Newton proposed that all matter is attracted to all other matter via a force that is proportional to the mass of both bodies and which diminishes with the inverse square of the distance.
He called this force gravity and used it to explain both the cohesion of the Earth and all Earthbound matter and the orbital motion of the planets. His proposal was controversial because it violated the principle of localism, which requires that objects cannot interact without being nearby to one another. People had always understood that objects have weight that causes them to fall towards the Earth from any height that anyone had experienced, people did not imagine that this force extended far outwards into the heavens. Although Newton's gravity fell off with distance, in theory it extended to the stars without limit.
By contrast, Descartes had attempted to explain the orbits of the planets by vorticies that spun around the sun and pushed on the planets like a kind of wind and he criticized Newton's theory as a kind of mathematical abstraction which did not necessarily pertain to the fundamental nature of the univrse. Newton's famous statement "I frame no hypothesis" was already a response to this objection. His laws were derived empirically from what had been observed in planetary motion and were not claimed to be statements about fundamental causes. Privately, however, Newton agreed with Descartes that a fundamental theory ought to be local.
As time went on, other phenomena came to be understood in terms of long-range forces similar to gravity. The electric force appears very similar to gravity but has a key difference: it is proportional to charge rather than mass.
This has two important implications. First, the electric force can be either attractive or repulsive. Second, an electric interaction with a distant body can be detected without a telescope because two bodies with a different ratio of charge to mass can be observed to accelerate differently. By contrast, a telescope is required to detect a gravitational acceleration because everything accelerates the same way in a gravitational field. You would need to see some distant body accelerating relative to you in order to detect gravity.
The magnetic force also obeys an inverse-square law, but unlike the electric and gravitation forces is proportional to current, which means that two charged bodies do not have a magnetic interaction unless they are both moving.
Despite Descartes, it appears that these laws are quite fundamental. They are still a part of the best currently known laws of physics, even while other qualities of matter that were unexplained in Newton's day, such as heat, have long since been explained in more fundamental terms. However, with Einstein's special and general theories, localism was resored and found not to be incompatible with long-range forces.
How is such a thing possible? At the very least, a local theory of long-range forces must admit waves. Imagine two bodies that interact via long-range force. If one of the bodies starts to oscillate, the other body must start oscillating as well because the force on it from the first body is changing with its positoin. However, it cannot start oscillating right away because that would be instantaneous communication with the distant other body. The second body must oscillate, but after a time delay. During this time delay, where are the oscillations? They must be travelling through space, between the two bodies.
Einsteinian theories, those compatible with the special or general theory, all work like this. The force is no longer a true interaction between distant bodies. It is a local interaction with a force field that fills all of space and admits waves. This revised understanding of long-range forces began with the discovery of Maxwell's equations of electromagnetism, a theory that unified the electric and magnetic forces with light, which turned out to be electromagnetic waves. Newton had, of course, studied light but had never imagined that it was so closely connected with his idea of force.
Einstein's special relativity was abstracted from Maxwell's equations and his general relativity was discovered by Einstein in his attempt to discover a set of equations that described gravity in analogy to those of Maxwell. Although Einstein's special and general theories are usually and were originally described as theories of relativity, relativity has been a part of physics since the time of Galileo. Before Einstein's general theory, the principle of relativity was understood to mean that the laws of physics are the same in any inertial reference frame, meaning that two scientists can pass each other and both think that they are standing still and that the other scientist is moving fast. No experiment they could do would elevate one perspective over the other, as long as neither was accelerating.
Einstein's special theory did not change the general idea but it did propose that the speed of light was an exception to the principle of relativity. Thus despite the fact that theories which incorporate Einstein's relativity are called "relativitstic", the special theory was actually less relativistic than the physics of Galileo. Einstein's general theory greatly expanded the principle of relativity to include accelerated reference frames, which could be understood as stationary motion with resistance to a gravitational field.
With the special theory, Einstein emphasized relativity because the principle of relativity had been temporarily disturbed due to an incompatability between Maxwell's equations and earlier mechanics. Both of these theories obeyed the principle of relativity on their own, but had different ways of changing perspectives which, when put together, destroyed relativity. For mechanics, a change in persective was called the Galilean transformation, and for Maxwell's equations it was called the Lorentz transformation. These two theories put together were not relativistic and they implied that some scientist could detect his velocity relative to the speed of light, which was absolute.
Einstein argued, however, that physics really is relativistic by noting that the Lorentz transform can be understood in terms of the principle of relativity, and that all mechanics could be reformulated to be compatible with the Lorentz transform. This argument not only restored relativity to physics but restored localism as well, which had been missing for a much longer time. Thus, relativity was a current crisis in physics in Einstein's day, but the true legacy of the Einstein's special theory was to vindicate Descartes on locality.
The compatibility of long-range forces with localism is, in large part, the story of modern physics. The theories which have made this possible are called gauge theories. Gauge theories describe not just electromagnetism and gravity, but the strong and electroweak forces as well. The gauge theory idea is so big that it bridges classical and quantum physics, all the way to gravitation and curved space. Ideas that go beyond known physics, such as grand unified theories and string theory are also gauge theories. Physics has yet to really move past the gauge theory idea, and it all begins with Maxwell. It is arguably the biggest idea in physics and if not it is certainly one of the two biggest ideas in physics, the other being quantum mechanics.
Arguments about locality returend to physics with quantum mechanics. Einstein, Podosky, and Rosen made the first argument that quantum mechanics was nonlocal by establishing entangled states, which are the basis for quantum computation. However, they misattributed the source of nonlocality in quantum mechanics to its indeterminism. Bell's theorem showed that the observed results of quantum mechanics are inconsistent with localism regardless of whether the explanation is deterministic or non-deterministic. Thus, locality appears to be false. However, nothing in quantum mechanics enables us to send signals faster than the speed of light, which suggests that locality is true. The resolution to this paradox is that nonlocality in quantum mechanics has to do with events that are never directly observed and may not be real, specifically the collapse of the wave function. If you believe in the collapse of the wave function, than quantum mechanics is necessarily non-local. However, it is possible to avoid the collapse of the wave function entirely, such as with the many-worlds interpretation.
An early controversy in mechanics concerned the motion of an isolated body, or in other words, a body that is free from all interaction, such as gravity and friction against the ground and air. If a body could be truly isolated, how would it move? Early physicists were not able to actually go somewhere that was isolated from outside influence. They were not even sure whether such a thing was possible for some, such as Descartes, believed with Aristotle that "nature abhores a vaccuum". Thus, the idea of an isolated body may have seemed to be quite esoteric at the time. Nonetheless, the imaginative powers spent on it were well worth it, and it turned out to be a crucial prerequisite to understanding interacting motion.
Galileo proposed to describe the motion of some body as more and more sources of friction were removed and extrapolate to guess at what its motion would be like if it were truly isolated. He asked his readers to imagine a ball rolling on a plane. As more sources of interaction were removed or minimized, the ball would roll farther. Galileo argued that a ball which could roll without friction would do so forever. Later, Newton extended this princple in his first law of motion to isolated bodies moving in three-dimensional space and said that they would maintain their state of motion unless an external force acted upon it. Today, Newton's first law is unfortunately called Principle of Inertia, which is a word that does not truly belong in modern physics.
Inertia is a word invented Kepler, a contemporary of Galileo who spent his life attempting to discover harmonic tones in the orbits of the planets. The purpose of the world was to extend Aristotle's theory of Earthly matter to heavenly bodies. Kepler had to explain why heavenly bodies would not come to a state of rest, as Aristotle had predicted of all Earthly matter. The concept of inertia was not intended to refute Aristotle, but to make the observed motion of celestial bodies consistent with him. Kepler proposed that bodies resisted coming to a state of rest because of some force called inertia would cause it to maintain its state of motion.
Today, the word inertia seems to mean something different, something that Newton referred to as the vis insista: "a power of resisting by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest or of moving uniformly forward in a right line." Newton's second law means that every body resists acceleration from a force in proportion to its mass, so is the vis insista really any different from mass? Seemingly not in Newtonian physics, but a distinction could be made in Einsteinian physics, where objects become heavier as they move faster. In Einsteinian physics, we speak of the rest-mass of an object, in contrast to the observed mass that would be inferred from applying Newton's second law. The rest-mass is understood to be an innate quality of an object irrespective of its motion, so maybe it could be called the inertia.
Hereafter, inertia should be understood to mean referring to uniform motion without any other connotation. It is just jargon that physicists use, a vestage of old arguments from a man who thought the solar system was a musical instrument.
The principle of relativity states that the laws of physics are the same in every inertial reference frame. This principle was originally proposed by Galileo as part of his argument for heliocentrism. Galileo wanted to argue that the Earth could orbit the Sun without producing any obvious effects in the direction of the Earth's motion, such a kind of wind tearing away the atmosphere. He imagined a laboratory in a moving ship on a very calm sea and asked whether a scientist inside would be able to tell that the ship was moving.
Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need to throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction.
Thus, a scientist in laboratory that was moving uniformly from the perspective of someone standing on the shore would not know how he was moving relative to the shore without being able to look out at it. If that is the case, then can we truly say that the laboratory is moving and the shore is standing still in any absolute sense? Not if the Principle of Relativity is true. If there was no ship, no Earth, no Sun, nothing to take as a standard reference of motion, nothing but scientists floating out in space, then each of them would naturally take themselves as their own bodies as a reference frame. All of them might do the same kinds of experiments as those proposed by Galileo and none of them would be able to detect any absolute state of motion.
At the time, the prevailing theory was that the natural motion of all Earthly matter is to approach a state of rest. However, if we start with the Principle of Relativity, clearly it is impossible for bodies to approach a state of rest on their own. From its own perspective, an isolated body is already at rest. If it were, from our perspective, to slow down towards a state of rest, then from its own perspective, it would be from a state of rest and speeding up in some arbitrary direction.
This is an example of a symmetry argument, which have come to take a central position in modern physics due to the amount of predictive work they accomplish for relatively low intellectual effort. In order to say absolutely that uniform motion is the only possible motion of an isolated body, we need to add rotational symmetry to the argument as well. Rotational symmetry means that there is no special direction in the universe either, just as there is no special velocity. Without rotational symmetry we could say that all bodies accelerate in some special direction.
The Principle of Relativity is an example of a spacetime symmetry, which means that the laws of physics do not change under some transformation of space and time. Although it was not understood until much later, the Principle of Relativity can be treated as foundational to mechanics. From it follows the conservation of mass and momentum, which together imply all three of Newton's laws of motion. This can be established directly using Noether's theorem, but that requires Lagrangian mechanics. Instead, we will use informal reasoning.
First, I would like to consider the question of whether the first law is actually necessary because it appears to be a special case of the second law. If the force is zero, then so will the acceleration and therefore the motion will be uniform. Thus, the first law appears to follow from the second law. This is true in modern restatements of Newton's laws which are often found in textbooks because the force in the second law is clarified as referring to the net force. Under this interpretation, zero force implies zero acceleration.
However, from Newton's wording of the second law, it is clear that he is talking about a single force and not the net force. Thus, the first law establishes what happens to a body when there is no force, and in doing so establishes the nature of force as having to do with an interaction. The second law establishes the effect of a force on the body. The third law
Let us now observe that Newton's three laws can be replaced by two principles:
- Local conservation of mass
- Conservation of momentum
From these, all three laws follow. Consider Newton's first law. If an isolated body has mass
Next, what about the second law? A force is simply an instantaneous change in momentum.
However, what of the possibility that two bodies could interact by transferring mass instead of momentum?
Finally, let us think about
Quantum mechanics is a theoretical approach in which the laws of physics are probabilistic rather than deterministic. Probability is not new in physics, but before quantum mechanics it was understood to represent in imperfection in our methods rather than an inherent property of the universe. A quantum mechanical theory will never predict the result of any given experiment. It will predict only a probability distribution. Repeated results of the experiment will adhere to the predicted distribution if the theory is correct.
Like many ideas in physics, quantum mechanics is poorly named due to having been discussed before being understood. What is truely new and different in quantum mechanics is a probability distribution that derives from wave mechanics. Quantization was the first characteristic of quantum mechanics to be observed and recognized as incompatible with what the prevailing framework of physics, which is now called classical mechanics in contrast to quantum mechanics.
Quantization can be seen everywhere in physics in contexts in which quantum mechanics can be ignored. It is always a result of wave mechanics. Every object has resonant frequencies. The quantization of quantum mechanics has to do with resonant frequencies of probabality waves. Furthermore, quantization is not always characteristic of quantum mechanics. The energy levels of electrons bound to atomic nuclei are quantized, but the electrons of free electrons are not quantized. Probability wave mechanics explains both the bound and the free electron. Wave mechanics as an explanation for quantization was first proposed by Schrodinger long after it was observed and the probabilistic interpretation of these waves came even later, from Max Born.
Before this time, physicists were confused by an idea known as wave-particle duality. This concept was proposed by de Broglie, building on work from Einstein. According to this idea, phenena which previously had been understood as waves, such as light, could behave more like particles (called photons) under some circumstances and phenomena previously understood as particles, such as electrons, would behave like waves under some circumstances. These wave-particle composites obeyed the relation
Although this relation holds true for photons and electrons, it is pointing in the wrong direction because the wavelength in each case has to do with different kinds of waves. In the case of the electron, its wave-like nature is due to the probability wave of quantum mechanics, whereas for the photon, it is the electromagnetic field. It is only once probability waves are placed on top of electromagnetic waves that the quantization of the electromagnetic field is explained. Once this is understood, then the idea of wave-particle duality is revealed to be ambiguous.
The best way to begin getting a feel for quantum mechanics is through the double split experiment. A particle is released on one side of a room. A row of detectors are placed on the other side of the room. In the middle of the room is a screen with two slits, which can be open or closed. Detectors are also be placed in front of the slits. When the particle is released, it may be detected at one of the slits and may be detected at the other end of the room somewhere. It may also be lost. We only care about the subset of trials in which the particle is detected at the other end of the room. In this set of trials, the particle may have been detected earlier at one of the two slits, or it may not have been. For each of these cases, we plot the frequency of particles detected at each location at the other end of the room. If the particle was not detected earlier at either of the two slits, the plot will show a diffraction pattern. Otherwise, it will not.
Thus, the particle appears to take the form of a wave that propagates outward from the point it was last observed. The wave is never seen as a wave, only inferred from many trials. If the particle was observed at one slit, then it appears to spread outward around that one slit. If it was not observend around either slit, then it appears to have gone through both of them.
Einstein, Podosky, and Roson argued that quantum mechanics violates locality and therefore must be incomplete. They did this by establishing the concept of entangled states, which are the basis for quantum computers. An entangled state in quantum mechanics is one that predicts a correlation between observable features of spacially separated bodies. For example, two distant particles may be required to show opposite spins along any given axis of measurement, as a result of having been produced together from a state having a total spin zero in the past. Einstein, Podosky, and Rosen argued that quantum mechanics violates locality because observable quantities in quantum mechanics may be superpositions that do not take on a definite value until after an observation is made. In order to take on a definite value when observed, seemingly the two particles would have to confer with one another to ensure that their spins would be opposite. A wave function that represents true indeterminacy, therefore, must arise from our lack of knowledge about the real state of things, and cannot be a feature of the real world. The two particles must posess some quantity that we do not know how to observe that determines the observed value of their spins beforehand.
The probability wave is never observed. It can only be inferred. Furthermore, it appears that it cannot be explained in terms of anything observable. It was always understood that a theory that correctly predicted experimental results did not necessarily describe the true nature of the universe. Although both Newton and Descartes agreed that a long-range force could not describe the true nature of the universe, they had no trouble imagining such a thing. With quantum mechanics, the theory, seemingly, cannot be described in terms of our experience.
In particular, what is the nature of the probabilty wave wave that is never observed? This idea has been understood broadly along four different philosophical approaches. The first approach is to point out, correctly, that the question is metaphysical rather than scientific and therefore not worth worrying about. As scientists, this is the correct approach. Of course, nothing is stopping a scientist from being a philosopher in his spare time.
The second is the "hidden variable" approach, which means that we say that the probability wave is not truly fundamental, but an emergent quality of some more fundamental process that is not truely probabilistic. This approach was pioneered by De Broglie with his pilot wave theory. This approach appears to be unscientific because hidden variables are by definition hidden, which means we cannot distinguish them from true probability waves. Of course, one could always say that new observations will expose those hidden variables, but if you tried to
However, what is known about hidden variable theories is that they must violate Einstein's special relativity, due to the possibility of preparing entangled states that have correlations over large distances.
The second is the wave function collapse approach, which propases that the wave function is a real thing that
Finally, there is the many worlds approach, which was expounded by ... In this approach, the wave function is real and it does not collapse. A superposition of states continues on and the appearance of a collapse of the wave function is an illusion. This makes perfect sense due to the linearity of quantum mechanics. Linearity means that the sum of two valid solutons is also a valid solution. Some superposition of states would be able to exist on top of each other without interacting. They would all disagree about what they had abserved. They would all think that they are the real exerience. This is the approach that is most concordant with the mathematics of quantum mechanics. It does not require the introducion of new concepts that are not already in the equations. It also involves another universe for everything that could happen different or might have happened differently.
Finally, let's discuss quantum mechanics and consciousness. There is certainly an interesting analogy between the two but consciousness is not genuinely quantum mechanical. It's just that your mind on psychadelics can feel like a probability wave. However, you could not compute Shor's algorithm efficiently with just your imagination.
I think a better analogy with quantum mechanics is to be found in biology. Eukariotic species follow a cycle of splitting and merging, which results in a continuous mixing of their DNA. The gene pool is like the wave function and each individual is like one of the many worlds. An individual is produced from many past parents that come together, and his future are more individuals that proceed from him. Of course, in real life, most animals do not have children, which is not like quantum mechanics at all. Quantum mechanics obeys a principle of unitarity, which means probability conservation, and that prevents a Darwinian evolution of quantum states.
However, with both these analogies, the true connection is the statistical ensable, which is probabilistic but not necessarily quantum mechanical. Stastical ensables are used in thermodynamics and artificial intelligence. For example, the thought of a cat could be understood as a statistical ensable of all cats that have been experienced, including live and dead cats, as well as cartoon cats and cats from stories and books.
It is hard to explain what a gauge theory is without a lot of mathematical preparation, but physicists like to describe the world in terms of quantities that are physically real only in the way that they change but not in their absolute value. These quantities are integrals, which means that there is an arbitray additive constant to them, as anyone would have got used to seeing in a calculus class. Energy is an example of such a quantity. The potential energy is a spacial integral, and any additive constant can be added to it to produce identical physics. Physicists typically define the potential energy to ensure that bound states correspond to negative values of potential energy and unbound states to positive values, but this is done for convenience, not because it is required physically.
To put it vaguely, when the geometry of a theory works out just right, there can be integral quantities that that support a larger range of parameters than simply an additive constant. In a gauge theory, the potentials can be changed by an abstract rotation over all of space and time. Gauge invariant quantities obey partial differential equations that are elliptic and hyperbolic in different derivatives, which means that they describe an evolution of oscillation in time while at the same time obeying constraints ensuring that distant objects always remain connected by a long-range force.
An implication of gauge theories is that all long-range forces in three special dimensions are inverse square with the distance, as we have observed. Although Newton studied other kinds of forces that he could imagine, such as an inverse cube force, these kinds of forces cannot be reconciled with localism. Quantum mechanical gauge theories can lead to confinement, which means that the force is inverse square for short distances but eventually becomes stronger with distance. This is why quarks are confined to hadrons. Gauge theories also predict a conservation of charge. In the case of gravity, the charges are energy and momentum, so general relativity predicts that these are both conserved, something that had been observed long before Einstein explained it.
The breakthroughs that made Galileo the father of modern mechanics had to do with the way he approached time. Previously, physicists had generally attempted to describe the motion of bodies in terms of geometric curves, which would tend to abstract away a notion of where the body was at a given time. The biggest successes in physics prior to Galileo had to do with motion that was so fast that it could be taken to be instantaneous, as in optics, or motion that could be measured in terms of natural cycles, as in astronomy. Galileo developed new experimental methods to measure small amounts of time and a theoretical approach to motion that treated time as the independent variable.
In other words, Galileo conceptualized the motion of a body as a position function
Galileo attempted to describe physics in terms of what would now be called differential equations. Earlier researchers did not consistently take this view and instead tended to look at motion in terms of geometric curves, in the style of classical geometry. The problem with a purely geometric viewpoint is that it matters where the body is at any given time, but the geometric view abstracts this information away. For example, Galileo observed not only that free-falling bodies trace out parabolas, but that their motion parallel to the surface of the Earth is uniform. This observation adds a nuance that prefigures the later notion of a force vector which might have been missed by a classical geometer.
Galileo's breakthrough was not purely theoretical, but experimental. It began with the invention of the pendulum clock, which could be used to measure small amounts of time. Previously, researchers had success in branches of science in which there were natural clocks, such as in astronomy, or in which time can be abstracted away, as in optics, where a geometric view is appropriate because the motion of light can be taken to be instantaneous.
Since Galileo, the most fundamental known laws of physics have had the form
where
Differential equations of this form are called second-order, which means that all derivatives of
Why is physics second-order? The answer is not known, but geometry points to a possible answer. Second-order differential equations naturally arise in the theory of geodesic curves, which are the curves that minimize distance in a curved space. When space is curved, there is no such thing as a straight line but there is such a thing as the shortest curve between two points. The differential equation that characterizes such a curve is second-order, and is equivalent to saying that the curve has no acceleration tangent to the space.
In Einstein's gravity, motion in a gravitational field is geodesic, but not all motion in physics is known to be geodesic. Thus, the connection between physics and geometry is tantalizing but incomplete. The universality of second-order differential equations is certainly a very important and deep principle that remains unexplained.
In physics, he first derivative of the position is called the velocity and the second is called the acceleration.
Velocity is not the same thing as speed. Velocity is speed together with direction. There can be negative velocities. Speed is the absolute value of velocity. Thus, a law of physics looks like
If a function like
the position is determined from the acceleration up to two arbitrary additive constants of integration.
Here
Of course, the integral cannot be performed because initially we do not know acceleration as a function of time, other than at
Galileo attempted to imagine what motion without interaction would be like by extrapolating from what he could observe. He could not prepare a frictionless vaccuum, but he could observe what happened as friction and other interactions were minimized. He imagined a ball that rolled along an infinite flat plane without friction and argued that it would roll at the same velocity forever.
Motion at constant velocity is called inertial and is described by the law
Evidently, if there is no interaction then there is no acceleration. If we want something interesting to happen, we need more than one body. We will do this with an index on
Where
Spacetime is a concept that merges Galileo's viewpoint with that of the classical geometer by treating time and space together by treating time as a coordinate like those of physical space. We are only thinking about one dimension of physical space for now, so space time is a two-dimensional space with coordinates
The importance of spacetime is that it simplifies physics conceptually and mathematically, but should not be taken as having any metaphysical meaning. We are not saying that the past and future exist in places outside our experience or that we could move around in time as we do in space.
The Galilean transform is a coordinate transform of spacetime that brings us from one inertial reference frame to another. If we are on the shore watching the ship, which is moving at velocity
Every position function which undergoes a Galilean transformation must be equally valid under the laws of physics as it was before. The laws therefore must satisfy
This can be accomplished by saying that only relative positions and velocities matter.
Now that we know what non-interaction is like, what is interaction like?
Linear momentum was originally conceived by Descartes as a quantity of motion that would be conserved in all interactions. He observed that a moving ball that strikes a stationary ball will be slower after the collision whereas the ball that was struck will be faster. He proposed that some quantity of motion must have been transferred from one ball to another.
We call this quantity linear momentum because, as we will see, later physicists generalized the concept of momentum to something that is not always conserved and that is quite different from the original idea.
Linear momentum is usually written with the letter
It must be proportional to velocity, by the principle of relativity.
If this is so, then any change in velocity of a body must correspond to a change in momentum. Since changes in velocity correspond to interactions and since momentum is conserved, all interactions can be understood as transfers of momentum from one body to another. The constant of proportionality is called the mass, and it can be measured by testing how easy it is to change the velocity of an object.
If it satisfied
then it would always be conserved because every force would be cancelled out by an equal and opposite force on another body, implying that the sum of all changes to momentum would be zero. Therefore, momentum must be
This quantity we call linear momentum because, as we will see, later on in physics the word momentum was used in a generalized sense to include other quantities that are not necessarily conserved.
We will see that the law of inertia is more correct than Newton's second law. When Einstein's special theory is introduced, the principle of inertia will remain and
Having established that no interaction implies no acceleration, is it also the case that no acceleration implies no interaction? No, for two reasons. The first is that two opposite forces can cancel each other out. This is why modern textbooks often change Newton's first law to say that a body does not divert from inertial motion unless there is a net force. Second, a body can have internal motions that that change without affecting its net motion. For example, a body could grow hotter or colder. As we will see, light and other things that move at the speed of light cannot change their speed, but they can still react to forces along their direction of motion. They all have a wavelength which can widen or shrink.
Newton's second law is a more concrete version of the law of inertia. It says
where
Of course, the second law is more commonly written as
We can combine this law with the first law with
where
Newton's third law establishes the connection between a force and an interaction. It says that every interaction between two bodies
$$F_{i j} (x_i(t), v_j(t), x_j(t), v_j(t))$
where
We define a space
There are two kinds of Euclidian space. One of them is the space that was actually studied by Euclid and the other is what modern mathematicians call Euclidian space, which is quite different although superficially similar. We will use the later but explain the differences with the former. The essentials of Euclidian space, from the modern perspective, are that it comes with notions of distance and angle and that it is flat, like the tables on which geometric constructions were originally performed.
Euclid's original space had these properties but it also was two-dimensional and relied entirely on compass and straightedge constructions. A compass is a device for drawing circles and a straightedge is for drawing straight lines. It is through the use of these devices that consistent notions of distance and angle emerged in Euclid. However, they also resulted in a different notion of magnetude than we use today. Euclid did not have a concept of the real numbers, which are foundational in modern treatments. All magnetudes in Euclid are what we would now call algebraic numbers, which means that they correspond to solutions of rational polynomials. And not just any polynomial either. The intersections of straight lines with straight lines, straight lines with circles, and circles with circles correspond to basic polynomials that can be combined with one another to produce more polynomials via methods that correspond to geometric constructions. The set of polynomials that can be constructed in this manner is not the full set of rational polynomials. Consequently there are many numbers that seem quite natural to us that Euclid could not construct, for example
The limitations of compass and straight edge constructions were understood by ancient mathematicians in terms of three famous problems that they could not solve: the trisection of an angle, the doubling of the cube, and the construction of a circle with the same area as a given square. Much of modern mathematics was founded through investigations into what made these problems unsolvable by Euclid. However, physicists don't care about any of that and don't know anything about compass and straight edge constructions. Modern physicists use the real numbers, which are defined in terms of limits of rational numbers. They have no touble trisecting an angle if they feel like it.
However, the real numbers come with their own difficulties. Any sequence of rational numbers with a limit is a real number. There is a problem with determining whether two real numbers are equal, but we do not have to imagine that we ever have a set of all real numbers so that is not a problem. However, there is a potential problem due to a consequence of Godel's theorems, which is that it is not necessarily the case that two convergent sequences of rational numbers can be proven to be either equal or unequal.
Real numbers turn into a big problem in physics later when we get to quantum field theory.
Thus, the modern notion of space as continuous needs to be viewed as an idealization that we cannot currently avoid, not knowing enough about what it is really like on a small scale.
Although Euclid's work is today considered to be a foundational work of mathematics, in some ways it is better understood as a work of physics. Euclid originated the axiomatic method in mathematics, which was so influential that virtually all mathematicians have used it since then.
The flatness of Euclidian space is a consequence of Euclid's fifth postulate, which says that a point can be constructed from any two lines that are not parallel. The fifth postulate is disturbing because it implies knowledge of space over distances much bigger than any real table. The intersection of two nearly parallel lines could be arbitrarily far away. How would we have abstracted that such a point exists from our experience of real tables, compasses, and straight edges? This may have been a mistake because, as it turns out, curved spaces approach flat space over short distances. Thus, space over real tables may have been imperceptively curved the whole time.
Nowadays,
As to Euclidian space being flat, that is intuitive for two-dimensional space, but what does it mean to be flat in higher dimensions? For our purposes, flatness can be understood in terms that are intuitive to imagine. One is that all triangles have
Given that we have already discussed Newton's three laws, what is left to put in this chapter on Newtonian physics? In addition to establishing a clear concept of force, Newton also formulated physics in a way that was suitable for bodies moving freely in three dimensional space, whereas the physics of the last chapter was one-dimensional.
In modern language, Newton formulated his physics in terms of vectors.
The word vector means something different to physicists as to mathematicans. To mathematicians, a vector is an element of a vector space, whereas to physicists a vector is a field that transforms as the vector representation of the rotation group.
Field is another word that means something different to mathematicians vs physicists. To physicists, a field is a function over space. To mathematicians, a field is a set, such as R or C that supports addition, subtraction, multiplication, and division.
Rotational physics has a theory that is exactly analagous to Newtonian physics.
orientation - position angular velocity - velocity angular acceleration - acceleration moment of inertia - mass angular momentum - momentum torque - force
The complexity lies in understanding the form of rotational quantities, which are more complicated than in the Newtonian theory. In
In two dimensions, a rotation is entirely described by an angle.
In three dimensions, a rotation must be specified by both an axis and an angle.
I lied above when I said a three-dimensional rotation is a vector. It is actually a pseudo-vector.
This chapter has to do with the elliptic partial differential equations that are obeyed by gravitational, electric, and magnetic forces and it is the first step in restoring localism to physics.