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there is a universal relationship between momentum, mass, and velocity, which always takes precisely this form for every object.
When a force does act on an object, for example, by bumping into another object, the total momentum of the entire system remains conserved.
“Lagrangian mechanics” and “Hamiltonian mechanics,” which are mathematically equivalent to Newtonian mechanics
if you had perfect knowledge of the information and arbitrarily accurate calculational abilities. Laplace imagined a “vast intellect” with such capabilities, but later commentators called this hypothetical being Laplace’s Demon.
The formula[*] for the kinetic energy of an object of mass m and speed v is
physicist Émilie du Châtelet. She had translated Newton’s work into French and appreciated the conservation of momentum but argued that energy was a distinct conserved quantity.
Noether’s theorem states that every smooth, continuous symmetry transformation of a system is associated with the conservation of some quantity.
Invariance under spatial shifts leads to conservation of momentum, and invariance under temporal shifts leads to conservation of energy.
change is described using a specific framework we will call the Laplacian paradigm, after Pierre-Simon Laplace.
fact: using “derivatives” to calculate the rate of change of something, and “integrals” to calculate the total amount of change.
Huygens derived a formula for the amount of “centripetal force” you needed to exert to make this happen. (He also proposed the wave theory of light, invented the pendulum clock, and discovered Saturn’s moon Titan.
Consider two celestial objects with masses m1 and m2, and a distance r between them. Let be a “unit vector”—a vector of length 1, in whatever units we are using to measure distance—pointing from object 2 to object 1. Then Newton says that the force acting on object 2, due to the gravitational pull of object 1, is given by
The Laplacian paradigm holds that all the information we need to determine what will happen to the system, or what did happen to it in the past, is contained in the state of the system at each moment in time.
But we can take the limit of Δt and Δx as they both individually approach zero, and their ratio—the velocity, v—is well defined. We call this the derivative of the function x(t), and write it in a suggestive notation as .
We can think of differentiation and integration as operators on functions: maps from functions to other functions.
Or in symbols, =
The integral of the infinitesimal interval itself equals a finite interval: .
whereas dynamics is specifically concerned with changes that obey the equations of physics.
acceleration is the second derivative of position: .
Velocity = first derivative of position (with respect to time) Acceleration = derivative of velocity = second derivative of position Jerk = derivative of acceleration = third derivative of position Snap = derivative of jerk = fourth derivative of position Crackle = derivative of snap = fifth derivative of position Pop = derivative of crackle = sixth derivative of position
That’s because there is a preferred acceleration: zero. There is a special class of paths, known as inertial trajectories,
words, on extremely general grounds we know that the shape of the potential for an oscillating system in the vicinity of equilibrium takes the approximate form
perturbation theory: We write the equations governing the system as the sum of something simple plus a tiny perturbation, solve everything exactly for the simple part, then add in the rest bit by bit. Sometimes (not always) the universe helps us along in our attempts to understand it.
Phase space = {} for all objects i. (Curly braces {} are often used to denote a set of things.) Specifying where the system is in phase space at any moment is enough to fix its entire evolution according to Newtonian mechanics; in other words, phase space is the set of all possible states the system could be in.
The action is the integral of the Lagrangian over time, and the real physical motions are those that minimize the action.
Rate of change of momentum with time = Minus the slope of potential energy with respect to position; Rate of change of position with time = Slope of kinetic energy with respect to momentum. Together, these two relations are known as Hamilton’s equations.
“symplectic geometry”
all oscillators behave harmonically at small amplitudes.
Individually, C, P, and T are all violated in elementary-particle physics. But the combination of transformations, CPT, is conserved.
τ2 = (Δt)2 - (Δx)2.
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There is no notion of simultaneity between two separated events. There is only whether two events are inside each other’s light cones. If they are, we say the events are “timelike separated,” while if they’re outside, they are “spacelike separated.” Those are well-defined notions that every observer in the universe will agree on, while “simultaneous” is not.
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So the time component of the four-momentum is just the mass of the object divided by a velocity-dependent factor.
Energy and momentum aren’t two distinct ideas; energy is the timelike version of momentum.
The correct formula for an arbitrary infinitesimal segment is ds2 = dr2 + r2dθ2.
This is the famous metric tensor, the object that will be at the center of our attention when we turn to general relativity. Each entry in the matrix, known as a component of the metric, is a separate function of position.
Knowing the metric is precisely equivalent to knowing the line element. The relationship is simply
The idea of a function on a manifold is simple enough. It’s a map from points on the manifold to the real numbers: an assignment of a number (the value of the function at that point) to each point.
Tensors are geometric quantities that contain the requisite information to address these more involved questions. Functions and vectors are kinds of tensors,
So a vector is just a tensor with a single index, while the metric is a tensor with two indices. An ordinary function is a tensor with zero indices.
inner product, or “dot product,” between the two vectors:
Two lines intersecting at a point are perpendicular when the inner product of two vectors pointing along them at that point vanishes.
Using cosine
Any index that is summed over is called a dummy index, while one that is not summed over is a free index. Free indices can take any value—as long as the same index takes the same value in every term of an equation—while dummy indices don’t have a “value.” They are just shorthand for “add up every possible value of this index.”
the study of arbitrary curvatures on manifolds is called differential geometry.
But we already have a name for maps between collections of vectors: They are tensors. So what we really have is the Riemann curvature tensor; it inputs two vectors to define a loop and a third vector to be transported, and outputs a fourth vector to characterize the change around the loop.
defining a metric, then parallel transport and geodesics, then curvature. Those will be the mathematical tools necessary to make sense of Einstein’s general relativity.
Relativity is built on the idea that spacetime has a particular kind of metric, one with a minus sign for the timelike direction. A metric with this kind of minus sign is called Lorentzian, and if spacetime is perfectly flat we have the Minkowski metric.
