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4

The other answers are correct. I would like to add to them with an example. Take a spring, with spring constank $k$, with a mass, $m$, at one end and fixed to a large immovable object at the other. Let the only force acting on the mass be due to the spring and the difference from the equilibrium position to be $x$, which can be positive and negative. This ...


1

This is a usual term about solving differential equations. By "analytic" (or mathematical analysis), we mean finding an algebraic expression like $y=f(t)$ which satisfy the desired differential equation. But sometimes we solve the equation only at some special points. The latter method is called "numerical". Since the Newton's law (and other principal ...


2

A multi-body problem consisting of $N$ objects requires $N$ coupled differential equations that need to be solved simultaneously (if you want to find the objects trajectories in time with known initial positions) When you solve $x + 2y = 3 \text{and} x + y = 2$, this is what is known as mathematical analysis. The exact solution can be found: $$x = 1, y = ...


4

The word 'Physics' comes from the Greek Word for 'Nature' (written as 'φύση'). From Google: 'etymology of physics' physics - ˈfɪzɪks noun: physics the branch of science concerned with the nature and properties of matter and energy. The subject matter of physics includes mechanics, heat, light and other radiation, sound, electricity, magnetism, ...


2

The eigenvalue is something physicists should be familiar with. For some matrix, $A$, multiplied by some vector $\mathbf x$, we get $$ A\mathbf x=\lambda\mathbf x \tag{1} $$ where $\lambda$ is the eigenvalue, a characteristic of $A$ on $\mathbf x$. An eigenfunction is related to Equation (1). Given an operator (a differential operator in the case of quantum ...


1

Acceleration is simply a rate of change of velocity. So the magnitude tells you, how quickly velocity changes.


-2

In physics, magnitude is the size of a phusical object, a property by which the object can be compared as larger or smaller than other objects of the same kind. More formally, an object's magnitude is an ordering (or ranking) of the class of objects to which it belongs.


0

Weak interactions with $W$ and $Z$ gauge bosons violate parity simply because the righ-handed and the left-handed fermions coupled differently to $W$ and $Z$. For example, the $W$'s couple only to the left-handed fields. A parity inviariant dynamics would require that both left- and right- fields couple in the same way to the gauge vector since they get ...


0

Astronomy is mostly concerned with observing the night sky, calculating the positions and movements of the heavenly bodies and identifying objects. Astrophysics is concerned with figuring out how stars form, studying the chemical reactions within the stars, calculating what elements the stars contain and so on.


0

An ideal gas is treated as a set of $N$ indistinguishable particles with no interactions, which means the partition function of a single particle is simply, $$Z = \int d^3p \, d^3 q \, \exp \left\{ -\beta\frac{p^2}{2m}\right\}$$ and the partition function of the entire system is $Z^N/N!$. From this, we can obtain the ideal gas law, $$PV = N k_B T$$ Of ...


0

I guess it is because you first of all change sign of $\vec x$ to $- \vec x$ in physical space(this is parity transformation in a nutshell). All this peculiar algebra concerning left and right chirality fields comes from $J = 1/2$ Lorenz group representation, so transformation rules are defined as representatives of parity transformation of physical space.


-1

Okay, I think I have an idea why the terminology is used, but I think this argument makes little sense: The Lagrangian term describing weak interactions is of the form $$ \bar \Psi \gamma_\mu P_L \Psi W^\mu $$ Under parity transformations $ \Psi \rightarrow \gamma_0 \Psi$ and $ \bar \Psi \rightarrow \bar \Psi \gamma_0 $, therefore $$ \bar \Psi ...


0

An electromagnet is a made coil associated with a ferromagnetic core. This way, the strength of the magnet is controlled by the input current. A solenoid is a simple shape used in magnetostatics or magnetics. Like the plane or the sphere in electrostatics, the 1-turn coil in magnetostatics, its study is interesting because the calculus of the magnetic ...


2

An electromagnet is an electromagnet. i.e.: a (insulated) wire wrapped around an iron core that produces an electromagnetic field when current is passed through it. A solonoid uses an electromagnet to perform a mechanical function.


0

Ok, there are a lot of points here. 1) First of all, an operator in Hilbert spaces is not defined only by its action (e.g. the operation of derivation for the momentum), but also by the so-called domain of definition, i.e. the subspace of vectors of the Hilbert space where it can act. Unbounded operators are not defined for every vector of the Hilbert ...


1

A fermion is just a particle of half-integer spin. Being a lepton for a particle is a matter of definition of global symmetries of the theory. This means that a lepton can in principle be both a fermion or a boson, although all known leptons are fermions (electron, muon, tau and their neutrinos). One example of bosonic lepton is the weak triplet Higgs ...


2

If you have different Hilbert spaces, you cannot say it is the same operator on them, since operators are defined on the Hilbert space. The momentum operator is a tricky one for many systems, and rigor requires the discussion of concepts like rigged Hilbert spaces. A nice introductory discussion of this is "Mathematical surprises and Dirac's formalism in ...


15

A fermion is any particle, elementary or composite, that obeys Fermi-Dirac (as opposed to Bose-Einstein) statistics relating to how identical particles behave when you swap two of them. Due to an important but complicated result, this is taken to amount to having half-integer spin. A lepton is one type of elementary particle with spin 1/2. The only leptons ...


9

A fermion is any particle characterized by Fermi–Dirac statistics and obeying the Pauli exclusion principle. So for example quarks are fermions, as are Helium-3 atoms. A fermion does not have to be an elementary particle. I'm not even sure that it has to be spin $\tfrac{1}{2}$, though I can't think of any fermions that aren't. A lepton is a spin ...


4

The Standard Model includes 12 elementary known as fermions that respect the Pauli exclusion principle. They include six quarks (up, down, charm, strange, top, bottom), and six leptons (electron, electron neutrino, muon, muon neutrino, tau, tau neutrino) (ref) All leptons are fermions, but not all fermions are leptons.


4

It is a "crank and slider" or "slider-crank" mechanism.


1

The name you're looking for is ubiquitous gaze, or also pursuing eyes. This is an optical illusion (which means that it is exclusively a function of how our brain interprets its visual inputs) and it can be realized on completely flat canvas by appropriate handling of perspective. For more information see e.g. this HowStuffWorks page.


1

If you were looking at a real person, as you move, their images changes: from the left, you see their left side image, from the right, you see their right-side image. This phenomenon is call parallax, and is partly responsible (in addition to binocular vision), for depth perception. In monocular vision, parallax is the only cue. In a painting, there is ...


0

would notEssentially, what is going on is that the light, shadow and perspective depicted in a painting are fixed, meaning they don't shift. If a real person stared forward and you walked from side to side, his or her eyes will not follow you because the light and shadow, as well as perspective you see, actually change. Features that were close to you as ...


2

In the simplest sense of it, the Free Energy is the heat of the system minus the compulsory heat loss due to entropy. So, in short, it is the amount of "energy" left over in the system, after we consider losses due to entropy. So basically some amount of heat is wasted, and the remaining amount is useful. And this remaining amount is the Gibbs Free Energy. ...


5

Sanaris's answer is a great, succinct list of what each term in the free energy expression stands for: I'm going to concentrate on the $T\,S$ term (which you likely find the most mysterious) and hopefully give a little more physical intuition. Let's also think of a chemical or other reaction, so that we can concretely talk about a system changing and thus ...


2

There are two forces inluencing the spontaneity of a reaction: (1)The tendency of a system to attain a state of minimum energy and maximum orderedness, or stability. (2)The tendency of a system to attain a state of maximum energy and minimum orderedness , or entropy. If a system attains maximum stability, it attains mininum entropy; and if it attains ...


9

Short answer: Gibbs free energy $G = U + PV - TS$ combines internal energy $U$, pressure $P$, volume $V$, temperature $T$, and entropy $S$ into a single quantity that measures spontaneity. With that, I mean that processes that lower the Gibbs free energy of your system will spontaneously occur, and equilibrium is reached when the Gibbs free energy reaches ...


12

First, you have system with some energy, named $U$ by physicists. You think you have all the information you need to characterize the system but then some guy comes near and says: "Whoa, that's bad, the volume of your system can change." You say: "No problem, we just add here $pV$. Our new energy is $H=U+pV$." "But hey," they say, "your temperature can ...


1

is a thermodynamic potential that measures the "usefulness" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure (isothermal, isobaric). Just as in mechanics, where potential energy is defined as capacity to do work, similarly different potentials have different meanings. The Gibbs free energy is the ...


1

This is d'Alembert's principle. The basic, very general idea is to take Newton's second law applied to an accelerating mass, and write it as $F-ma=0$. That is, we take the $ma$ term and pretend it's another force balancing the $F$ term. This allows us to think about the dynamic, accelerating mass as if it's a static system. The $ma$ term is what's referred ...


0

In this case, by inertial force, they do not refer to the pseudoforce from an non intertial reference frame. Instead, by inertial force, they refer to the force due to the momentum of the fluid. This is usually expressed in the momentum equation by the term (ρv)v. So, the denser a fluid is, and the higher its velocity, the more momentum (inertia) it has.


0

In the context of e.g. a pseudo-orthogonal Lie group $$\tag{1} O(p,q)~:=~ \{\Lambda\in {\rm Mat}_{n\times n}(\mathbb{R}) ~|~\Lambda^T\eta\Lambda= \eta \} $$ of pseudo-orthogonal matrices $\Lambda$ for the metric $$\tag{2} \eta_{\mu\nu}~=~{\rm diag} (\underbrace{1,\ldots,1}_{p~\text{times}},\underbrace{-1,\ldots -1}_{q~\text{times}}), \qquad n~=~p+q,$$ ...


3

The equation you gave is indeed the definition of matrix multiplication, applied to a $d\times d$ matrix and a $d\times 1$ matrix. But the underlying concept is something more. The thing about vectors is that they exist, in some sense, independent of the numbers used to represent them. For example, an ordinary 3D displacement vector represents a physical ...


0

Yes, he defined the vector as behaving that way (a vector rotation is equivalent to a change of basis), otherwise it would not be a vector. A tensor is a different kind of object, it has at least two indexes and behaves different that a vector under transformation of coordinates (as defined in your book). You might probably read more about linear algebra up ...


2

Torque is not a force. You can say there is a torque caused by normal forces, but there is no special name for that. A normal force comes from acting with a force on an object resting next to a surface. The surface prevents the object from moving through it by producing a reaction force that is necessarily normal (perpendicular) to the surface (parallel ...


2

Both $h$ and $\tilde{h}$ are usually called weights. Their sum, $\Delta=h+\tilde{h}$ is the (scaling) dimension of the operator, while the difference, $s=h-\tilde{h}$ is called the spin. This is due to their association with scale transformations (dilatations) and rotations, respectively. To see this, note that the dilatation operator is given by ...


-1

For a real particle to be off-shell, it is sufficient to be in an external field of some sort. It is not necessary to be "absorbed". "Absorbed" are virtual particles, for example, "virtual photons" who describe non propagating fields like a Coulomb one. For a Compton scattering, the real electron is "off-shell" during interaction with the real photon, i.e., ...


1

Richard Feynman has nice words about science. It is not bad to read chapter 3 of "Feynman Lectures on Physics". The main point of his lecture is that "there is no strict boundary between different fields of science", "nature doesn't concern what we call its parts!" So, we can't look for a line that divides celestial works into astronomical or astrophysical ...


1

Astronomy talks about celestial objects (such as stars, galaxies, nebulae etc) and celestial phenomena (such as gamma ray bursts etc), their position, motion, evolution, chemistry, physics. Astrophysics is a sub-branch of astronomy to deal with physics of celestial objects and phenomena. The first encounter: A guy put microscope in front of rainbow spectrum ...


3

An ADS search for "star formation" turns up about 142,000 articles with "star formation" in the title or abstract. The first article is a 43 page review paper of Star Formation in Galaxies in the Hubble Sequence, written by Robert Kennicutt, Jr, one of the leaders of the field. He never defines anything else to mean star formation and one of the "key words" ...



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