# The Theoretical Minimum  ## Metadata - Author: [[Leonard Susskind and George Hrabovsky]] - Full Title: The Theoretical Minimum - Category: #books ## Highlights - The term classical physics refers to physics before the advent of quantum mechanics. Classical physics includes Newton’s equations for the motion of particles, the Maxwell-Faraday theory of electromagnetic fields, and Einstein’s general theory of relativity. But it is more than just specific theories of specific phenomena; it is a set of principles and rules—an underlying logic—that governs all phenomena for which quantum uncertainty is not important. Those general rules are called classical mechanics. The job of classical mechanics is to predict the future. ([Location 106](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=106)) - Tags: [[blue]] - In classical physics, if you know everything about a system at some instant of time, and you also know the equations that govern how the system changes, then you can predict the future. That’s what we mean when we say that the classical laws of physics are deterministic. If we can say the same thing, but with the past and future reversed, then the same equations tell you everything about the past. Such a system is called reversible. ([Location 116](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=116)) - Tags: [[blue]] - A collection of objects—particles, fields, waves, or whatever—is called a system. A system that is either the entire universe or is so isolated from everything else that it behaves as if nothing else exists is a closed system. ([Location 120](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=120)) - Tags: [[blue]] - A world whose evolution is discrete could be called stroboscopic. ([Location 138](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=138)) - Tags: [[blue]] - A system that changes with time is called a dynamical system. A dynamical system consists of more than a space of states. It also entails a law of motion, or dynamical law. The dynamical law is a rule that tells us the next state given the current state. ([Location 139](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=139)) - Tags: [[blue]] - The variables describing a system are called its degrees of freedom. Our ([Location 153](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=153)) - Tags: [[blue]] - Because in each case the future behavior is completely determined by the initial state, such laws are deterministic. All the basic laws of classical mechanics are deterministic. ([Location 163](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=163)) - Tags: [[blue]] - According to the rules of classical physics, not all laws are legal. It’s not enough for a dynamical law to be deterministic; it must also be reversible. The meaning of reversible—in the context of physics—can be described a few different ways. The most concise description is to say that if you reverse all the arrows, the resulting law is still deterministic. Another way, is to say the laws are deterministic into the past as well as the future. Recall Laplace’s remark, “for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.” ([Location 191](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=191)) - Tags: [[blue]] - There is a very simple rule to tell when a diagram represents a deterministic reversible law. If every state has a single unique arrow leading into it, and a single arrow leading out of it, then it is a legal deterministic reversible law. Here is a slogan: There must be one arrow to tell you where you’re going and one to tell you where you came from. ([Location 208](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=208)) - Tags: [[blue]] - The rule that dynamical laws must be deterministic and reversible is so central to classical physics that we sometimes forget to mention it when teaching the subject. In fact, it doesn’t even have a name. We could call it the first law, but unfortunately there are already two first laws—Newton’s and the first law of thermodynamics. There is even a zeroth law of thermodynamics. So we have to go back to a minus-first law to gain priority for what is undoubtedly the most fundamental of all physical laws—the conservation of information. The conservation of information is simply the rule that every state has one arrow in and one arrow out. It ensures that you never lose track of where you started. ([Location 210](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=210)) - Tags: [[blue]] - When the state-space is separated into several cycles, the system remains in whatever cycle it started in. Each cycle has its own dynamical rule, but they are all part of the same state-space because they describe the same dynamical system. ([Location 254](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=254)) - Tags: [[blue]] - Whenever a dynamical law divides the state-space into such separate cycles, there is a memory of which cycle they started in. Such a memory is called a conservation law; it tells us that something is kept intact for all time. ([Location 259](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=259)) - Tags: [[blue]] - Laplace may have been overly optimistic about how predictable the world is, even in classical physics. He certainly would have agreed that predicting the future would require a perfect knowledge of the dynamical laws governing the world, as well as tremendous computing power—what he called an “intellect vast enough to submit these data to analysis.” But there is another element that he may have underestimated: the ability to know the initial conditions with almost perfect precision. Imagine a die with a million faces, each of which is labeled with a symbol similar in appearance to the usual single-digit integers, but with enough slight differences so that there are a million distinguishable labels. If one knew the dynamical law, and if one were able to recognize the initial label, one could predict the future history of the die. However, if Laplace’s vast intellect suffered from a slight vision impairment, so that he was unable to distinguish among similar labels, his predicting ability would be limited. In the real world, it’s even worse; the space of states is not only huge in its number of points—it is continuously infinite. In other words, it is labeled by a collection of real numbers such as the coordinates of the particles. Real numbers are so dense that every one of them is arbitrarily close in value to an infinite number of neighbors. The ability to distinguish the neighboring values of these numbers is the “resolving power” of any experiment, and for any real observer it is limited. In principle we cannot know the initial conditions with infinite precision. In most cases the tiniest differences in the initial conditions—the starting state—leads to large eventual differences in outcomes. This phenomenon is called chaos. If a system is chaotic (most are), then it implies that however good the resolving power may be, the time over which the system is predictable is limited. Perfect predictability is not achievable, simply because we are limited in our resolving power. ([Location 270](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=270)) - Tags: [[blue]] - To describe points quantitatively, we need to have a coordinate system. Constructing a coordinate system begins with choosing a point of space to be the origin. ([Location 290](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=290)) - Tags: [[blue]] - The next step is to choose three perpendicular axes. Again, their location is somewhat arbitrary as long as they are perpendicular. The axes are usually called x, y, and z but we can also call them x1, x2, and x3. Such a system of axes is called a Cartesian coordinate system, as in Figure 1. ([Location 293](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=293)) - Tags: [[blue]] - When we study motion, we also need to keep track of time. Again we start with an origin—that is, the zero of time. We could pick the origin to be the Big Bang, or the Birth of Jesus, or just the start of an experiment. But once we pick it, we don’t change it. Next we need to fix a direction of time. The usual convention is that positive times are to the future of the origin and negative times are to the past. We could do it the other way, but we won’t. Finally, we need units for time. Seconds are the physicist’s customary units, but hours, nanoseconds, or years are also possible. Once having picked the units and the origin, we can label any time by a number t. ([Location 308](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=308)) - Tags: [[blue]] - A vector can be thought of as an object that has both a length (or magnitude) and a direction in space. An example is displacement. ([Location 374](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=374)) - Tags: [[blue]] - Vectors can also be described in component form. We begin with three perpendicular axes x, y, z. Next, we define three unit vectors that lie along these axes and have unit length. The unit vectors along the coordinate axes are called basis vectors. The three basis vectors for Cartesian coordinates are traditionally called , , and (see Figure 14). More generally, we write , , and , when we refer to (x1, x2, x3), where the symbol ^ (known as a carat) tells us we are dealing with unit vectors. ([Location 393](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=393)) - Tags: [[blue]] - An important property of the dot product is that it is zero if the vectors are orthogonal (perpendicular). Keep this in mind because we will have occasion to use it to show that vectors are orthogonal. ([Location 429](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=429)) - Tags: [[blue]] - Let f(t) be a function of the variable t. As t varies, so will f(t). Differential calculus deals with the rate of change of such functions. The idea is to start with f(t) at some instant, and then to change the time by a little bit and see how much f(t) changes. ([Location 451](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=451)) - Tags: [[blue]] - The position of a particle is specified by giving a value for each of the three spatial coordinates, and the motion of the particle is defined by its position at every time. Mathematically, we can specify a position by giving the three spatial coordinates as functions of t: x(t), y(t), z(t). The position can also be thought of as a vector (t) whose components are x, y, z at time t. The path of the particle—its trajectory—is specified by (t). The job of classical mechanics is to figure out (t) from some initial condition and some dynamical law. ([Location 546](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=546)) - Tags: [[blue]] - The displacement is the small distance that the particle moves in the small time Δt. To get the velocity, we divide the displacement by Δt and take the limit as Δt shrinks to zero. ([Location 560](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=560)) - Tags: [[blue]] - Placing a dot over a quantity is standard shorthand for taking the time derivative. This convention can be used to denote the time derivative of anything, not just the position of a particle. For example, if T stands for the temperature of a tub of hot water, then Ṫ will represent the rate of change of the temperature with time. It will be used over and over, so get familiar with it. ([Location 565](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=565)) - Tags: [[blue]] - Acceleration is the quantity that tells you how the velocity is changing. If an object is moving with a constant velocity vector, it experiences no acceleration. A constant velocity vector implies not only a constant speed but also a constant direction. ([Location 576](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=576)) - Tags: [[blue]] - This shows an interesting property of circular motion that Newton used in analyzing the motion of the moon: The acceleration of a circular orbit is parallel to the position vector, but it is oppositely directed. In other words, the acceleration vector points radially inward toward the origin. ([Location 642](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=642)) - Tags: [[blue]] - Equations for unknown functions that involve derivatives are called differential equations. This one is a first-order differential equation because it contains only first derivatives. Equations like this are easy to solve. The trick is to integrate both sides of the equation: ([Location 810](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=810)) - Tags: [[blue]] - The analogous procedure to reversing the arrows when time is continuous is very simple. Everywhere you see time in the equations, replace it with minus time. That will have the effect of interchanging the future and the past. Changing t to −t also includes changing the sign of small differences in time. In other words, every Δt must be replaced with −Δt. In fact, you can do it right at the level of the differentials dt. Reversing the arrows means changing the differential dt to −dt. ([Location 828](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=828)) - Tags: [[blue]] - It is interesting that Aristotle’s equations do have an application—not as fundamental laws, but as approximations. Frictional forces do exist, and in many cases they are so important that Aristotle’s intuition—things stop if you stop pushing—is almost correct. Frictional forces are not fundamental. They are a consequence of a body interacting with a huge number of other tiny bodies—atoms and molecules—that are too small and too numerous to keep track of. So we average over all the hidden degrees of freedom. The result is frictional forces. When frictional forces are very strong such as in a stone moving through mud—then Aristotle’s equation is a very good approximation, but with a qualification. It’s not the mass that determines the proportionality between force and velocity. It’s the so-called viscous drag coefficient. ([Location 843](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=843)) - Tags: [[blue]] - Aristotle’s mistake was to think that a net “applied” force is needed to keep an object moving. The right idea is that one force—the applied force—is needed to overcome another force—the force of friction. An isolated object moving in free space, with no forces acting on it, requires nothing to keep it moving. In fact, it needs a force to stop it. This is the law of inertia. What forces do is change the state of motion of a body. If the body is initially at rest, it takes a force to start it moving. If it’s moving, it takes a force to stop it. If it is moving in a particular direction, it takes a force to change the direction of motion. All of these examples involve a change in the velocity of an object, and therefore an acceleration. ([Location 851](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=851)) - Tags: [[blue]] - The quantitative measure of an object’s inertia is its mass. ([Location 858](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=858)) - Tags: [[blue]] - force equals mass times the rate of change of velocity: no force—no change in velocity. ([Location 895](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=895)) - Tags: [[blue]] - Now let’s consider the unit of force. One might define it in terms of some particular spring made of a specific metal, stretched a distance of 0.01 meter, or something like that. But in fact, we have no need for a new unit of force. We already have one—namely the force that it takes to accelerate one kilogram by one meter per second per second. Even better is to use Newton’s law F = ma. ([Location 917](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=917)) - Tags: [[blue]] - This, incidently, is often referred to as Newton’s first law of motion: Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. Equations (1) and (2) are called Newton’s second law of motion, The relationship between an object’s mass m, its acceleration a, and the applied force F is F = ma. But, as we have seen, the first law is simply a special case of the second law when the force is zero. ([Location 934](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=934)) - Tags: [[blue]] - One condition for a local minimum is that the derivative of the function with respect to the independent variable at that point is zero. This is a necessary condition, but not a sufficient condition. This condition defines any stationary point, ([Location 1024](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1024)) - Tags: [[blue]] - The second condition tests to see what the character of the stationary point is by examining its second derivative. If the second derivative is larger than 0, then all points nearby will be above the stationary point, and we have a local minimum: ([Location 1027](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1027)) - Tags: [[blue]] - If the second derivative is less than 0, then all points nearby will be below the stationary point, and we have a local maximum: ([Location 1030](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1030)) - Tags: [[blue]] - If the second derivative is equal to 0, then the derivative changes from positive to negative at the stationary point, which we call a point of inflection: ([Location 1035](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1035)) - Tags: [[blue]] - There are other places where the ground is level. Between two hills you can find places called saddles. Saddle points are level, but along one axis the altitude quickly increases in either direction. Along another perpendicular direction the altitude decreases. All of these are called stationary points. ([Location 1049](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1049)) - Tags: [[blue]] - If the determinant and the trace of the Hessian are positive then the point is a local minimum. If the determinant is positive and the trace negative the point is a local maximum. If the determinant is negative, then irrespective of the trace, the point is a saddle point. However: One caveat, these rules specifically apply to functions of two variables. Beyond that, the rules are more complicated. ([Location 1083](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1083)) - Tags: [[blue]] - The fundamental forces are those that act between particles, like gravity and electric forces. These depend on a number of things: Gravitational forces between particles are proportional to the product of their masses, and electric forces are proportional to the product of their electric charges. Charges and masses are considered to be intrinsic properties of a particle, and specifying them is part of specifying the system itself. Apart from the intrinsic properties, the forces depend on the location of the particles. For example, the distance between objects determines the electric and gravitational force that one particle exerts on another. ([Location 1116](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1116)) - Tags: [[blue]] - the state of a system of particles consists of more than just their current locations; it also includes their current velocities. For example, if the system is a single particle, its state consists of six pieces of data: the three components of its position and the three components of its velocity. We may express this by saying that the state is a point in a six-dimensional space of states labeled by axes x, y, z, vx, vy, vz. ([Location 1156](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1156)) - Tags: [[blue]] - Since velocity and momentum are so closely linked, we can use momentum and position instead of velocity and position to label the points of the state-space. When the state-space is described this way, it has a special name—phase space. The phase space of a particle is a six-dimensional space with coordinates xi and pi (see Figure 1). ([Location 1187](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1187)) - Tags: [[blue]] - Configuration space plus momentum space equals phase space. ([Location 1196](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1196)) - Tags: [[blue]] - The principle of least action—really the principle of stationary action—is the most compact form of the classical laws of physics. This simple rule (it can be written in a single line) summarizes everything! Not only the principles of classical mechanics, but electromagnetism, general relativity, quantum mechanics, everything known about chemistry—right down to the ultimate known constituents of matter, elementary particles. ([Location 1390](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1390)) - Tags: [[blue]] - The process of minimizing the action is a generalization of minimizing a function. The action is not an ordinary function of a few variables. It depends on an infinity of variables: All the coordinates at every instant of time. Imagine replacing the continuous trajectory by a “stroboscopic” trajectory consisting of a million points. Each point is specified by a coordinate x, but the whole trajectory is specified only when a million x’s are specified. The action is a function of the whole trajectory, so it is a function of a million variables. Minimizing the action involves a million equations. Time is not really stroboscopic, and a real trajectory is a function of a continuously infinite number of variables. To put it another way, the trajectory is specified by a function x(t), and the action is a function of a function. A function of a function—a quantity that depends on an entire function—is called a functional. Minimizing a functional is the subject of a branch of mathematics called the calculus of variations. Nevertheless, despite the differences from ordinary functions, the condition for a stationary action strongly resembles the condition for a stationary point of a function. In fact, it has exactly the same form as Eq. (4) in Interlude 3, namely δ A = 0. ([Location 1446](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1446)) - Tags: [[blue]] - There are two primary reasons for using the principle of least action. First, it packages everything about a system in a very concise way. All the parameters (such as the masses and forces), and all the equations of motion are packaged in a single function—the Lagrangian. Once you know the Lagrangian, the only thing left to specify is the initial conditions. That’s really an advance: a single function summarizing the behavior of any number of degrees of freedom. In future volumes, we will find that whole theories—Maxwell’s theory of electrodynamics, Einstein’s theory of gravity, the Standard Model of elementary particles—are each described by a Lagrangian. The second reason for using the principle of least action is the practical advantage of the Lagrangian formulation of mechanics. ([Location 1531](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1531)) - Tags: [[blue]] - There is really nothing very general about Cartesian coordinates. There are many coordinate systems that we can choose to represent any mechanical system. For example, suppose we want to study the motion of an object moving on a spherical surface—say, the Earth’s surface. In this case, Cartesian coordinates are not of much use: The natural coordinates are two angles, longitude and latitude. Even more general would be an object rolling on a general curved surface like a hilly terrain. In such a case, there may not be any special set of coordinates. That’s why it is important to set up the equations of classical mechanics in a general way that applies to any coordinate system. ([Location 1612](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1612)) - Tags: [[blue]] - All systems of classical physics—even waves and fields—are described by a Lagrangian. ([Location 1623](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1623)) - Tags: [[blue]] - There are two ways to think about a change of coordinates. The first way is called passive. You don’t do anything to the system— just relabel the points of the configuration space. For example, suppose that the x axis is labeled with tick marks, x = . . ., −1, 0, 1, 2, . . . and there is a particle at x = 1. Now suppose you are told to perform the coordinate transformation   x' = x + 1. (4) According to the passive way of thinking, the transformation consists of erasing all the labels and replacing them with new ones. The point formerly known as x = 0 is now called x' = 1. ([Location 1734](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1734)) - Tags: [[blue]] - In the second way of thinking about coordinate transformations, which is called active, you don’t relabel the points at all. The transformation x' = x + 1 is interpreted as an instruction: Wherever the particle is, move it one unit to the right. In other words, it is an instruction to actually move the system to a new point in the configuration space. ([Location 1743](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1743)) - Tags: [[blue]] - In general, when we make a transformation, the system actually changes. If, for example, we move an object, the potential energy—and therefore the Lagrangian—may change. Now I can explain what a symmetry means. A symmetry is an active coordinate transformation that does not change the value of the Lagrangian. In fact, no matter where the system is located in the configuration space, such a transformation does not change the Lagrangian. ([Location 1747](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1747)) - Tags: [[blue]] - Since finite transformations can be compounded out of infinitesimal ones, in studying symmetries it’s enough to consider transformations with very small changes in the coordinates, the so-called infinitesimal transformations. ([Location 1800](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1800)) - Tags: [[blue]] - Now we can re-state the meaning of a symmetry for the infinitesimal case. A continuous symmetry is an infinitesimal transformation of the coordinates for which the change in the Lagrangian is zero. ([Location 1839](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1839)) - Tags: [[blue]] - For any system of particles, if the Lagrangian is invariant under simultaneous translation of the positions of all particles, then momentum is conserved. In fact, this can be applied separately to each spatial component of momentum. If L is invariant under translations along the x axis, then the total x component of momentum is conserved. Thus we see that Newton’s third law—action equals reaction—is the consequence of a deep fact about space: Nothing in the laws of physics changes if everything is simultaneously shifted in space. ([Location 1875](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1875)) - Tags: [[blue]] - The symmetry connected with energy conservation involves a shift of time. ([Location 1950](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1950)) - Tags: [[blue]] - Imagine an experiment involving a closed system far from any perturbing influences. The experiment begins at time t0 with a certain initial condition, proceeds for a definite period, and results in some outcome. Next, the experiment is repeated in exactly the same way but at a later time. The initial conditions are the same as before, and so is the duration of the experiment; the only difference is the starting time, which is pushed forward to t0 + Δt. You might expect that the outcome will be exactly the same, and that the shift Δt would make no difference. Whenever this is true, the system is said to be invariant under time translation. ([Location 1951](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1951)) - Tags: [[blue]] - Time-translation invariance does not always apply. For example, we live in an expanding universe. The effect of the expansion on ordinary laboratory experiments is usually negligible, but it’s the principle that counts. At some level of accuracy, an experiment that begins later will have a slightly different outcome than one which begins earlier. ([Location 1956](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1956)) - Tags: [[blue]] - How is time-translation symmetry, or the lack of it, reflected in the Lagrangian formulation of mechanics? The answer is simple. In those cases where there is such symmetry, the Lagrangian has no explicit dependence on time. ([Location 1963](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1963)) - Tags: [[blue]] - we can now give a very succinct mathematical criterion for time-translation symmetry: A system is time-translation invariant if there is no explicit time dependence in its Lagrangian. ([Location 1977](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=1977)) - Tags: [[blue]] - In the Hamiltonian formulation, the focus is on phase space. Phase space is the space of both the coordinates qi and the conjugate momenta pi. In fact, the q’s and p’s are treated on the same footing, the motion of a system being described by a trajectory through the phase space. Mathematically, the description is through a set of functions qi(t), pi(t). Notice that the number of dimensions of phase space is twice that of configuration space. What do we gain by doubling the number of dimensions? The answer is that the equations of motion become first-order differential equations. In less technical terms, this means that the future is laid out if we know only the initial point in phase space. ([Location 2054](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=2054)) - Tags: [[blue]] - If at any time you know the exact values of all the coordinates and momenta, and you know the form of the Hamiltonian, Hamilton’s equations will tell you the corresponding quantities an infinitesimal time later. By a process of successive updating, you can determine a trajectory through phase space. ([Location 2094](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=2094)) - Tags: [[blue]] ## New highlights added June 10, 2025 at 1:00 PM - A field is nothing but a function of space and time that usually represents some physical quantity that can vary from point to point and from time to time. ([Location 2592](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=2592)) - Tags: [[blue]] - nonsense. A field that consists of only one number at each point of space is called a scalar field. ([Location 2596](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=2596)) - Tags: [[blue]] - All magnetic fields have one characteristic feature: Their divergence is zero. Thus it follows that any magnetic field can be expressed as a curl of some auxiliary field: ([Location 2648](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=2648)) - Tags: [[blue]] - The vector potential is a peculiar field. In a sense it does not have the same reality as magnetic or electric fields. It’s only definition is that its curl is the magnetic field. A magnetic or electric field is something that you can detect locally. In other words, if you want to know whether there is an electric/magnetic field in a small region of space, you can do an experiment in that same region to find out. The experiment usually consists of seeing whether there are any forces exerted on charged particles in that region. But vector potentials cannot be detected locally. ([Location 2654](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=2654)) - Tags: [[blue]] - If the vector potential is ambiguous but the magnetic field quite definite, why bother with the vector potential at all? The answer is that without it, we could not express the principle of stationary action, or the Lagrangian, Hamiltonian, and Poisson formulations of mechanics for particles in magnetic fields. It’s a weird situation: The physical facts are gauge invariant, but the formalism requires us to choose a gauge (a particular choice of vector potential). ([Location 2688](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=2688)) - Tags: [[blue]] - Since there are many possible choices of vector potentials that describe the same physical situation, a specific choice is simply called a gauge. ([Location 2764](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=2764)) - Tags: [[blue]] - The physical principle that the result of any experiment should not depend on the gauge choice is called gauge invariance. ([Location 2766](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=2766)) - Tags: [[blue]] - The correct definition is that the momenta are the derivatives of the Lagrangian with respect to the components of velocity. This does give p = m v with the usual particle Lagrangians, but not with a magnetic field. ([Location 2775](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=2775)) - Tags: [[blue]] - The mechanical momentum is not only familiar; it is gauge invariant. It is directly observable, and in that sense it is “real.” The canonical momentum is unfamiliar and less “real”; it changes when you make a gauge transformation. But whether or not it is real, it is necessary if you want to express the mechanics of charged particles in Lagrangian and Hamiltonian language. ([Location 2814](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=2814)) - Tags: [[blue]] - Gauge fields and gauge invariance are not minor artifacts of writing the Lorentz force in Lagrangian form. They are the central guiding principles that underlie everything, from quantum electrodynamics to general relativity and beyond. They play a leading role in condensed matter physics—for example, in explaining all sorts of laboratory phenomena such as superconductivity. ([Location 2878](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=2878)) - Tags: [[blue]] - The simplest meaning of a gauge field—the vector potential is the most elementary example—is that it is an auxiliary device that is introduced to make sure certain constraints are satisfied. In the case of a magnetic field, not any is allowed. The constraint is that should have no divergence: To ensure that, we write the magnetic field as the curl of something——because curls automatically have no divergence. It’s a trick to avoid having to worry explicitly about the fact that is constrained. But we soon discover that we cannot get along without . There is no way to derive Lorentz’s force law from a Lagrangian without the vector potential. That is a pattern: To write the equations of modern physics in either Lagrangian or Hamiltonian form, auxiliary gauge fields have to be introduced. But they are also nonintuitive and abstract. Despite their being indispensable, you can change them without changing the physics. Such changes are called gauge transformations, and the fact that physical phenomena do not change is called gauge invariance. Gauge fields cannot be “real,” because we can change them without disturbing the gauge invariant physics. On the other hand, we cannot express the laws of physics without them. ([Location 2882](https://readwise.io/to_kindle?action=open&asin=B00HTQ31PE&location=2882)) - Tags: [[blue]]