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  1. Wow, that's a good place to start. First, the law itself was established experimentally. When no one was aware of modern theoretical physics at all, but they could get involved in simple experiments. In addition, this pattern can be noticed by us. For example, the phone itself does not charge, nothing can get hot or move by itself, and so on. With the advent of theoretical physics, all modern laws began to have a strict mathematical justification. Any conservation laws are associated with some kind of symmetry, this symmetry is tied to mathematics and at the same time to space. Symmetry sounds very abstract, but in relation to the law of conservation of energy (hereinafter referred to as the Law of conservation of energy), it is a symmetry with respect to time shifts. That is, if you perform an experiment now, and then a day later, and then a year later, the results should match (and they will match if all other conditions are equivalent). In addition, in the equations describing any physical system, you can also make a shift in time and this will not break anything.

    As for me, this law can become most understandable if we realize that energy and mass are two interrelated and equivalent concepts. Any massive particle (with mass m) represents energy E = �m*c^2. Roughly speaking, a spoonful of sugar contains such an amount of energy that it's creepy. Nooo, don't be so happy that oil and even solar panels are no longer needed, because to extract it, we need to find anti-sugar. And if we combine them, then we will get the release of this energy (and the destruction of sugar (both), because the law of conservation of energy also works here). In addition, this ratio has been confirmed in other experiments. If a chemical reaction occurs and a new product is formed from two reagents, then once it is formed, the binding energy in its molecules is greater than in the previous reagents (and it is negative, so the modulo energy is greater, but it is smaller in magnitude). And from this formula above, it follows that the mass of the new product becomes smaller (and no, not because something “evaporated” or was lost). The number of molecules and atoms is still the same, but their total mass in this product is less, because ZSE.

    Well, now about the violations, it's even more fun. In short, violation is the whole of quantum mechanics on small time scales. For I recently said that particles are just “felt clumps” of energy, but everyone has heard that they can be born on their own, and there are even phenomena that confirm this. For example, Hawking radiation in black holes. They are born, as a rule, in pairs, so as not to violate the laws of conservation of charges and aromas (electric, lepton, beauty, top, bottom, strangeness, baryon, and so on). Well, you can immediately say that nothing works and in general horror. And you will be right, because here is a direct violation of the law of conservation of energy, and then we learn that in general, thanks to this violation, we exist and our world exists, but everything in order. �

    Quantum physics and uncertainty come into play. The fundamental mathematical explanation for its presence lies in the theory of operators, on which quantum physics is based, and their eigenvectors. But too small a percentage of people understand this, so if someone is very interested and he is a little fumbling in the linal, then he can write to me in the BOS. The logical explanation in my head is still very vague, and each physicist probably has his own and is built on associations and imagination. If it's very priblezhenno, then on a small scale some kind of fucked-up is happening. For example, if there is no energy at all and we look at a single molecule, then there will still be an oscillatory motion called zero vibrations, which does not lend itself to logic at all and is a violation of the law of conservation of energy.
    By the way, an important note – uncertainty applies only to pairs (!) of some (!) quantities. For example, you can find out where an electron is located by putting a plate that will record the place where it fits (well, only before the collision, the place where it fits will not be specific, as with the ball that you threw, but will be described by probability and have a considerable spread, but you can find out the place in the end). But to find out the coordinate and speed at the same time will not work, alas. The explanation for this again lies in the operators and the relations between them.

    Well, we are interested in a specific uncertainty – the uncertainty between time and energy. Mathematically, its essence is: the product of a time interval and the change in energy within this interval is always greater than a certain small number. The logical point: for short periods of time, the energy of the system can be disturbed by itself. The shorter this time interval, the greater the change in energy can be. That is, the law of conservation of energy exists, but with an amendment: for a small period of time, it is violated. After this period of time, everything falls into place. These stretches of time are incredibly small scales that are simply not imaginable to our imagination.
    Actually, from this indefiniteness,for example, the birth of pairs of �particles occurs(and instant death, so that the ZSE works). Such particles that are born due to this indefiniteness are called virtual.

    Now it's time to talk about why this is so important.
    We all know (or are about to learn) that all interactions are transmitted by carrier particles. Consider only the gravitational and electromagnetic (in the strong and weak the same with the help of transporters, but the story is more complicated). Electromagnetic interaction occurs due to the fact that particles exchange gamma quanta. Gravitational – due to the exchange of gravitons (which have not yet been discovered, but whatever you call them, they still exist, and with something massive particles must exchange). Without these fundamental interactions, there would be no molecules, no planets, no atoms, no nuclei, nothing.
    And the question arises: why on earth does each particle give birth to them and surround itself with their cloud, thanks to which the particles learn about each other and interact? But with the fact that there is uncertainty. At each moment of time, the energy of the particle can change, at the moment of change, the energy of the particle increases sharply, and since this energy is enough to give birth to, for example, a gamma quantum, it successfully does this, returning to its previous state. Then these gamma quanta or gravitons fly away from the particle until they meet another one. When a collision occurs, an interaction occurs, and the intensity of collisions determines the magnitude of this interaction (force). Moreover, maybe someone at school asked the question: why do we have a square at the bottom in the formulas of universal gravitation and Coulomb's law? And the answer is that if you draw straight lines from a sphere, then the number of these lines passing through the unit area of the sphere that surrounds this point will decrease back to the radius squared. That is, Coulomb's law and the law of universal gravitation can be deduced from the fact that there are carriers, although initially these are also experimental laws (I explained this law in more detail in my other answer).

    Another surprising phenomenon that follows from this uncertainty is the presence of “virtual fur coats” in particles. Imagine a particle. Let's say we are already in the 100th century and have an Iphone XXXX, which has an electron microscope instead of a camera (probably an electronic one will not be enough, but some worse, but let it be so) with a time dilation of 10^34 times. And here we are looking at a particle. At such scales, we will be able to notice this change in energy, due to which the particle will turn into some other ones (because energy allows it, and if energy allows it, then this will happen). �All these particles will in turn break down into others (because they are fat and cannot last long). As a result, observing a neutron, for example, we will see not a neutron, but a fur coat of various particles that are born and die, are born and die. Весело Fun, isn't it?
    If this is translated into classical physics, then imagine that you have a ping-pong ball on the table. At some point in time, it turns into a bowling ball, which immediately breaks up into 10 tennis balls, then 5 tennis balls into 7 darts, and so on (but on average it will still be the same ping-pong ball�:) ). And all this in dynamics, with random transformations(so that only other conservation laws are fulfilled), in general, fun and fervent.

    The fun is that a bunch of these virtual particles will still have on average the same mass as the original particle, absolutely all the time will have the same electric charge and other charges. And, for example, if this happens to a proton in a hydrogen atom, then since the electron is very far away, it will not notice this and it will not affect the properties of objects around us in any way.

    Here is such a fun world we have 🙂

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