/sci/ - Science & Math

>>10873150
It’s all just an abstraction.

It’s also important to know there’s a difference between available energy and absolute energy. Available energy, energy you can use to do useful work, is always going down thanks to thermodynamics. The amount of absolute energy never changes. All forms of energy are slowly becoming heat energy.

>>10873150
the faster something moves the more energy it takes to move it even faster

I thought "energy" was just the general term for the "stuff" matter is made from

>>10873150
Just read the chapter on energy for any year 1 undergrad physics text. Not even trying to be rude, but it's hard to answer in a single post. Know the absolute basics of calculus.

>>10873151
>It's all just an abstraction.
I had suspected as much. If I want to understand it better, is analytic mechanics the right rabbithole to start with? I gather that's where this all started.
>>10873157
>know the absolute basics of calculus
I do. And of course the obvious pattern-match for something like [math]{1\over 2}mv^2[/math] is that it's the integral of mv. But why? Is there an intuition behind it, that you could explain to soneone like Euclid? Or is it just a hard-won scientific fact?

>>10873150
Good video for you OP

https://youtu.be/PUn2izowBkw

What you seem to percieve as strange is a direct consequence of the definition of energy. Energy is defined as a force acting upon an object along a distance (Nm). The "definition" of energy, which I suspect seems more natural to you is a force acting upon an object for a time. But that is exactly what Impulse is (Ns).

The best I can offer is to visualize that engery by comparing it to potential energy, because potential energy is linear again (in a homogeneous gravitational field).

Maybe think about it this way: Dropping a 1kg object and a 4kg object from 1m height. The 4kg object experiences 4 times the force than the 1kg object, after the fall it has 4 times the energy.

Now consider the situation with impulse. Then time of the fall is exactly the same, so they both have exactly the same impulse, although the 4kg object has 4 times the energy.

A nice way to see where this difference comes from is to make this little exercise in calculus:

Let x be and object of 1kg mass and y be an object of 4kg mass. If they have the same kinetic energy, what is the relation between their impulses?

A last thought on the matter: Energy is usually the more important notion in everyday life, e.g. a 4kg object hitting your foot after falling 1m hurts much more than a 1kg object. However, if two such objects collide elastically (without deforming and stuff, like billiard balls) then impulse is the more important notion.

>>10873150
>>10873162
It's because that's what countless amounts of experimental data tell us. Remember, you're doing science, not first-order logic. If course we want our scientific models to exhibit logical consistency, but at the end of the day, we perform experiments and they tell us what the world is like. It will serve you well to keep this in mind as you progress your studies and career.

Just to elaborate a little more: if the work--kinetic energy theorem is unsatisfactory to you (i.e. integral of p dv = KE), then what *will* satisfy you? I could move to Hamiltonian mechanics, and talk about symplectic manifolds and cotangent bundles and Hamiltonian flows, and it will all be very nice and elegant and beautiful. But you won't gain any "deeper" intuition about what energy *is* and *why* it satisfies the properties it does. It just does, and those are axioms we set for our theories. Instead, you'll learn a lot more about how those properties create the world we see, to a good approximation. And you'll probably learn a lot more about the subject in a general sense. But those properties, these axioms, are fixed, at least in the context of that theory. Where did these axioms come from, you say? Experiments. Lots of them. Perhaps not necessarily in that chronological order, as certainly theories can be thought of before experiments are done. But always, *always*, experiments must be performed, if you want to learn about the world.

So to answer your question: what is kinetic energy? It's the thing that doubles when you double your mass, but quadruples what you double your speed. Or perhaps it's one of the leading order terms in the series expansion of Einstein's dispersion relation. Or maybe it's the spectrum of the Laplacian. No matter how deep you go, none of these levels will tell you *what* it is. That question is simply not within the domain of science.

>>10873205
Bingo. Physics is in the business of predicting the universe, anything beyond that is philosophy

>>10873205
>why do planets move in ellipses?
>lmao they just do you can't know why
Stfu smooth brain

>>10873205
>>10873198
Pretty much perfect responses right here.

A neat thing I learned recently is that the transfer of power - energy per time - determines how badly you get hurt in a given interaction. This is what I always go back to when I try to conceptualize energy as a physical property.

>>10873221
Thats literally how it is

>>10873225
No, it's the result of a more fundamental law of physics. The same could be true of energy, and why it acts as it does. Again, stfu smooth brain.

>>10873231
>Regurgitates the same witless insult
>Doesn't even make an attempt at a real argument
You literally look like a retard.

>>10873237
Did I hit a nerve, smooth brain?

>>10873205
>It's because that's what countless amounts of experimental data tell us. Remember, you're doing science, not first-order logic. If course we want our scientific models to exhibit logical consistency, but at the end of the day, we perform experiments and they tell us what the world is like. It will serve you well to keep this in mind as you progress your studies and career.
Fuck you, you pompous little faggot. I know it took a lot of work and creativity and years to figure out the rules of energy. This is acknowledged in the questions that you obviously didn't read.

Thanks for the career advice, though. I hope to god your experiments are as cash money as you claim, because you're not squeezing by the tenure committee on the basis of teaching or affability, you condescending piece of shit.

>>10873198
A few corrections here:
>The "definition" of energy, which I suspect seems more natural to you is a force acting upon an object for a time. But that is exactly what Impulse is (Ns).
This is a definition of (change in) momentum, not energy. Newton actually did it the other way around: he defined force as change in momentum over time.
>Now consider the situation with impulse. Then time of the fall is exactly the same, so they both have exactly the same impulse, although the 4kg object has 4 times the energy.
The time is the same, but the force is 4x stronger for the 4 kg object, so it has 4x the impulse, or 4x the change in momentum.

>>10873538
If you ask what "energy" is in the title of your thread, people are going to talk down to you.

And lmao at the irony of you attacking that guy's affability.

>>10873150
Think about fuel in a car. You need to consume fuel in order to move the car. Energy will be like a universal currency for the fuel needed to run the car at such and such velocity and such and such distance. More than that, energy will tell you how much charge an electrical car should have in order to do the same stuff. Thinks in terms of money/currency.

>>10873581
>If you ask what "energy" is in the title of your thread, people are going to talk down to you.
Asking if there is a simple physical intuition behind a concept, to help visualize or understand, is perfectly fine. If you dropped "kinetic energy" onto Archimedes, he would ask the same question. As for affability, nice try faggot. He was rude and deserved no better.
>>10873600
>Energy will be like a universal currency
I understand that much -- an intriguing and beautiful and very very useful unification of the various known kinds of "work" and "power". What I don't have is an intuitive grasp of the formal equations:
>Why is something moving at 5m/s five times harder to accelerate than something moving at 1m/s? And why is acceleration at small speeds like 0.001 so VERY cheap?
The fact that uncreative people are calling this a stupid question, suggests to me that creative people have already thought long and hard about it and turned up nothing. But I thought I'd put it out there anyway. There are, after all, some geniuses on the chans.

Your daily reminder that conservation of energy is a mere approximation.

https://motls.blogspot.com/2010/08/why-and-how-energy-is-not-conserved-in.html

>>10873184
Not them, but much appreciated. Subbed to that channel for further digging around as well, looks to be a breddy gud resource.

>>10873752
>harder to accelerate
Gonna have to expand on what you mean by that a bit. Are you referring to a consequence of E = (1/2)mv^2?

>>10873150
Why does 1kg moving at 2m/s have the same energy as 4kg moving at 1m/s?

Toque vs speed, it's all about how the object is expressing itself and its inertia

>My intuition, however, is that the latter could do twice as much "work" as the former. Is there any physical or mathematical intuition behind energy that could help here?

Energy is just a change in something being released into equilibrium. If you release the pressure from a compressed air tank what happens? It just releases the potential of "more air" into "less air". The intermingling of those pressures are the "energy", not the compressed tank or the atmosphere outside of it. Basically it's the motion of "stuff" from where it is to where it isn't.

>Is any form of energy other than heat a real "thing" that you can visualize, or is it all just a useful abstraction?

What would "change" itself be determined as? It's constantly changing. That *is* energy. Action of something, not a "thing" itself. If you want "more energy" all you do is spin the magnet faster or make the wheel of magnets larger.

>>10873225
Are you sure a star's field of magnetism isn't just elliptical in nature and planets rotate along in a certain path due to the vacuum of space pulling with an equal but slightly weaker level of force than the force of gravity exerted by the center star?

And perhaps the field of magnetisms observed are elliptical in nature due to other trials and errors of different form not lasting long enough to be observed by man?

>>10873783
>Are you referring to a consequence of E = (1/2)mv^2?
Yes, to its derivative.
>>10873811
OK, that's a fair qualitative account of what energy behaves and what NOT to expect from it, specifically perpetual motion. But it isn't enough to predict any particular quantitative property. I suppose I should have phrased the question:
>Why is it that an object moving twice as fast, can boil four times as much water?
Not that I am really expecting a full, easy answer -- if one were known, we would all have been taught it in school -- but maybe some of the more imaginative people here have at least a little trick or analogy to help make it click.

>>10873877
So essentially you want to know why v is squared in the classical kinetic energy equation? There's this:
https://en.wikipedia.org/wiki/Kinetic_energy#Derivation
And it mentions the dot product, which gives me the impression any question like "Why is kinetic energy so much more dependent on velocity than mass?" is impossible to answer intuitively.

>>10873877
The Pythagorean theorem means that the kinetic energy of each component of velocity adds to the kinetic energy of the total velocity. [math] v^2 = v_x^2 + v_y^2 + v_z^2 [/math]

>>10873902
That doesn't really explain much, though, it just expresses [math]{1 \over 2} mv^2[/math] as the obvious integral. (The inclusion of the dot product and of vectors in general is important for dealing with curved paths. Their importance diminishes when everything is in a straight line -- which is what I was thinking of when I asked.)
>>10873905
>the energy of v is the sum of the energies of its orthogonal components
Is this really a fact that helps you understand kinetic energy? I don't immediately see how, but I'll think it over. Or can you explain in more detail?

>>10873877
I understand kinectic energy as the price you pay to maintain movement. Like an object free of external interactions - a metal ball over a frictionless surface sliding with constant speed - has to have that speed from something, it cannot sponteneously start moving. If it is moving, some force - change of momentum - in the past acted on it during an interval of time. This force, while acting, did work. Energy is grossly speaking the ability to do work. When you compute the work done for a free body in motion you get the square velocity dependance. Notice that is the scalar product of the velocity vector with itself, which is a scalar and is invariant in different coordinate systems. You can also think thermodinamically, considering the equipartition theorem, which among other stuff, relates temperature to the average kinectical energy of the molecules of a gas. Temperature is related to the average velocity squared, if you tried some other power you wouldn't have agreement with experimental data.

If you think in rotational kinectic energy, the formulas are analogous, there is a dependance to the square of the angular velocity. And it represents the work done by a torque to generate that rotation.

Notice that kinectical energies of free particles moving at constant speed can be "cancelled" - linear kinectical energy is zero in a reference frame moving with the same speed as the particle and rotational kinectic energy is zero in a reference frame rotating with the same angular velocity as the particle.

There is of course a difference, since rotation is a non-inertial motion, so you have a centrifugal force that must be balanced by a centripedal force. But centripedal force is perpendicular to the rotation motion, thus making no work and we have no contradiction.

Idk if this will get you more confused, but thinking in terms of costs/gains is more intuitive for me and I hope it can help you to think about energy in different ways.

E=mc^2

aka
light has 0 energy according to einstein.

people pretending to be smart is the reason why we can't have nice things

>>10873986
E=p.c for massless particles

>>10873989
>stipulations

>>10873969
>Is this really a fact that helps you understand kinetic energy?
Think of it this way: imagine you're trying to move a body up to a certain speed at a 45 degree angle between x and y. Let's say it takes 10 joules of energy to do this by directly accelerating in that direction, leaving it with 10 joules of kinetic energy. We could also bring it to this speed by accelerating it with 5 joules of energy in the x direction, and adding 5 more joules in the y direction. If energy is conserved, it should take 10 joules either way. For the energy to add componentwise like this, it must be proportional to the square of velocity.

>why does the world behave the way it does?
>we don't know that's just how experimentation tells it does
>okay but WHY lmao fuckin BRAINLET
Learn what science even /is/ before you speak.

>>10873877
OK, that's a fair qualitative account of what energy behaves and what NOT to expect from it, specifically perpetual motion. But it isn't enough to predict any particular quantitative property.

Exactly. "neither created nor destroyed" doesn't imply a quantity nor a change in quantity. Where does the "quantity" come from? It can't, it can only be a change in quality.

>Why is it that an object moving twice as fast, can boil four times as much water?

Because motion is already impelled into it, it doesn't have to work as much because it's displacing the energy by means of traversing through the environment (it's in motion). It's like saying "well a drop of blue dye doesn't make already blue dyed water as blue as undyed water. Well of course, it already has the quality of blue dye in it, it is closer to "maximum blueness" than the regular water.

>>10873986
That equation is only a special case of this: [math] E^2 = (mc^2)^2 + (pc)^2 [/math]

A photon has no mass but has momentum, so has energy.

>>10873972
>Temperature is related to the average velocity squared
That's very interesting, thank you. Maybe the way to understand (electrical, kinetic) energy better is to look closely at the foundations of temperature itself. I will look at that.
>>10874013
Interesting perspective. Conservation of energy logically entails [math]v^2[/math]?? I dunno, seems a bit fishy. At a minimum you have to make sure your vectors are all orthogonal. I might have to sleep on it.
>>10874036
Sorry, you've lost me.

>>10874066
>Sorry, you've lost me.

"energy cannot be created not destroyed"

Where does the quantity come from? There is none, it's a displacement of what is already there (the original quantity if you want to call it that). There cannot be a "new quanity". From where? From what? Water is turned into steam and the pressure turns a turbine, then the steam condenses and turns back into water, it can even turn into ice. All that's occurred is a change in state of the water, using the QUALITY of "heat". "steam, ice and water" are all just forms of "water". There is no new "quantity" of water.

If there is "quantity" then it's just that. One quantity whose quantity could not be changed without violating the laws of thermodynamics. You cannot magically conjure up and "create" more out of nothing. You have that "something" that was always there and you make it last forever.

>>10874066
In classical/quantum mechanics the Hamiltonian - which in most cases represents total energy - is the generator of displacements in time, similarly to linear momentum being the generator of spatial displacements. If you have no linear momentum you have no change in your position. If you have no energy, your system does not change in time. To "turn time on" you need energy.

So, you can think of action as the relevant quantity here. Action = position.momentum = energy.time. Momentum generates alteration in position and vice-versa. Energy generates alteration in time and vice-versa. In quantum mechanics action is quantized, while in classical mechanics it is not.

A good system to think about energy is the harmonic oscillator, since the potential energy depends on (displacement)^2. So zero displacement means maximal kinectic energy, while zero velocity means maximal potential energy.

>>10874066
>>10874152
Another good system to think about energy is free fall near Earth's surface - without air resistance. The equations of motion do not depend on the falling particle's mass. Now it is easy to see from Newton's second law that gH+(1/2)v^2 is constant for every instant of time. You can extrapolate this for a particle sliding through a frictionless surface of arbitrary shape. Hence this quantity has nothing to do with the way you do the motion, only with the gravitational force and the speed^2 and has the same value for every instant of time. It is deductible from Newton's laws, but actually it is more general than Newton's laws, since conservation of total energy is never violated by any system. So it makes more sense to define this invariant quantity, calling it energy due to its relation to heat, and derive all laws of physics in all domains from it.

"Energy" is just a word idiot.
What "Energy" is is an abstract concept explaining a pattern in reality. "Meaning" as you are searching for is pointless. You are just mapping "meaning" to some other abstractions of patterns.

Energy is a word used to describe a pattern. That is the meaning. What you are thinking is "meaning", is just what maps well to you intuitive understanding of the world based on patterns you have experienced most.

What you want to do is this:

Gain an INTUITIVE understanding of the patterns defined by Energy, such that you can reason about it the same way you reason about say gravity.

>>10873150
The voltage of the anus is highly positive relative to the rest of the body. The voltage of the penis is somewhat negative. This is why they subjugate young males by anally raping them, it's an energy harvesting and brain scrambling ritual.

Better Explained for brainlet meatbags

Meaning as you define it just means the following:

"This makes sense to me on an intuitive level because I have seen the pattern a lot and can predict it well."

Asking what the meaning of something is and expecting some deep truthful is stupid. If you want "meaning" all you do is gain more experience with the pattern so it becomes intuitive.

Intuition = Meaning

Humans are just fucking pattern lookers you fucking moron. There is no great meaning. All it is you discerning patterns in reality and becoming more intuitive with reality such that they feel normalized and predictable. There is no such thing as "meaning" at a deep level.

>>10874202
Clamped.

>>10874202
The words

Energy, Particle, could just as easily be "Pattern 1", "Pattern 2" etc. It doesn't matter. The reason things seem "real" or "meaningful" is just familiarity.

>>10873154
Energy is a quantification of a bodies ability to do work

>>10874202
>>10874194
based
>>10874021
based
>>10873538
cringe

>>10874045
disingenuous.

>>10873150
>what is energy
energy is a number derived from plugging other numbers into a formula. Don't get confused just because it sounds deep. Brainlets and non STEM people use the term all the time to describe things that are not energy. Before science was invented, people used energy as a hocus term to describe how much OOMPH something has, without a precise definition, so it was meaningless.

> the latter could do twice as much "work" as the former
there is a precise scientific definition for "work" also.

>>10873767
It’s my favourite YouTube channel. Everything is fantastic.

>>10873152
your statement is true and false

>>10873184
>>10874379
I hate these hand-talking nu-males tbqh senpai desu

Phiysical energy is like money budgeting - what one side loses, another gains. We count approximations for things so we don't have to deal with reality in detail.

>>10874454

What energy really is though? It is the ability to do work, to change the state of any system, to inflict a force.

E = INT F ds for conservative fields.

>>10874423
cringe. He's obviously pretending to grab big milkers

>>10873205
Quite frankly, you are the one who doesn't understand it. We got the concept of energy because that is what our mathematical system (our basic axioms) give us easily. If we were to have other groundwork for the maths we use, we would probably end up with an whole other way of representing the same thing because it IS an abstraction.

The best exemple I can give is exergy, 20-30 years ago it was all the rage in (applied) thermodynamics and many scientist wanted it to be placed at the same level of importance as energy. Obviously not used in the same way though. But since we already had the concept of energy and exergy being easily related to energy we just stuck with the latter.

>>10873759
Conservation is only an approximation because the observer is part of the system. There is no way in principle to isolate observer from the physical system, since, at the end of the day, observations constitute the primary object of physical research.

OBSERVER MAKES OBSERVATIONS BY MEANS OF INTERACTING WITH SYSTEM, THUS SERVING AS ONE OF SOURCES AND/OR RECIPIENTS OF SYSTEM'S ENERGY, AT THE VERY LEAST. THERE IS AND CAN'T EVER BE SUCH A THING AS NON-PARTICIPATING OBSERVER. MEASUREMENTS ARE MADE VIA INTERACTION.

>>10874629
>THERE IS NOT AND CAN'T EVER BE
FIX

>>10874629
Observation is a particular case of interaction. Physics researches interactions.

>>10874658
>Observation is a particular case of interaction. Physics researches interactions.
Oh jesus christ this fucking thread

>>10874751
But he's actually correct, you stupid frog

>>10873151
Just use steam engine lol duh energy crisis solved

>>10874975
He's not correct, he's in Not Even Wrong territory. How do I know? Because
>physics is the study of interactions
is no more true than
>accounting is the study of interactions.
It's a stick of doggerel that then has to be decoded by the reader, and only by a reader who already knows exactly what he means. Scrolling through this thread I can find people who take PARAGRAPHS to say:
>energy is that which always goes down
>it's hopeless to try to understand things
>energy cannot be created or destroyed, just like water molecules cannot be create or destroyed
I swear this board is full of IFLS numale cucks who think pic related is someone to look up to. Shoutout though to >>10873205 who just seems purely autistic and is probably a nice enough, knowledgeable guy in person.

>>10875506
>physics is the study of interactions
>accounting is the study of interactions.
Obviously, in these two examples the word "interaction" related to two utterly different concepts.

You, however, have a point. The word "interaction" >>10874658 here DOES require a definition, and without it those two statements can be interpreted whatever which way.

Now, a question, though.

WHAT IF the meaning of the word "interaction" itself changed over the course of the history of physics in accordance with certain crucial experiments, that ended up demonstrating the inadequacy of the previous understanding of that word to the problems at hand?

>>10875506
I think this board in general reeks of inadequacy. Not just in 'knowledge' of scientific models but also inadequate understanding of how to apply those models and what they actually describe. Pop-sci needs to pivot towards this end of the spectrum at some point or we'll end with tons of pseuds who know all the results but don't realise they're misusing them horribly.

Also as much as I agree with you fuck you Sagan was fine.

>>10875506
And yet Feynman is apparently considered absolutely fine.

>>10875555
Feynman confirmed a fine man

>>10875561
That tangential attentionwhore? Seriously?

>>10875555
Feynman was based tho

>>10873150
Energy is an invariant quantity computed from the parameters of a physical system (when the constraints it is submitted to are invariant with time), in a given coordinate system.
Actually energy doesn't exist physically, it is really a mathematical construct that will change if you change the coordinates.
Check "Noether theorem" and "conservation laws" on the web. There is also a very good discussion in Landau-Lifshitz: Mechanics/

A mathematical abstract quantity that is conserved in all processes. Noether's theorem demonstrates that energy and momentum must be conserved.

Beyond that? Nobody really knows what energy ''is'' or even if asking ''what is energy?'' is even a proper question.

>>10875555

The process by which the collective consciousness produces change in the realm of information, aka the physical world, which propagates to the personal consciousness.

>>10873150
think about the concept of torque

>>10873150
>What is energy?
Energy is a conserved quantity in our universe. Classically, we have conservation of momentum and energy, which are consequences of the fact that everything is relative in time and space (move the whole universe by some amount in time and space and nothing will have changed, physically speaking), due to Noether's theorem. It's advanced stuff, but the upshot is that "conservation of energy" isn't a dogma, it's a theoretical consequence of our universe.

>What is kinetic energy?
Applying force (over the course of some distance) on a massive object costs energy. The force accelerates the object, and the energy (called work) put into the object becomes its kinetic energy. If you look up the derivation of the formula for kinetic energy, you'll find that it is defined as the integral of a force over a distance. By using various definitions, this can be shaped into the familiar E_k = 1/2 mv^2. As for intuition, imagine a car with some velocity slowing down over a fixed distance. Velocity appears twice: a force must be applied to take away velocity, but the break distance is covered in short time (car goes fast), so extra force is needed due to the smaller time frame.

>What about other forms of energy?
Other Anons are correct to point out that energy is a "currency". You can give something more energy, or take energy from it. For example, in the case of an object in a gravitational field: energy can be extracted by dropping it, and it can have energy put into it by elevating it.

>Is it just abstraction?
Energy is an abstract concept, but it can be easily understood in everyday scenarios, see the example of gravity above. Early on you can trust on your intuitive understanding, and when you feel confident that "calculating the system's energy" is what you must do (energy is conserved, so knowing something at one point in time tells you something about another point in time), you can take the jump to abstraction.

>>10874423
>i have no argument

>>10873184
>meme analogies
>cringe shirt and posture

never exited a video so quickly

>>10875486
ive developed this net to catch hot air, maybe we could co-lab

>>10873150
Given a symplectic manifold [math](M,\omega)[/math], a Hamiltonian vector field [math]V_f\in TM[/math] associated to [math]f\in C^\infty(M)[/math] is a vector field such that [math]df + \iota_{V_f}\omega = 0[/math], where [math]\iota[/math] is the inner product on [math]TM[/math].
Let [math]G[/math] be a compact Lie group such that [math]\operatorname{Lie}G[/math] is isomorphic as a differential Lie algebra to [math]TM[/math], graded by [math]d[/math], [math]\iota[/math] and the Lie derivative [math]\mathcal{L}[/math]. Let this isomorphism be [math]\xi \mapsto a_\xi[/math], then [math]f[/math] is said to be symmetric/invariant under [math]G[/math] if [math][V_f,a_\xi] = 0[/math] for all [math]\xi \in\operatorname{Lie}G[/math].
Now given a moment map [math]\Phi: M\rightarrow \operatorname{Lie}G[/math], we can perform symplectic reduction invariant sets [math]T[/math], where [math]t\in T[/math] implies [math]a_{\Phi(m)}t \in T[/math] for all [math]m\in M[/math]. Now as [math][V_f,a_{\Phi}] = 0[/math], [math]\mathcal{L}_{\Phi(m)}V_f = \iota_{\Phi(m)}d_\text{hor}V_f = 0[/math] for all [math]m \in T[/math], where [math]d_\text{hor}[/math] is the differential horizontal to the foliation of [math]M[/math] by invariant sets [math]T[/math]. As [math]m[/math] varies, we have [math]d_\text{hor}f = 0[/math] so [math]f[/math] is covariantly constant on [math]T[/math]. "Energy" by definition is this constant.
Hope this helps.

>>10879072
where to learn more about this ?

>>10879083
here

http://webmath2.unito.it/paginepersonali/sergio.console/lee.pdf

>>10879096
thanks

>>10873150
Take F=ma, OP.

Suppose I apply the same force on each, 2 newtons. This force applied to the first one accelerates by 2m/s^2, so in 1 second it reaches the speed I want. On the second one, 2=4a, so a = 1/2 s^2, thus it takes 2 seconds for it to reach 1 m/s. 1:2, hence your confusion.

However, energy = work = F x d, and not energy = F x t. Why is it times distance? Intuitively, I understand that it is because the effect is Spatial, not Temporal, and while it takes me longer, in seconds, to accelerate the second body the distance displaced is the same( V^2 = 2ad ). Because of physical symmetry, this distance is equal to the displacement when decelerating the body with a -2 newton force. So when concerned with the spatial consequences of a movement, it makes sense to think of energy as force x distance and not time.

>>10874045
>it has no mass yet somehow has momentum

>>10873150
It's simple. try to think of something that isn't energy.

You can't. That's because everything is energy.

>>10873150
Don't believe the bullshit in here. Energy is one component of the energy-momentum tensor. Not abstract at all.