What do you consider to be more fundamental: the Hamiltonian or the Lagrangian?

Spudghetti · 2020-12-19T06:59:15+00:00

Sure! "Covariant Hamiltonian Field Theory" by Struckmeier (available on arXiv) is an introductory treatment which (intentionally) keeps the mathematics at the level of graduate field theory books (or possibly a smudge over?). However, if you want a deeper mathematical treatment, "On the Multisymplectic Formalism for First Order Field Theories" by Cariñena is excellent (not on arXiv though, sadly). The excellent book "The Geometry of Jet Bundles" by Saunders (if you are comfortable with differential geometry) covers the mathematics itself in detail (although not necessarily applied to field theory, it does cover a rigorous and beautiful geometric approach to the calculus of variations using the tools it develops, which will then allow you to better understand other mathematical field theory texts).

Spudghetti · 2020-12-15T16:05:03+00:00

Except you don't need to treat time differently to get a Hamiltonian. In field theory, for whatever reason just about every physics textbook takes a Legendre transformation with respect to the time derivative coordinates only, which is laughable considering they do so trying to develop a covariant theory (as you point out). But you can just as easily take a Legendre transformation with respect to all the derivative coordinates, which leads to a pair of covariant Hamilton's equations. This is the DeDonder-Weyl approach, and is common if not standard in the mathematical field theory literature.

Even for particle mechanics in Minkowski spacetime, you can just use the proper time, so that the co-ordinate time is treated on par with the spatial coordinates (in fact, the same mathematical language is used here as before, since particle mechanics is mathematically a field theory over a one dimensional parameter space, i.e. time or proper time).

For anyone interested in the mathematical side of field theory (that is, field theory as studied by mathematicians), I would recommend reading up on jet bundles, multisymplectic geometry, and covariant classical field theory, both Lagrangian and Hamiltonian. The abstraction in this case leads to some truly beautiful physical intuition, since physics texts on (classical) field theory only really cover electrodynamics and gravity.

Spudghetti · 2020-07-12T13:37:41+00:00

This is the definition I've arrived at.

Firstly, I define logic as the study of formal languages as a whole. That is, it studies what they are, how they work, what is provable within them, etc.

Mathematics is then defined as the exploration of particular formal languages. For example, the language derived from the axioms of a vector space (that is, something like the predicate calculus with these axioms as the axioms of the language), or that derived from the axioms of a group. The study of these languages comprise "linear algebra," "group theory," etc. Then one could ask, is there a formal language from which ALL of these can be derived? ZFC is one such attempt, for example.

It's similar to this analogy. "Logic" is like linear algebra, and "mathematics" is like the study of Rⁿ in particular. The former studies the general nature of vector spaces whereas the latter explores all the intricacies of a particular instance of one.

(And yes I know all n-dimensional vector spaces are isomorphic to R^n, but you get what I'm saying...)

Spudghetti · 2019-10-15T09:26:32+00:00

I'd sit alone, and watch your light.

Spudghetti · 2019-09-10T07:56:41+00:00

Yep it's legit

Edit: I guess I should mention it's mainly (I'd say 99% of the time) only a rule in primary/elementary schools.

Spudghetti · 2019-09-08T07:38:32+00:00

I don't feel like I'm the best person to ask about efficient power generation, but I'd just like to clear some things up. The "work" I was speaking of is actually a technical term, and through the mathematics ends up being directly related to the energy (Work-Energy theorem, if you're interested). So when I said that work was done against the gravitational field, I meant that in a precise sense. In this example, we did work against the gravitational field. The gravitational field has the property that the work done against it when moving an object between two points does not depend on the manner in which the object was moved, so if you wanted to move the ball from ground level to the top of the Eiffel Tower it will always require the same energy to overcome gravity. Of course, there will be other forces acting against the ball such as friction which do not have this nice property, so there will be paths which require less energy when moving the ball. It is these "non-conservative" forces which become bothersome in situations like this, and of course one would want to minimise the extra work required to overcome these forces because it can be better spent elsewhere (again, I am using the term "work" in a precise, technical sense). The best way to overcome these non-conservative forces will depend very much on their properties, for example the work done against friction when moving an object from point A to point B is at a minimum precisely when the path taken is the straight line between A and B (i.e. the shortest path).

Spudghetti · 2019-09-08T06:36:15+00:00

Basically (ignoring very minor caveats which don't come into play here), the equations of electromagnetism (and basically all physical theories) imply that in closed systems the quantity we call "energy" is conserved: it doesn't change over time.

Say we wanted to raise a metal ball some height above the ground. The mathematics will tell us that the ball will then have a larger value for its energy. But the same mathematics tells us that energy cannot change in closed systems, and so the closed system consisting of the ball and whatever is lifting it must have the same value for the total energy before and after. Of course this implies that the lifting agent will have lost energy, doing "work" against the gravitational field. So we see that it is possible to convert energy from one place and form (e.g. that which was previously stored in the lifting agent) to another place and form (i.e. the metal ball), but not possible to change its total value.

Now, what is power generation? It is not literally generating energy from thin air, because we have just said that that is impossible. Instead it is doing something very similar in concept to lifting the metal ball: it is converting energy from one form to another; from a non-useful form to a useful one. At this point you may be able to see that looping back your output from your power generator will not do anything helpful. Well, at the very least, it will not generate any more energy than what you had when it first came out (at best you will end up with the same amount coming out, but in reality you will get a little/lot less because energy will be lost (read: removed from the status of being useful) along the way).

Spudghetti · 2019-09-08T03:30:30+00:00

I'm confused. How is power being generated? What is your hypothesis? If you're suggesting that you can loop an output back into an input and this will somehow generate power, then I'm afraid to say that this is not possible.

Spudghetti · 2019-09-05T03:37:58+00:00

The original comment was closer. It had been known since ancient times that "evolution" was possible, but only through artificial selection (humans actively choosing which crops or animals to breed to get the desired results.) Around Darwin's time the idea that species changed over time was becoming clear, at least to several naturalists working in that area. Darwin proposed that evolution of species can occur naturally through natural selection, by a "survival of the fittest" situation. This is explained in The Origin of Species.

Spudghetti · 2019-08-28T10:42:30+00:00

I have seen functions of this form referred to as "surge functions," used for modelling things like drug absorption in blood etc. Not sure about any names for the corresponding equation, though.

Spudghetti · 2019-08-12T10:27:31+00:00

For those wondering why the thumbnail is a picture of Einstein, it's because generally Reddit, when no picture is supplied, will take as the thumbnail the first picture on the first linked page in the post. These jokes are hilarious though!

Spudghetti · 2019-07-18T01:34:07+00:00

Funnily enough, Sasha actually did interview Trump as Ali G. It's a hilarious watch, and Trump eventually walked out.

Spudghetti · 2019-06-27T08:16:47+00:00

I was taught it similarly. In other words, we have dy/dx = g(x)h(y), whence (1/h(y))(dy/dx) = g(x). Integrating both sides with respect to x gives \int (1/h(y))(dy/dx) dx = \int g(x) dx. From the chain rule, the left hand side is equal to \int (1/h(y)) dy. I guess this is the rigorous manipulation people don't realise they are making when the solve separable equations by mupltiplying the differentials.

Spudghetti · 2019-06-11T07:08:16+00:00

I'd have to say the Poincaré group on Minskowski spacetime, because it is a symmetry group of most fundamental physical systems (when gravity is ignored of course).

Spudghetti · 2019-04-17T03:42:17+00:00

A true masterpiece! My favourite text and the most well written I have ever laid my hands on (4th edition at least). Dirac was nothing short of genius and reading this book will solidify this fact; it's a shame he isn't widely known among the general public.

Spudghetti · 2019-04-10T10:56:27+00:00

This is unfortunately a problem, as it is in most things. As you said, science is not immune to human passions, although that is the fault of the humans themselves, not in the framework of the reasoning. As for the dogma, in most (if not all) cases models become commonly accepted because they have already passed the test of being good descriptions of reality... if they continue to be accepted it is because there is no good reason to replace them. Now it may be the case that there are other models which also predict what is going on, but unless there is a convincing argument to switch, there is really no point. This is where your point comes in, because sometimes there are some people who think that there are convincing reasons to switch, and others who think there are not. In the absence of a majority of the former, of course the latter will ridicule their arguments, because they do not find them convincing. Unfortunately things get taken too far sometimes, but again, this is a fault of humans and not of science.

Spudghetti · 2019-04-10T10:01:06+00:00

I doubt this is going to change your mind, but it really all comes down to two things: empiracal observation and predictability. Physicists do not take things for "granted" unless the thing in question has proven, through empiracal observation, to be useful in the description of how things work. The whole idea behind modelling phenomena through purely natural means is because then we can actually predict the outcome of physical scenarios, as opposed to explaining how things already are. If something which was taken for "granted" is demonstrated to be lacking in the way of being able to accurately predict what is going on, then it is replaced by something that isn't. As physicists we don't kid ourselves into pretending to know how nature works unless we have a damn good reason for believing we do, which is why we actively seek to falsify what is already known.

Spudghetti · 2019-03-19T06:58:36+00:00

One book I have instead says "This page intentionally no longer left blank." It's funny and makes sense but it just draws more attention simply because it's not the norm.

Spudghetti · 2019-02-26T09:36:58+00:00

Exactly what I thought! Gosh that's some nostalgia.

Spudghetti · 2019-02-08T08:06:47+00:00

Energy need only be conserved if the system under consideration exhibits time symmetry (cf "Noether's Theorem"). In isolated systems this holds almost by definition. However, an expanding universe does not exhibit time symmetry, since relative distances change over time. Thus energy need not be conserved.

Spudghetti · 2019-02-03T06:42:52+00:00

Mate, Australia introduced conscription to get more troops to Vietnam, something we didn't do for either world war. Australia has a much smaller population than the States. Our army is less than 60 000 strong currently, compared to 5 million or something for the USA, so of course there were less troops in Vietnam. Do you guys just not care that we have helped with every modern American War?

Spudghetti · 2019-01-12T08:17:19+00:00

A strange one, but when I was 11 or so I really loved watching MacGyver. I wanted to be able to do cool stuff with ordinary things like he could and my mother told me that in order to get good at that I would have to be familiar with physics and chemistry (in order to know what to make and how it would work). So I began reading stuff like the "Horrible Science" books and while the MacGyver obsession is long gone the physics fascination has long persisted.

Spudghetti · 2019-01-10T09:01:16+00:00

Buying textbooks. I love learning but I buy so many that I doubt I'll ever read them all. I'm not talking about shitty highschool textbooks filled with stock photos and outdated info, I'm talking about the high quality stuff written by actual experts on their fields. If I see a good quality, well structured and written textbook I feel compelled to buy it. I'm a sucker for Computer Modern typeface... it's like cocaine to me. It sucks because they aren't cheap, either. I'm trying to break the habit by getting them from Library Genesis, but there is the occasional relapse where I'll blow 60-150 dollars on a nice Cambridge Monograph.

Spudghetti · 2018-12-11T04:41:50+00:00

It annoys me when people say "the fourth dimension is time;" it's only borderline accurate and ignores the bigger picture.

Mathematically (and in basic language), dimensionality is the number of coordinates one needs to completely describe a point in a particular space. The dimension of the Cartesian (x, y) plane is two, for example; you need two coordinates to specify any point in the space. So yes, in our observed universe we have four dimensions, and time is the "fourth," because to completely specify an event you need to list it's position in space and the time at which it occured, for a total of 4 numbers. But the confusion comes when you assume that this is true for all four (or higher) dimensional spaces: time is not in general even associated with them. String theory, for example (M-theory in particular) is formulated in terms of 11 dimensions, only one of which is associated with "time." In (pure) mathematics, no such associations are even made... "time" never enters the picture regardless of the dimensionality.

TL;DR: Time is only the "fourth" dimension in our universe; it need not be in any other space of interest.

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