, but there sequences are at random, certain sequences are more equal then others in H2O absorption (without blowing up). In halve the vesicles, in a given population, these sequences would be above average, any catalyst that can get somehow faintly inspired by what ever floats around, instead of doing original work every time, will be a further improvement for this half of the population at the expense of the other half. This is a fractionairy genetic transfer, from the crystals only, to the mixture crystals and polymer sequences. I assume a linear no-threshold model of catalysis, theres no threshold of efficiency bellow which they are no catalysts at all.

=(^.^)=

This inspiration, can have extremely diverse mechanisms, but because this is a "replication" of extremely low fidelity(barely above random, modern DNA replication error rate 1/10⁹), no 1 bothered to study it, so i'm attempting to make some educated guesses. Before anything else, there's going to be a selection of copyability. The lowest of all lows, is to influence the proportion of monomers composing the polymers. More serious business, as the polymers float around and fold in various ways randomly, they get the chance, eventually to present all of them selves to the surface of the crystals, including parts that on average are hidden. With this information, the crystals, could produce small pieces that resemble sections of the polymers, with some luck, as they polymerize between them, the end result will have some family resemblance to the parent polymer. Alternately, negative parts of polymer can be produced, that in turn are used to produce positive pieces. The mixture of positive and negative chunks, as they overlap, help the polymerization of longer chunks of correct sequence. Related to copying, its reading, a certain category of polymer fibers tend to be "copied", but useless on there own. These are "read", very imperfectly, according to some arbitrary rules, and a polymer of an other category is produced. This last 1 is actually useful, "reading" has the extra overhead of a second copying, but the first polymer can now be more suited for copying, and the second for catalytic action. Other way, is through indirect catalysis, the crystals are 1 purpose, hard build, to generate specific copy-able polymers. Of course these systems are very restricted, they work beater for certain compounds, on average the catalysis is closer to copying, while for the rest the average is closer to random. They are probably certain similarities with, not understood yet, prion replication mechanisms. If the filaments can do something more to help, then just siting there and sucking H2O it would be nice, maybe some kind of catalysis, or something structural or some assistance to the crystals or by cooperating with each other or some kind of cooperation between encapsulated vesicles or simply avoiding toxic products (example: avoid spiky molecules that can tear the membrane) or what ever that can help, no mater how small, it's probable, that any functionality will outweigh quite easily the base function. Other then copied filaments, they are also the indirect polymers produced by hard-build 1 purpose crystals. Crystals are rather not good as catalysts, they introduce a lot of mistakes, even there own replication is plagued with a bunch of mistakes. But mistakes at this stage in the replication of the filaments is very well tolerated, since it doesn't correspond with virtually 0 functionality as in modern cells, they are still useful for H2O absorption, there main function. There extra functionality, that at our stage, is really a bonus, doesn't need either to be always present, at the start they come and go intermittently, as they are produced randomly, but still the photocell is favored by natural selection. The crystals will tend to "copy" those functioning filaments in to something "resembling" them, photocells that contain the most filaments that "resemble" the functioning filaments, will also tend to produce the most functioning filaments, thus constituting an advantage,that will be favored by natural selection. Since theres a higher probability that a crystal produces some correct filaments by wrongly copying a huge number of resembling it, filaments, rather then by wrongly copying huge numbers of some completely random filaments, thus overcoming somewhat the low quality of the crystalline catalysis.

2.3.2 autocatalytic

monster eats his tail

you're fired monkey at a typewriter From these functioning catalytic polymers, it's not the 1s that do something new that are of great interest, but the 1s that do something old, in particular the catalytic function of the crystal that created them. Simply as, helper compounds, or out right doing the entire catalysis on there own. There addition in the mixture of crystals will lower the level of mistakes in polymers production ("copying"). In turn the quantity of this polymer will increase, since it catalyzes its own production, bringing the level of mistakes further down. Starting a positive feed back loop, incrementally reducing the level of mistakes up to a level. At some point, a second polymer appears, of greater still quality then the first because of the lowering of mistakes brought by the first polymer, its addition in the mixture of catalysts will bring the level of mistakes further down. Numerous successions of superior versions get added through this mechanism, coexisting with or replacing old 1s or with some complicated helpful interaction. The reduction of mistakes permits to increase the quantity, quality and diversity of polymers that have a bonus function, giving an edge to the protocell. At some point, the quality of the mixture of catalyzing polymers becomes so good, that it breaks even. It is barely above replacement rate and it no longer depends on the self replication capacity of the crystals, and they are dumped, but the mixture still depends on the lower quality but easier to copy polymers. The first such "self replicating" polymers or group of polymers, would still "copy" them selves in a very erratic way, producing a gigantic number of failures, from the point of view of copying, but good enough from the point of view of H2O absorption.

some alternates of nucleic acids The group of polymers will continue to evolve higher quality polymers, that in turn will permit higher still quality of polymers in a succession of generations. Complexity increase incrementally, with the succession of multiple generations of replicators: increases in size, utilization of other monomers that need more precise handling(that the previous generation couldn't achieve) more complex cooperation among the polymers, production de novo of more and more compounds, and what ever else can help, no mater how small. After an unspecified number of biological generations, we arrive at some alternate simple nucleic acid, that is used somehow, probably catalytically in parallel with other creatures. Evolving gradually, to more and more complex nucleic acids, until we finally arrive to the RNA world, from here what happens next is far less speculative. Examples of alternate nucleic acids: GNA, GmNA, PNA, TNA, p-RNA; with alternate bases, huge number of combinations, PNA seems particularly interesting because it resembles somewhat a protein. RNA have in it self enzymatic properties, acting like proteins, the ribosomes, are mainly RNA, with some proteins attached, presumably, ribosomes at first where RNA only(and precursors?), hinting that proteins came after RNA. DNA can not be a precursor of anything, DNA is derived from RNA, the 2 are almost identical, the tiny changes gives to DNA greater rigidity, thus being a more reliable longterm carrier of information, while RNA is flexible and can fold in complex 3D structures(uridine turn). Most probably the protein world began, with an enzyme(RNA?) attaching an amino acid on a proto-t-RNA, later a proto-ribosome transfered an amino acid from 1 proto-t-RNA on the amino acid of an other proto-t-RNA, crudely building a first tiny protein(2 amino acids). A first arbitrary proto-m-RNA precursor(imagine something rather folded) was added, so that a different type of proto-t-RNA can attach at 1 of the binding sites(a trace in modern translation code?), producing a micro-protein slightly longer. Proto-m-RNA, expanded with the ability to bind a second proto-t-RNA, giving it higher probability that it will be used because of its spacial proximity, producing a random chain, but with certain preferred proportions in amino acides. As proto-m-RNA expanded more and more to bind more and more proto-t-RNA, the proto-ribosome got refined in to processing the proto-m-RNA. For example adopting a ring configuration around the proto-m-RNA(can we check that on modern ribosomes?) would aloud it at first to translate preferably from the same proto-m-RNA, but in a random back and forth reading(with misses). A second ring around the forming amino acid chain would increase the chance that it will be adding on the same chain.The addition of a ratchet mechanism(like backwards looking hairs) will allow a 1 way reading, but still with misses. Proteins got gradually longer and more complex as errors went down. The whole process got refined to todays mechanism, spelling errors got way down and in particular m-RNA precursor got simplified and straighten up.

2.3.3 protocel cooperation left: messy vesicles, center: a cell, right:a cell, the theory multivesicular vesicles 	cell electron micropgraph 	cell diagram More macroscopically, a detail of vesicle configuration is potentially relevant, in general vesicles are rather messy, multiply encapsulating vesicles in various complex ways(hmm, splitting could still work for inner vesicles?). Initially cooperating among them, after heavy reorganizations, most of them would eventually be merged together. Functionalities would be moved around so that vesicles get very specialized and at the same time systems for coordination and transports would coevolve between them. For some the cooperation could have become entrenched, but still there genetic system would be completely absorbed by the ancestor of the nucleus(contains the DNA today). These would be the ancestors of several cellular organelles. And finally for some, the inner genetic system them selves would be entrenched(for regulatory reasons). This would be an alternative explanation to the theory of the origin of mitohondria and chloroplasts(a bacteria got domesticated like a cow by a bigger cell inside it), but without precluding this kind of event from happening. Following this thinking, the bacteria would have evolved from these early inner vesicles. Giving an explanation why the lineage between archea(like bacteria, but different), bacteria and eucariotes become blurred very early in there evolution(apart because its only 3 billion year ago). They would exchange gene between them, more intensely then if they where always separate as the official explanation. The dates given for the formation of eucariotes would simply be the time of the formation of the nucleus. The bacteria/archea would be too miniaturized(read advance) and less fault tolerant, then bulkier cells, for natural selection to start from scratch with them(because of the "copying"). Once the biological technology reaches a certain threshold, some inner vesicles are more allowed to become outside symbionts and eventually independent. Above a certain threshold, the components have become too integrated to be able to separate. The bacteria and archea lineages, would correspond to 2 distinct populations of "organelles" that became vagabonds. Perhaps 2 picks of opportunity for independence, 1 at the start of the window(minimum technology, low entrenchment) giving the more versatile archea, and a second at the end(more advanced and entrenchment) giving the bacteria. Viruses would have emerged at a third stage(even more advanced and entrenched), a virus is so minimalistic that it needs good quality industry to usurp for it self, but at this point entrenchment is so great, that only a very simple organelle has a reasonable chance to become independent. Theres enough holes in the raw data to accommodate this upside down theory. Other proposed theories its that the independent bacteria/archea merged in various ways to set up the eucariotes, with the addition of my theory all the possibilities about who get inside of what are covered, at least 1 has to be true. All these theories have in common that they see eucariotes as a kind of extremely centralized bacterial colony.

2.3.4 paleontology pink microscope Apart from trying to replicate the process in a lab, there is some hope that we can extract archaeological evidence. Even, maybe they still can form, or found refuge, in weird special places, and we simply didn't looked for them. The organic parts, of the protocells, have decayed to oblivion eons ago, but there's a chance that there gene-crystals, didn't. It should be feasible to recover a fossilized signature of some kind. Say if we find significant quantities, with a shape/defects distribution that strangely deviates from normal, packed in a certain geological layer billions of years old. I'm not aware that any systematic study of this kind, on minerals, is done at all. We would have access to a sequence of fossils extremely early in evolution, and some study on there catalytic properties would give hint on the chemistry of there environment, and from that we could extrapolate on potential decedents. Specificities from todays cells, like: the genetic code (triplet synonyms, triplet amino acid correlation, start/end sequences), the amino acid in use, ribosome structure, the sugars in use, lipids in use, etc ; could be used to bridge the gap between the crystal era and the RNA era. Of course, if the cool mars theory is true, we should find no crystal-fossils at all on earth, and we would have to wait for a Martian colony to settle the issue :(.

2.4 All work and no play, make johnny a dull boy

phases bricabrac H2O H2 C

Fe Bi UF6

We should be happy that spontaneous assemblage of life, isn't too improbable, like requiring the entire universe so that theres a reasonable probability to found just our selves. How would you prove anything if that was the case? In theory at least, it should be possible to build a working protocell in a lab today, with some figling of this basic blue print. It seems reasonable to assume that there are more then 1 recipes, and once you know the recipe it should be reproducible quite easily. Having a model of a protocel would be nice, but it would be far cooler if we tried with more exotic recipes, using other liquids (dihydrogen monoxide, alcohols, oils, Hg, melted metals, H2O2, liquid He, molten glass, H2SO4, NH3, plasma, what ever), with the appropriate surfactant molecules ,molecules with a liquid-phobic tail and a liquid-phile head, with crystals and other molecules that dissolve willingly in that liquid, all that at some unspecified temperature and pressure. For example, alcohol and NH3 should be straight forward, an inverted vesicle in oil with some organic crystal, for liquid metals maybe some sort of ceramic particles. Apart for H2O, knowledge for other liquids is almost completely inexistent. All money is spend on carbon based molecules, that work in H2O, or other organic liquids. Why would anyone spend money in to surfactants that work in liquid He? No need to be C chauvinists, if you play with temperature and pressure, a lot of atoms should adopt carbon like properties, capable of forming long chains. Under high enough pressure H2 becomes metallic, while we stay alive inside a very narrow window of pressure and temperature; these examples are just to show that the possibilities are almost infinite. For example the other atoms of the C group in the periodic table, have the possibility to have 4 bonds, for the N2 group atoms it's 3 bonds, but all are less stable then carbon INSIDE OUR WINDOW OF SURVIVABILITY. Naively, by changing the temperature and pressure, exotic complex compounds should be more usable. Compounds usable at very low/high temperatures, should look extremely fragile/rigid at our temperatures. In general they should exhibit complex behavior inside some narrow window of survivability(like C), the least exotic would be Si. Alternatives to O2, it could be simply electricity given by us, chemical alternative should be very reactive molecules, for example halogens, like Cl.

pop out and full size continuous artificial evolution of bacteria since 1988(21 years as i wright this). A total of 44.000 generations, from generation 31.500, they unexpectedly evolved in eating there citrate substrate your browser does not support the video tag, install

A really exotic biology would be dust particles suspended in plasma. Say, the use of big enough, specially manufactured dust particles, that act as "molecules" but can be seen in the naked eye or a simple magnifier, in real time, in room temperature. Amphiphilic dust particles, with a very inert chain, and a easily excitable head, with the correct geometry. Other "molecules" with carefully manufactured shape and certain of it surfaces, either inert or excitable or magnetized so that they stick together and simulate polymers and catalysts.

home of plasma space amoebas? a nebula Of course they are interesting theoretical ramifications about extraterrestrial life forms, of a different biology, at protocell or advanced state. From liquid He plutonians to plasma space amoebas. But they would also be interesting as nanomachines working for us, for impossible tasks to natural living cell, with some accelerated artificial evolution. For example Molten-metal-cells(metaloids) for the metallurgic industry, H2SO4-cells(sulfuroids) for the chemical industry, liquid-N2-cells(nitrogenoids) for the cryogenic industry, liquid-glass-cells for the glass industry(glassoids?), etc. Maybe, even primitive protocells, would permit the creation of weird materials, or the concept could be used for some cheaper synthesis, or for new material research(looking at the products).

Practically: the exotic cells, could be "frozen" or "frozen dried", for easy storage and manipulation. No need to keep them active all the time. For example, metaloids should be revivable after a deep "freeze" at room temperature, corresponding to a virtual -3000°C freeze of normal cells. Stored simply as ingots or pebbles on a shelf, ready for melting. Nitrogenoids could be "freeze dried" of there N2 and stored as powders at room temperature, just need to add liquid N2 to revive them. They could also be 3D printed in complex "organs", as active or as frozen/frozen-dried, instead of use in simple bioreactors(we shouldn't be doing that with normal cells?).

They could be used for more plausible terraforming, a succession of cell biologies, that incrementally come closer and closer to earth standards, this way we could start even with an irradiated dead rock bringing it the state of a lush jungle.

zerg evolution factory

This accelerated artificial evolution, of course will cheat a lot, she will have a lot of input from us(we don't want to wait millions of years). We could do this with evolution factories at an industrial scale(read, absolutely huuuuge), not just in a lab by hand on petri dishes, as pore Zachary. Something like several times an Olympic sized swimming pool, extracting periodically samples and freezing them, studying wants going on, and injecting "GMO" protocells with some engineered by hand "genes". The handcrafted "genes", will survive or not, survivors will evolve on there own, globally evolution will be speeded up substantially. A more automatic process would be to extract some protocells, pass them through harsh mutating environment, and then reinject them in the main pool. Continuous more smart automated machine selection of individuals. Periodically, kill all and reseed with a selected individual. Alternatively, gradually modifying the environment in some smart and complex way, could induce certain desired evolution. Materials that are considered as waste, could be used in the soup, it would be cheaper and maybe it would be possible in the future to "recycle/reuse" that waste material with the newly evolved cells. To reduce costs and increase speed, the evolution could be carried out on some extraterrestrial location, with an environment very close to the desired, while supplementing artificially what ever is missing. All the above, could be done with normal bacteria. Maybe we should start with them as a proof of concept experiment and for experience building before attempting with new biologies. I expect that early stages of evolution in protocells happen exponentially, the pressure to get beater is huge, and when you start from zero improvements are easy, after the initial stages evolution speed slows down considerably.

More down to earth, a related practical issue, is the experimental demonstration of the concept. Any initial experiment should target a potentially extremely cheap biology, at the exclusion of all other considerations. Theres no even the need to attempt to use natural plausible materials. We can try to simulate the primordial soup with something like raw petrol or a simple commercially available monomer, use some kind of commercial soap or detergent(shampoos, soap bars) as amphiphile and some kind of commercial clay, Montmorillonite seams to be popular. All that in an agitator, in the kitchen, hopefully sterilization will not be needed and we could detect something under an inexpensive microscope. Hopefully as easy and cheap as cooking.

Using half arsed chemistry for a first cheap and sloppy experiment is budgetary good, but we can get even more cheap and sloppy then that. I'm over-stretching the concept of biology and experiment, but an abstract non-realistic computer simulation should illustrate the concept even more easily and cheaply. We are forced to use an abstract chemistry, otherwise, the problem with realistic chemistry rules, is not tractable. The use of a simple desktop computer(my nice quad core :D ) would give a moderately impressive result, because the programmer could anticipate the results. The use of a system like BOINC, should permit the exploration of a sufficiently complex abstract chemistry, with out permitting to the programmer to anticipate the results and being accused of framing the question, while remaining computationally tractable.

Of course nothing stop us from using supercomputers and simulate arbitrary chemistries of serious complexities. Why restrict at 3D? You want 5D? why not 13D? Lets use element n°999 instead of C(n°6). Why restrict our selves at 118 elements? lets use 1189. What? they can only be 173 elements? no problem, just increase the speed of light and your fine. You find that electron mass and charge are ugly? Lets put 7 on both....

In any case, it will take us, a considerable amount of time, before we exhaust all possibilities.

title: User:Super Quantum immortal/Abiogenesis


text:
User:Egm4313.s12.team4.blj

Brittany Jacobs. Team 4

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Egm4313.s12.team4.blj


text:
User:Egm4313.s12.team13.r1

=Report 1=

Problem R1

Derive the equation of the motion of a spring dashpot system in parallel with a mass and applied force $$ f(t)\,\!$$

First we examine the figure, and declare constants. $$ y \,\!$$ is the distance the coil expands, and k is the spring constant. Therefore, we are able to define a force $$ F_1\,\!$$ as

(1.0) $$ F_1= -ky\,\!$$.

The force is negative because in this model, the up position is negative and the down position is positive. This is a force opposing the motion of the mass, thus it is negative. Thereafter, we know Newton's second law as

$$ Force= mass* acceleration \,\!$$, or otherwise known as

$$ F=m\,\!$$$$ \frac{dV}{dt}\,\!$$

From calculus, it is also known that $$ \frac{dy}{dx}\,\!$$ can also be written as $$ y',\!$$. Furthermore, acceleration is known as

$$ \frac{dV}{dt}\,\!$$ $$ =\!$$ $$ \frac{d^2y}{dx^2}\,\!$$ $$ = \,\!$$ $$ y'',\!$$ (1.1)

If we sum the forces and equal them to zero we ca write Newtons second law for this system as shown below

$$ \sum{F}\!$$ $$ = \!$$$$ m\,\!$$$$ \frac{dV}{dt}\,\!$$ (1.2)

$$ my''+ky=0 \!$$ (Notice it is important to have all terms on the same side of the)

It is important to note, this force only includes the restoring force of the spring. It is now necessary to include the damper term from the dashpot. The damping constant will be defined as $$ C\!$$. It is also assumed the damping force is proportional to the velocity (Kreyszig, 2011). Therefore, we are able to write

$$ F_2= -c y',\!$$ (1.3)

and by summing forces one more time:

$$ \sum{F}\! = $$ $$ \frac{dV}{dt}\,\!$$ $$ \longrightarrow\!$$ $$F_1 + F_2=my \!$$ $$ \longrightarrow,\!$$$$ my+ cy' +ky =0 \!$$ (1.5)

To get the differential equation in standard form, the only thing required is to divide the equation by the mass.

Solution
$$ y''+ \frac{c}{m}y' +\frac{c}{m}y =0 \!$$

Since this is a parallel system, we are able to say the spring force relation and the damper force relation are equal to each other, which is the reason we don't differ from the $$ y's,\!$$ with subscripts.

Picture obtained from here Section 1 Lecture Notes

Problem R1.2
Derive the equation of motion of the spring-mass-dashpot in fig. 5.3 in K 2011 p 85 with an applied force r(t) on the ball.

In order to derive the equation of motion of the spring-mass-dashpot with an applied force of $$ \displaystyle r(t)$$, the formula of a non-homogenous equation must first be understood.

In the non homogenous equation:
 * {| style="width:100%" border="0"

$$  \displaystyle y'' + p(x)y' + q(x)y = r(x) $$     (2.0) The $$\displaystyle y $$ represents position, $$ \displaystyle y' $$ represents velocity, and $$ \displaystyle y'' $$ represents acceleration.
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Figure 5.3 consist of a mass $$\displaystyle m$$, a spring constant $$\displaystyle c$$, and a damper coefficent $$\displaystyle k$$ These components create 3 internal forces- intertia $$\displaystyle my''$$, damping force $$\displaystyle cy'$$, and restoring force $$\displaystyle ky$$. Along with these 3 forces is also a driving force of $$ \displaystyle r(t)$$.

Solution
Using these 4 components, equation 1 becomes:
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$$\displaystyle my'' + cy' + ky = r(t)$$
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Section 1 Lecture Notes

Problem Statement
For the spring-dashpot-mass system on p.1-4 (Section 1-4 Notes) draw the FBDs and derive the equation of motion.

A free body diagram of the spring-mass-dashpot found in Sec 1 of the notes must be broken up into three different parts. The spring, the dashpot, and the mass.

The positive y direction equals $$ \overrightarrow{y+} \! $$

The spring's free body diagram is: $$ F_k \longleftarrow\! $$ $$ \longrightarrow\! F_k $$

The dashpot's free body diagram is: $$ F_k \longleftarrow\! $$ $$ \longrightarrow\! F_c $$

The free body diagram of the mass is: $$ F_c \longleftarrow\! $$ $$ \longrightarrow\! F(t) $$

Since there are only two opposing forces on the dashpot those two forces must be equal. :{| style="width:100%" border="0" $$  \displaystyle F_c = F_k = F_i $$     (3.0)
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By definition
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$$  \displaystyle ma=my'' $$     (3.1)
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Using Newtons second law :
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$$  \displaystyle \sum F = ma $$ (3.2)
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using substitution and equations (3.2), (3.1), and (3.0) we get
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$$  \displaystyle \sum F_y = F_(t) - F_i = my'' $$     (3.3)
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Rearranging equation (3.3) gives us the solution

Solution
$$ \displaystyle F(t) = F_i + my''$$

Section 1-4 Notes

Problem Statement
Using the circuit given, the governing circuit equation (4.0), and the capacitance equation (4.1), derive alternate equations (4.2) and (4.3).

Circuit Equation

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$$  \displaystyle V=LC\frac{{d}^{2}{v}_{c}}{d{t}^{2}}+RC\frac{d{v}_{c}}{dt}+{v}_{c} $$     (4.0)
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Capacitance Equation

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$$  \displaystyle Q = C{v}_{c} \rightarrow \int idt = C{v}_{c} \rightarrow i=C\frac{d{v}_{c}}{dt} $$     (4.1)
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Alternate Equation (4.2),

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$$  \displaystyle LCI''+RCI'+\frac{1}{C}I = V' $$ (4.2)
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Alternate Equation (4.3),

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$$  \displaystyle LCQ''+RCQ'+\frac{1}{C}Q = V $$ (4.3)
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Solution
First take the derivative of the governing circuit equation (4.0) to get

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$$  \displaystyle V' = LC\frac{{d}^{3}v_{c}}{d{t}^{3}}+RC\frac{{d}^{2}v_{c}}{d{t}^{2}}+\frac{{d}^{}v_{c}}{d{t}^{}} $$     (4.4)
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Since L, R, and C are constants with respect to time

Then, since

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$$  \displaystyle I = C\frac{dv_{c}}{dt} $$     (4.1)
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that means

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$$  \displaystyle I' = C\frac{{d}^{2}v_{c}}{d{t}^{2}} $$     (4.5)
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And

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$$  \displaystyle I'' = C\frac{{d}^{3}v_{c}}{d{t}^{3}} $$     (4.6)
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 *

 * }

Substituting these into the derivative of the circuit equation (4.4) gives the first alternate equation (4.2)

$$  \displaystyle V' = LCI''+RCI'+\frac{1}{C}I $$

Then, since

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$$  \displaystyle Q=Cv_{c} $$     (4.1)
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 * }

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$$  \displaystyle Q' = C\frac{d{v}_{c}}{dt} $$     (4.7)
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 *

 * }

 * {| style="width:100%" border="0"

$$  \displaystyle Q'' = C\frac{{d}^{2}{v}_{c}}{d{t}^{2}} $$     (4.8)
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 *

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Now substituting these into the appropriate places in the circuit equation (4.0) gives the second alternate equation (4.3)

$$ LQ''+RQ'+\frac{1}{C}Q = V $$

IEA notes section 2

Problem Statement
4.) Find the general solution to the following ODE and check the result by substitution.

$$ y'' + 4y' + (\pi^2 + 4)y = 0\,\!$$

Solution
The homogeneous characteristic equation for linear 2nd order ODE's with constant coefficients is

$$ \lambda^2 + a\lambda + b = 0\,\!$$
(5.1a)

For the ODE in this problem, the characteristic equation becomes

$$ \lambda^2 + 4\lambda + (\pi^2 + 4) = 0\,\!$$

To solve for the solutions, the discriminant to the quadratic equation must first be calculated.

$$ \Delta = a^2 - 4b\,\!$$
(5.2a)

$$ \Delta = 16 - 4(\pi^2 + 4)\,\!$$ $$ = 16 - 4\pi^2 - 16\,\!$$ $$ = -4\pi^2\,\!$$

Since the discriminant is negative, there will be two imaginary solutions to the ODE. The solutions can be obtained by solving the remainder of the quadratic formula.

$$ \lambda_{1,2} = \frac{-a \pm \sqrt{a^2 - 4b}}{2}\,\!$$
(5.3a)

$$ \lambda_{1,2} = \frac{-4 \pm \sqrt{-4\pi^2}}{2}\,\!$$ $$ = \frac{-4 \pm 2\pi i}{2}\,\!$$ $$ = -2 \pm \pi i\,\!$$

Therefore,

$$ \lambda_{1} = -2 + \pi i \,\!$$
(5.4a)

$$ \lambda_{2} = -2 - \pi i \,\!$$
(5.5a)

The two distinct, linearly independent, homogeneous solutions for the case with two imaginary roots are

$$ y_{h,1}(x) = e^{a_1 x} \cos b_1 x\,\!$$
(5.6a)

$$ y_{h,2}(x) = e^{a_2 x} \sin b_2 x\,\!$$
(5.7a)

The homogeneous solution is

$$ y_{h} = c_1 y_{h,1} + c_2 y_{h,2}\,\!$$
(5.8a)

$$ y_{h}(x) = c_1 e^{a_1 x} \cos b_1 x + c_2 e^{a_2 x} \sin b_2 x\,\!$$

$$ y_{h}(x) = c_1 e^{-2 x} \cos \pi x + c_2 e^{-2 x} \sin \pi x\,\!$$

The final general solution is therefore

$$ y_{h}(x) = e^{-2 x}(c_1 \cos \pi x + c_2 \sin \pi x)\,\!$$
(5.9a)

To check this result using substitution, we must take the first and second derivatives of the general solution to substitute back into the original ODE.

$$ y'(x) = -2e^{-2x}(c_1 \cos \pi x + c_2 \sin \pi x) + e^{-2x}(-\pi c_1 \sin \pi x + \pi c_2 \cos \pi x)\,\!$$
(5.10a)

$$ y''(x) = 4e^{-2x}(c_1 \cos \pi x + c_2 \sin \pi x) - 2e^{-2x}(-\pi c_1 \sin \pi x + \pi c_2 \cos \pi x) - 2e^{-2x}(-\pi c_1 \sin \pi x + \pi c_2 \cos \pi x) + e^{-2x}(- \pi^{2} c_1 \cos \pi x - \pi^{2} c_2 \sin \pi x)\,\!$$
(5.11a)

$$ = 4e^{-2x}(c_1 \cos \pi x + c_2 \sin \pi x) - 4e^{-2x}(-\pi c_1 \sin \pi x + \pi c_2 \cos \pi x) + e^{-2x}(- \pi^{2} c_1 \cos \pi x - \pi^{2} c_2 \sin \pi x)\,\!$$

Substituting back into the original ODE gives

$$ y'' + 4y' + (\pi^2 + 4)y = 0\,\!$$
(5.12a)

$$ 4e^{-2x}(c_1 \cos \pi x + c_2 \sin \pi x) - \cancelto{0}{4e^{-2x}(-\pi c_1 \sin \pi x + \pi c_2 \cos \pi x)} + e^{-2x}(- \pi^{2} c_1 \cos \pi x - \pi^{2} c_2 \sin \pi x) - 8e^{-2x}(c_1 \cos \pi x + c_2 \sin \pi x) + \cancelto{0}{4e^{-2x}(-\pi c_1 \sin \pi x + \pi c_2 \cos \pi x)}$$

$$ + (\pi^{2} + 4)(e^{-2x})(c_1 \cos \pi x + c_2 \sin \pi x) = 0\,\!$$

$$ \cancelto{0}{-4e^{-2x}(c_1 \cos \pi x + c_2 \sin \pi x)} + \cancelto{0}{e^{-2x}(- \pi^{2} c_1 \cos \pi x - \pi^{2} c_2 \sin \pi x)} +

\cancelto{0}{(\pi^{2} + 4)(e^{-2x})(c_1 \cos \pi x + c_2 \sin \pi x)} = 0\,\!$$

Since all terms cancel to 0, $$ y_(x) = e^{-2 x}(c_1 \cos \pi x + c_2 \sin \pi x)\,\!$$ is a general solution to the original ODE.

Section 2 Lecture Notes

Problem Statement
5. Find a general solution. Check by substitution.

$$  \; {y}''+2\pi {y}'+\pi ^{2}y=0 $$

Solution
$$  \; {y}''+2\pi {y}'+\pi ^{2}y= \frac{d^2y}{dx^2}\ + 2 \pi \frac{dy}{dt}\ + \pi ^{2}y=0   $$
(5.0b)

If $$ \; \frac{d}{dx}\ = \lambda $$,

The homogeneous linear ODE with constant coefficients becomes:

$$ \; \lambda ^{2}+2\pi \lambda +\pi ^{2}=0 $$
(5.1b)

By using the quadratic formula, or in this case factoring, we find the solutions of the equation:

Solving for $$ \; \lambda $$,

$$ \; (\lambda +\pi) (\lambda +\pi)=0 $$
(5.2b)

$$ \; \lambda =-\pi, -\pi $$

General Solution:

General solution in the case of distinct real roots:

$$  \; y=c_1e ^ { \lambda _1 x} + c_2xe ^ { \lambda _2 x}  $$
(5.3b)

So the general solution becomes:

$$  \; y=c_1e ^ {-\pi x} + c_2xe ^ {-\pi x}= e ^ {-\pi x} (c_1+c_2x) $$
(5.4b)

Check by Substitution:

To check by substitution, you find $$  \; y, y', y'' $$ and plug into the original ODE.

$$  \; y= e ^ {-\pi x} (c_1+c_2x) $$

$$  \; y'= -\pi c_1e ^ {-\pi x} + c_2 (e ^ {-\pi x} -\pi xe ^ {-\pi x})= - \pi e ^{- \pi x} (c_1 + c_2x) + c_2 e^ {- \pi x} $$
(5.5b)

$$  \; y''= \pi ^{2} c_1e ^ {- \pi x} - \pi c_2e ^ {-\pi x} -\pi c_2(e ^ {- \pi x} - \pi xe^ {- \pi x} )= \pi ^ {2} e^ {- \pi x} (c_1+ c_2x) -2 \pi c_2 e^ {- \pi x} $$
(5.6b)

Plugging in  $$   \; y, y', y $$ into $$   \; {y}+2\pi {y}'+\pi ^{2}y=0 $$ :

$$  \; [ \pi ^ {2} e^ {- \pi x} (c_1 + c_2x) - 2 \pi c_2e^ {- \pi x}] + 2 \pi [ - \pi e^ {- \pi x} (c_1 + c_2x) + c_2e^ {- \pi x} ] + \pi ^ {2} [ e^ {- \pi x} (c_1+c_2x)]=0 $$

When plugging in, the general solution is confirmed because it equals zero.

Section 2 Notes

Problem Statement
For each ODE in Fig. 2 in K 2011 p.3 (except the last one involving a system of 2 ODEs), determine the order, linearity (or lack of), and show whether the principle of superposition can be applied.

Background Information
Superposition Principle

An ordinary differential equation in standard form:

$$ y''+p(x)y'+q(x)y=r(x)\!$$

The homogeneous solution of the function is in the form:

$$ y_h''+p(x)y_h'+q(x)y_h=0 \!$$

The particular solution of the function is in the form:

$$ y_p''+p(x)y_p'+q(x)y_p=r(x) \!$$

The function, $$ \bar y(x) \!$$, is the sum of the homogeneous solution, $$ y_h(x)\!$$ and the particular solution $$y_p(x)\!$$:

$$ \bar y(x) = y_h(x)+y_p(x) \!$$

To prove the superposition principle, we must prove that $$ \bar y \!$$ is a solution. We then plug in $$ \bar y \!$$ into the standard form of the ordinary differential equation:

$$ (y_h+y_p)+p(x)(y_h'+y_p')+q(x)(y_h+y_p)=r(x) \!$$

If the function is linear:

$$ y_h'+y_p'=\bar y' \!$$ and $$ y_h+y_p=\bar y'' \!$$

If the superposition principle holds, the function should yield:

$$ \bar y''+p(x)\bar y'+q(x)\bar y=r(x) \!$$

Falling Stone Problem
$$ y''=g=constant \!$$

Order: 2nd  Linearity: Linear

The homogeneous equation is:

$$ y_h''=0\!$$
 (6.1)

The particular equation is:

$$ y_p''=g \!$$
 (6.2)

Adding equations 6.1 and 6.2 yields:

$$ y_h+y_p=g \!$$
 (6.3)

Simplifying equation 6.3:

$$ (y_h+y_p)''=g \!$$
 (6.4) Which equals:

$$ \bar y''=g \!$$
 (6.5)

Superposition principle can be applied

Parachutist Problem
$$ mv'=mg-bv^2 \!$$

Order:1st Linearity Non-linear

The homogeneous equation is:

$$ mv_h'+bv_h^2=0 \!$$
 (6.6)

The particular equation is:

$$ mv_p'+bv_p^2=mg \!$$
 (6.7)

Add equations 6.6 and 6.7:

$$ (mv_h'+mv_p')+(bv_h^2+bv_p^2)=mg \!$$
(6.8)

Simplify equation 6.8:

$$ m(v_h+v_p)'+b(v_h^2+v_p^2)=mg \!$$
 (6.9)

Which equals: $$ m(\bar v)'+b(v_h^2+v_p^2)=mg \!$$
 (6.10)

Since $$ b(v_h^2+v_p^2)\ne b(v_h+v_p)^2\!$$ the superposition principle can not be applied.

Water Level Problem
$$ h'=-k\sqrt{h} \!$$

Order:1st  Linearity: Non-linear

The homogeneous equation is:

$$ h_h'+k\sqrt{h_h}=0 \!$$
 (6.11)

The particular equation is:

$$ h_p'+k\sqrt{h_p}=0 \!$$
 (6.12)

Adding equations 6.11 and 6.12 yields:

$$ (h_p'+h_h')+(k\sqrt{h_p}+k\sqrt{h_h})=0 \!$$
 (6.13)

Since $$ k\sqrt{h_h}+k\sqrt{h_p}\ne k\sqrt{h_h+h_p} \!$$ the superposition principle can not be applied

Vibrating Mass on a Spring Problem
$$ my''+ky=0 \!$$

Order: 2nd  Linearity: Linear

The homogeneous equation is:

$$ my_h''+ky_h=0 \!$$
 (6.14)

The particular equation is:

$$ my_p''+ky_p=0 \!$$
 (6.15)

Adding equations 6.14 and 6.15 yields:

$$ m(y_h+y_p)+k(y_p+y_h)=0 \!$$
 (6.16)

Simplifying the equation 6.16:

$$ m(y_h+y_p)''+k(y_p+y_h)=0 \!$$
 (6.17)

Which equals:

$$ m(\bar y'')+k(\bar y)=0 \!$$
 (6.18)

Superposition principle can be applied.

Beats of a Vibrating System Problem
$$ y''+w_0y=\!$$$$ \cos(wt) \!$$

Order: 2nd  Linearity: Linear

The homogeneous equation is:

$$ y_h''+w_0y_h=0\!$$
 (6.19)

The particular equation is:

$$ y_p''+w_0y_p=\!$$$$ \cos(wt) \!$$
 (6.20)

Adding equations 6.19 and 6.20 yields:

$$ (y_h+y_p)+w_0(y_h+y_p)=\!$$$$ \cos(wt) \!$$
 (6.21)

Simplifying equation 6.21:

$$ (y_h+y_p)''+w_0(y_h+y_p)=\!$$$$ \cos(wt) \!$$
 (6.22)

Which equals:

$$ (\bar y'')+w_0(\bar y)=\!$$$$ \cos(wt) \!$$
 (6.23)

The superposition principle can be applied

Current (I) in an RLC Circuit Problem

$$ LI''+RI'+ \frac{1}{C}I=E'\!$$

Order: 2nd  Linearity: Linear

The homogeneous equation is: $$ LI_h''+RI_h'+\frac{1}{C}I_h=0 \!$$
 (6.24)

The particular equation is: $$ LI_p''+RI_p'+\frac{1}{C}I_p=E' \!$$
 (6.25)

Adding equations 6.24 and 6.25 yields: $$ L(I_h+I_p)+R(I_h'+I_p')+\frac{1}{C}(I_h+I_p)=E' \!$$
 (6.26)

Simplifying equation 6.26:

$$ L(I_h+I_p)''+R(I_h+I_p)'+\frac{1}{C}(I_h+I_p)=E' \!$$
 (6.27)

Which equals:

$$ L(\bar I)''+R(\bar I)'+\frac{1}{C}(\bar I)=E' \!$$
 (6.28)

The superposition principle can be applied

Deformation of a Beam Problem

$$ EIy^{iv}=f(x)\!$$

Order: 4th  Linearity: Linear

The homogeneous equation is:

$$ EIy_h^{iv}=0 \!$$
 (6.29)

The particular equation is:

$$ EIy_p^{iv}=f(x) \!$$
 (6.30)

Adding equations 6.29 and 6.30 yields:

$$ EI(y_h^{iv}+y_p^{iv})=f(x) \!$$
 (6.31)

Simplifying equation 6.31:

$$ EI(\bar y^{iv})=f(x) \!$$
 (6.32)

The superposition principle can be applied.

Pendulum Problem

$$ L\theta''+g\sin(\theta)=0\!$$

Order: 2nd  Linearity: Non-linear

The homogeneous equation is:

$$ L\theta_h''+g\sin(\theta_h)=0 \!$$
 (6.33)

The particular equation is:

$$ L\theta_p''+g\sin(\theta_p)=0 \!$$
 (6.34)

Adding equations 6.33 and 6.34 yields:

$$ L(\theta_h+\theta_p)+g(\sin(\theta_h)+\sin(\theta_p))=0 \!$$
 (6.35)

Simplifying equation 6.35: $$ L(\theta_h+\theta_p)''+g(\sin(\theta_h)+\sin(\theta_p))=0 \!$$
 (6.36)

Since $$\sin(\theta_h)+\sin(\theta_p) \ne \sin(\theta_h+\theta_p) \!$$ the superposition principle does not apply.

Reference:

Section 2 Lecture Notes Photos taken from Kreyszig, Erwin. Advanced Engineering Mathematics. 10th. Hoboken: John Wiley & Sons, Inc., 2011. 3. eBook.

= Contributing Members =

Table code referenced from Team 1

title: User:Egm4313.s12.team13.r1


text:
User:Super Quantum immortal/Quantum immortality

(fix the formating latter)

Firs read Quantum suicide and immortality

The official quantum immortality say's that a sentient experimenter will always survive with some non zero probability. Hence he is immortal. You jump in a volcano, you survive, it doesn't try to look more then that.

According to me, this "immortality", is very constrained, they are still rules to be followed. Its not a blank check to do anything he wants, like a spoiled child. Its very interesting to note how probabilities get deformed from normal. There is no certainty in results, its only the certainty you survive, it doesn't tell you in what state you survive though. Furthermore, the probabilities you perceive are not simply the possibilities you survive at an event. You perceive probabilities that maximize your long term survival( you don't perceive anything when you die). Your present, makes sense, only if you exist (deep) in the future to remember it.

So a quantum immortal, most probably(no certainty), he will see in his past, probabilities that maximize his present and future survival. Normally ridiculously improbable events are still ridiculously improbable to a quantum immortal. They can happen, but probably, his life will be saved by something more plausible. By this mechanism, some information, can leak back to him from his future, at his past and present. Since he knows he will survive in the long run, with the most plausible events physically available.

With an example: A naive assumption, would be that if you play in the russian roulette 100 times, you win all of them. First, theres the very high possibility you "survive" an attempt. Even if you arrange a russian roulette variant where you die for "sure". The probability, that the "sure" mechanism fails and you "survive" is higher then going to the end of 100 russian roulettes. On top of that, for your long term survival, getting the message that its ok to casually play 100 time the russian roulette is fine, will make you extremely careless, thus reducing the futures where you are alive. The futures where you are alive are more numerous with presents where you "survived", making you prudent. We could even imagine that the immediate probability of surviving unharmed the russian roulette, is lower then normal for the quantum immortal, because theres less of him in the future to recall surviving unharmed, by his lack of prudence. In other words, in the reference frame of a quantum immortal, there will be a bias towards "surviving" over wining a russian roulette. Its not just, the possibilities that he dies in an attempt, that are discounted. This bias, is a form of leak from the future.

Quantum immortality is not simply a variant of quantum suicide. Quantum suicide is about surviving nuclear explosions, surviving jumping in volcanoes, etc... In quantum immortality, past events reduce the probability that you will get nuclear warheads exploding right next to you or jump in volcanoes. The most probable way to survive the experience of nukes and volcanoes, is simply not to experience them.

A quantum immortal, will perceive the universe, as "trying" to "survive him", through the above mechanisms. The rob is, that the "universe", will do this in an extremely lazy manner. The lazy, slight improbability mechanism, exhibit a god like behavior, over the quantum immortal. The mechanism, works through butterfly effect like biases. Spread deep in to the past, if its 10.000 years ago, it needs shallower interventions, then if its 10 seconds ago. If its 1000 years ago, for an event today, it doesn't matter. The mechanism, is super ubber smart, but super ubber lazy. Over 1000s of years, the biases get amplified to the required levels. This way, it can afford to be as little interventionist as possible. Events, seem, to naturally unfold, since time imemorable, to deliver there packages, like they always were.


text:
User:Egm4313.s12.team11.vargas.aa

EGM4313 Spring 2012

Team 11

Report 2, Problem 2.6

Report 3, Problem 3.5

Report 4, Problem 4.3

title: User:Egm4313.s12.team11.vargas.aa


text:
User:Egm4313.s12.team9.tjl

=R2.3: Find a general solution for problems 3 and 4 (K2011 pg. 59)=

Problem 3
Given: $$ y'' + 6y' + 8.96y = 0 $$ $$ \lambda^{2} + 6\lambda + 8.96 = 0 $$ Using the Quadratic Equation: $$ a=1, b=6, c=8.96 $$ $$ \frac{-b\frac{+}{-}\sqrt{b^{2} - 4ac}}{2a} $$ $$ \frac{-6\frac{+}{-}\sqrt{6^{2} - 35.84}}{2}$$ $$ \frac{-6 + 0.4}{2}$$ $$ \frac{-6 - 0.4}{2}$$ Root 1: -2.8 Root 2: -3.2 General Solution: $$ y = c_1e^{-2.8x} + c_2e^{-3.2x} $$

Problem 4
Given: $$ y'' + 4y' + (pi^{2}+4)y = 0 $$ $$ \lambda^{2} + 4\lambda + 13.87 = 0 $$ Using the Quadratic Equation: $$ a=1, b=4, c=13.87 $$ $$ \frac{-b\frac{+}{-}\sqrt{b^{2} - 4ac}}{2a} $$ $$ \frac{-4\frac{+}{-}\sqrt{4^{2} - 55.48}}{2}$$ $$ \frac{-4 + 6.28i}{2}$$ $$ \frac{-4 - 6.28i}{2}$$ Root 1: -2 + 3.14i Root 2: -2 - 3.14i General Solution: $$ y = c_1e^{-2x + 3.14ix} + c_2e^{-2x-3.14ix} $$

=R2.7: Develop the MacLaurin Series for $$ e^{t}, cos(t), and sin(t) $$= $$ P_n(t) = f(a) + f'(a)(t-a) + \frac{f''(a)}{2!}(t-a)^{2} +...+\frac{f^{n}(a)}{n!}(t-a)^{n} $$

MacLaurin Series(Taylor Series at t=0) for $$ e^{t} $$
$$ f(t) = e^{t}.....f(0)=1 $$ $$ f'(t) = e^{t}.....f'(0)=1 $$ $$ f(t) = e^{t}.....f(0)=1 $$ $$ f(t) = e^{t}.....f(0)=1 $$ $$ f'(t) = e^{t}.....f'(0)=1 $$ $$ P_5(t) = 1 + t + \frac{t^{2}}{2!} + \frac{t^{3}}{3!} + \frac{t^{4}}{4!} $$

MacLaurin Series(Taylor Series at t=0) for $$ cos(t) $$
$$ f(t) = \cos{t}.....f(0)=1 $$ $$ f'(t) = -\sin{t}.....f'(0)=0 $$ $$ f(t) = -\cos{t}.....f(0)=-1 $$ $$ f(t) = \sin{t}.....f(0)=0 $$ $$ f'(t) = \cos{t}.....f'(0)=1 $$ $$ P_5(t) = 1 - \frac{t^{2}}{2!} + \frac{t^{4}}{4!} $$

MacLaurin Series(Taylor Series at t=0) for $$ sin(t) $$
$$ f(t) = \sin{t}.....f(0)=0 $$ $$ f'(t) = \cos{t}.....f'(0)=1 $$ $$ f(t) = -\sin{t}.....f(0)=0 $$ $$ f(t) = -\cos{t}.....f(0)=-1 $$ $$ f'(t) = \sin{t}.....f'(0)=0 $$ $$ P_5(t) = t - \frac{t^{3}}{3!} $$

title: User:Egm4313.s12.team9.tjl


text:
User talk:Egm4313.s12.team9.tjl

Hello Tristan

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MaintenanceBot (discuss • contribs) 03:43, 11 November 2013 (UTC)

title: User talk:Egm4313.s12.team9.tjl


text:
User:Egm4313.s12.team16.standish

hey

title: User:Egm4313.s12.team16.standish


text:
Sample image-ideograms dictionary

(fix the formating latter)

Idea from Future of language (User:Super Quantum immortal).

Purpose is to build a little demonstration image-ideograms dictionary. The words are like ideograms but they are full color images instead of usual monochromatic. With 1000 it would cover 90% of normal needs, with a sample of 100 it would cover about 40% of language needs.

from

title: Sample image-ideograms dictionary


text:
User:Egm4313.s12.team20.ibrahim

R1.4

title: User:Egm4313.s12.team20.ibrahim


text:
User:Egm4313.s12.team20.ibrahim/Report 1

Given
Circuit Equation:
 * {| style="width:100%" border="0"

$$ V = LC\frac{d^2v_{c}}{dt^2} + RC\frac{dv_{c}}{dt} + v_{c}$$
 * style="width:95%" |
 * (4.0)
 * }

Capacitance:
 * {| style="width:100%" border="0"|-

$$ Q=Cv_{c} \rightarrow \int idt=Cv_{c} \rightarrow i = C\frac{dv_{c}}{dt}$$
 * style="width:95%" |
 * (4.1)
 * }

Solution
Alternate Circuit Equation 1

Current Equation:
 * {| style="width:100%" border="0"

$$ I = C\frac{dv_{c}}{dt}$$
 * style="width:95%" |
 * (4.2)
 * }

Derive Equation 4.2:
 * {| style="width:100%" border="0"|-

$$ I' = C\frac{d^2v_{c}}{dt^2}$$
 * style="width:95%" |
 * (4.3)
 * }

Rearrange the Current Intergral from Equation 4.1:
 * {| style="width:100%" border="0"|-

$$ \int idt=Cv_{c} \rightarrow v_{c}=\frac{1}{C}\int idt $$
 * style="width:95%" |
 * (4.4)
 * }

Substitute Equations 4.2, 4.3 and 4.4 into Equation 4.0:
 * {| style="width:100%" border="0"|-

$$ V = LC\frac{d^2v_{c}}{dt^2} + RC\frac{dv_{c}}{dt} + v_{c}$$ $$ \Downarrow $$
 * style="width:95%" |
 * }
 * {| style="width:100%" border="0"|-
 * {| style="width:100%" border="0"|-

$$ V = LI'+ RI + \frac{1}{C}\int idt $$
 * style="width:95%" |
 * (4.5)
 * }

Derive Equation 4.5 to find Alternate Circuit Equation 1:
 * {| style="width:100%" border="0"|-

$$ V = LI'+ RI + \frac{1}{C}\int idt $$ $$ \Downarrow $$
 * style="width:95%" |
 * }
 * }

Alternate Circuit Equation 2

Charge Equation:
 * {| style="width:100%" border="0"

$$ Q = Cv_{c} \ $$
 * style="width:95%" |
 * (4.7)
 * }

Derive Equation 4.7:
 * {| style="width:100%" border="0"|-

$$ Q'=C\frac{dv_{c}}{dt}$$
 * style="width:95%" |
 * (4.8)
 * }

Derive Equation 4.8:
 * {| style="width:100%" border="0"|-

$$ Q''=C\frac{d^2v_{c}}{dt^2}$$
 * style="width:95%" |
 * (4.9)
 * }

Rearrange the Charge equation from Equation 4.1:
 * {| style="width:100%" border="0"|-

$$ Q = Cv_{c} \rightarrow v_{c}=\frac{Q}{C}$$
 * style="width:95%" |
 * (4.10)
 * }

Substitute Equations 4.8, 4.9 and 4.10 into Equation 4.0:
 * {| style="width:100%" border="0"|-

$$ V = LC\frac{d^2v_{c}}{dt^2} + RC\frac{dv_{c}}{dt} + v_{c}$$ $$ \Downarrow $$
 * style="width:95%" |
 * }
 * }


text:
User:Super Quantum immortal/Colored alphabet for the partially sighted

(fix the formating latter)

The theory, instead of just magnifying the letters. Lets give them colors, so we don't need to see to sharply in order to reed them.

In Kate made a filter that configures each letter to a certain color and background color. http://kde-files.org/content/show.php/Colors+alphabet?content=140780 Made a grease-monkey script replacing individual letters with images. http://userscripts.org/scripts/show/96470

sample images http://thechurchofthequantumimmortal.blogspot.com/p/implementing_14.html


text:
User:Egm4313.s12.team19.Aburton426

My contribution to Report 1

For Report 1, I did problem 1.2.

title: User:Egm4313.s12.team19.Aburton426


text:
User:Egm4313.s12.team20.eizagaechevarria

R1.3


text:
User:Egm4313.s12.team20.eizagaechevarria/Report 1

Given
Free body diagram:

Solution
By definition:

$$\displaystyle a=y'' $$

(3.1)

$$\displaystyle f_k=ky_k$$ (3.2)

$$\displaystyle f_c=cy_c' $$ (3.3)

Since the spring, dashpot and mass are aligned, the forces on the dashpot are equal and opposite and can be described as the internal forces.

$$ \displaystyle f_k=f_c=f_I $$ (3.4)

The net force of the system is the difference between the applied force and the internal force.

$$ \displaystyle F_s=f(t)-f_I $$ (3.5)

Applying Newtons 2nd law:

$$\displaystyle F_s=m*a $$  (3.6)

Plug in equation (3.1) and (3.5) into equation (3.6):

$$\displaystyle

my''=f(t)-f_I $$  (3.7)

Rearrange:


text:
User:Egm4313.s12.team20.samartino

R1.6

title: User:Egm4313.s12.team20.samartino


text:
User:Egm4313.s12.team20.samartino/Report 1

Given
For each ODE in Fig.2 in Kreyszig 2011 p.3 (except the last one involving a system of 2 ODE's), determine the order, linearity (or lack of), and show whether the principle of superposition can be applied:

Falling stone
 * {| style="width:100%" border="0"

$$  \displaystyle y''=g=const. $$     (6.0)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Parachutist
 * {| style="width:100%" border="0"

$$  \displaystyle mv'=mg-bv^2 $$     (6.1)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Outflowing water
 * {| style="width:100%" border="0"

$$  \displaystyle h'=-k\sqrt{h} $$     (6.2)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Vibrating mass on a spring
 * {| style="width:100%" border="0"

$$  \displaystyle my''+ky=0 $$     (6.3)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Beats of a vibrating system
 * {| style="width:100%" border="0"

$$  \displaystyle y''+w_{0}^2y=\cos wt, w_{0}\approx w $$ (6.4)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Current I in an RLC circuit
 * {| style="width:100%" border="0"

$$  \displaystyle LI''+RI'+\frac{C}I=E' $$     (6.5)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Deformation of a beam
 * {| style="width:100%" border="0"

$$  \displaystyle EIy^{iv}=f(x) $$     (6.6)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Pendulum
 * {| style="width:100%" border="0"

$$  \displaystyle L\Theta''+g\sin\Theta=0 $$     (6.7)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Solution
The order of an ODE is determined by the order of the highest-order derivatives present in the equation.

A linear ODE is one in which the dependent variable, y in this case, and its derivatives in additive combinations of their first powers with coefficients that are constants or that depend on an independent variable. If the equation does not fit the above specifications, the equation is said to be nonlinear.

The superposition principle states that if $$y_p$$ (particular) and $$y_h$$ (homogeneous) are both solutions to a differential equation:

$$ \displaystyle ay_p'' + by_p' + cy_p=f_p(x)$$

$$ \displaystyle ay_h'' + by_h' + cy_h=f_h(x)$$

Then when the two equations are added together, substituting in:

$$ \displaystyle y=\bar{y} = y_p + y_h $$

You will obtain the original ODE.

6.0

 * {| style="width:100%" border="0"

$$  \displaystyle y''=g=const. $$     (6.0)     Order: $$2^{nd}$$
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Linearity: Linear

Superposition: Yes

Examining both particular and homogeneous solutions: Particular solution: $$ \displaystyle y_p'' = g$$

Homogeneous solutions: $$ \displaystyle y_{h1}'' = 0$$

$$ \displaystyle y_{h2}'' = 0$$

Adding the two homogeneous solutions together, results in the following: $$ \displaystyle y_h = (y_{h1} + y_{h2}) = 0$$

Let: $$ \displaystyle \bar{y} = y_h + y_p''$$

Adding $$y_p$$ and $$y_h$$ equations together results in: $$ \displaystyle (y_p + y_h) = g $$

Which simplifies to: $$ \displaystyle \bar{y}'' = g $$

Thus, the superposition principle can be applied.

6.1

 * {| style="width:100%" border="0"

$$  \displaystyle mv'=mg-bv^2 $$     (6.1)     Order: $$1^{st}$$
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Linearity: Nonlinear

Superposition: No

Homogeneous solution: $$ \displaystyle mv_h' + bv_h^2 = 0 $$  (6.1.1)

Particular solution: $$ \displaystyle mv_p' + bv_p^2 = mg $$  (6.1.2)

Let $$ \displaystyle \bar{v} = v_h + v_p $$

Adding (6.1.1) and (6.1.2): $$ \displaystyle m(v_h' + v_p') + b(v_h^2 + v_p^2) = mg $$

Likewise $$ \displaystyle m\bar{v} + b\bar{v}^2 = mg $$

However $$ \displaystyle \bar{v}^2 = (v_h + v_p)^2 != (v_h^2 + v_p^2) $$

Thus $$ \displaystyle \bar{v} $$ is not a solution and superposition does not apply.

6.2

 * {| style="width:100%" border="0"

$$  \displaystyle h'=-k\sqrt{h} $$     (6.2)     Order: $$1^{st}$$
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Linearity: Nonlinear

Superposition: No

Homogeneous solution: $$ \displaystyle h_{h}' + k\sqrt{h_{h}} = 0 $$

Particular solution: $$ \displaystyle h_{p}' + k\sqrt{h_{p}} = 0 $$

Adding $$h_p'$$ and $$h_h'$$ equations together results in: $$ \displaystyle h_{h}' + k\sqrt{h_{h}} + h_{p}' + k\sqrt{h_{p}} = 0 $$

Which can be rearranged to: $$ \displaystyle (h_{h}' + h_{p}') + k(\sqrt{h_{h}} + \sqrt{h_{p}}) = 0 $$

Simplify using: $$ \displaystyle \bar{h} = h_{p} + h_{h} $$

To: $$ \displaystyle \bar{h'} + k(\sqrt{h_{h}} + \sqrt{h_{p}}) = 0 $$

Since: $$ \displaystyle k(\sqrt{h_{h}} + \sqrt{h_{p}}) \not= k\sqrt{h_{h} + h_{p}} $$

Then: $$ \displaystyle \bar{h'} + k(\sqrt{h_{h}} + \sqrt{h_{p}}) \not= \bar{h'} + k\sqrt{\bar{h}} $$

So superposition can not be applied.

6.3

 * {| style="width:100%" border="0"

$$  \displaystyle my''+ky=0 $$     (6.3)     Order: $$2^{nd}$$
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Linearity: Linear

Superposition: Yes

$$ \displaystyle y_1'' + \frac{k}{m}y_1 = 0 $$  (6.3.1)

$$ \displaystyle y_2'' + \frac{k}{m}y_2 = 0 $$  (6.3.2)

$$ \displaystyle c_1 * (6.3.1) + c_2 * (6.3.2) = (c_1y_1 + c_2y_2)'' + \frac{k}{m}(c_1y_1 + c_2y_2) = 0 $$

Therefore, $$ \displaystyle y = c_1y_1 + c_2y_2 $$

Thus the principle of superposition applies.

6.4

 * {| style="width:100%" border="0"

$$  \displaystyle y''+w_{0}^2y=\cos wt, w_{0} \approx w $$ (6.4)    Order: $$2^{nd}$$
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Linearity: Linear

Superposition: Yes

Homogeneous solution: $$ \displaystyle y_h'' + w_{0}^2 y_h = 0 $$

Particular solution: $$ \displaystyle y_p'' + w_{0}^2 y_p = \cos{wt} $$

Examining homogeneous solution: $$ \displaystyle y_{h1}'' + w_{0}^2y_{h1} = 0 $$
 (6.4.1)

$$ \displaystyle y_{h2}'' + w_{0}^2y_{h2} = 0 $$
 (6.4.2)

$$ \displaystyle c_1 * (6.4.1) + c_2 * (6.4.2) = (c_1y_{h1} + c_2y_{h2})'' + w_{0}^2(c_1y_{h1} + c_2y_{h2}) = 0 $$

Therefore $$ \displaystyle y_h = c_1y_{h1} + c_2y_{h2} $$

And also $$ \displaystyle y(t) = y_h + y_p = c_1y_{h1} + c_2y_{h2} + y_p $$

Superposition is valid.

6.5

 * {| style="width:100%" border="0"

$$  \displaystyle LI''+RI'+\frac{C}I=E' $$     (6.5)     Order: $$2^{nd}$$
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Linearity: Linear

Superposition: Yes

Homogeneous solution: $$ \displaystyle LI_h'' + RI_h' + (1/C)I_h = 0 $$

Particular solution: $$ \displaystyle LI_p'' + RI_p' + (1/C)I_p = E' $$

Examining the homogeneous solution: $$ \displaystyle LI_{h1}'' + RI_{h1}' + (1/C)I_{h1} = 0 $$
 (6.5.1)

$$ \displaystyle LI_{h2}'' + RI_{h2}' + (1/C)I_{h2} = 0 $$
 (6.5.2)

$$ \displaystyle c_1 * (6.5.1) + c_2 * (6.5.2) = L(c_1I_{h1} + c_2I_{h2})'' + R(c_1I_{h1} + c_2I_{h2})' + (1/C)(c_1I_{h1} + c_2I_{h2})$$

Therefore $$ \displaystyle I_h = c_1I_{h1} + c_2I_{h2} $$

And $$ \displaystyle I = I_h + I_p = c_1I_{h1} + c_2I_{h2} + I_p $$

Superposition is valid.

6.6

 * {| style="width:100%" border="0"

$$  \displaystyle EIy^{iv}=f(x) $$     (6.6)     Order: $$4^{th}$$
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Linearity: Linear

Superposition: Yes

Homogenous solution: $$ \displaystyle EIy_h^{iv} = 0 $$

Particular solution: $$ \displaystyle EIy_p^{iv} = f(x) $$

Examining the homogeneous solution:

$$ \displaystyle EIy_{h1}^{iv} = 0 $$
 (6.6.1)

$$ \displaystyle EIy_{h2}^{iv} = 0 $$
 (6.6.2)

$$ \displaystyle EIy_{h3}^{iv} = 0 $$
 (6.6.3)

$$ \displaystyle EIy_{h4}^{iv} = 0 $$
 (6.6.4)

$$ \displaystyle c_1 * (6.6.1) + c_2 * (6.6.2) + c_3 * (6.6.3) + c_4 * (6.6.4) = EI(y_{h1} + y_{h2} + y_{h3} + y_{h4} )^{iv} = 0 $$

Therefore $$ \displaystyle y_h = c_1y_{h1} + c_2y_{h2} + c_3y_{h3} + c_4y_{h4} $$

Also $$ \displaystyle y(x) = y_h + y_p = c_1y_{h1} + c_2y_{h2} + c_3y_{h3} + c_4y_{h4} + y_p $$

Superposition applies.

6.7

 * {| style="width:100%" border="0"

$$  \displaystyle L\Theta''+g\sin\Theta=0 $$     (6.7)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Order: $$2^{nd}$$

Linearity: Nonlinear

Superposition: No

Let $$ \displaystyle \Theta_1 $$ and $$ \displaystyle \Theta_2 $$ be solutions to the homogeneous ODE above: $$ \displaystyle L\Theta_1'' + g\sin\Theta_1 = 0 $$
 (6.7.1)

$$ \displaystyle L\Theta_2'' + g\sin\Theta_2 = 0 $$
 (6.7.2)

For superposition: Let $$ \displaystyle \bar\Theta = c_1\Theta_1 + c_2\Theta_2 $$

$$ \displaystyle c_1 * (6.7.1) + c_2 * (6.7.2) = L(c_1\Theta_1 + c_2\Theta_2)'' + g(c_1\sin\Theta_1 + c_2\sin\Theta_2) = 0 $$

$$ \displaystyle L\bar\Theta'' + g\sin\bar\Theta = 0 $$

Comparing: $$ \displaystyle (c_1\Theta_1 + c_2\Theta_2) = \bar\Theta $$ BUT $$ \displaystyle \sin\bar\Theta != c_1\sin\Theta_1 + c_2\sin\Theta_2 $$

Therefore $$ \displaystyle \bar\Theta $$ is not a solution and superposition cannot be used.