Questions tagged [degrees-of-freedom]

This tag is for questions relating to the Degree of Freedom (DOF) of a mechanical system. It is the number of parameters that determine the state of a physical system and is important to the analysis of systems of bodies in mechanical engineering, aeronautical engineering, robotics, and structural engineering.

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Generalized coordinates of two unequal masses attached to a mass-less rigid rod

Consider a system of two particles of masses $m_1$ and $m_2$ moving in a plane. Let the respective position vectors be $\mathbf{r_1}$ and $\mathbf{r_2}$. The particles are attached at the end of a ...
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Is three dimensional supergravity dynamical?

So it is well known that standard $D = 3$ Einstein gravity is non-dynamical in the sense that the graviton has no on-shell degrees of freedom (d.o.f $= D(D-3)/2$ and the theory is topological). ...
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Gauge invariance and the degrees of freedom [duplicate]

I am currently starting to learn some QFT. And I am very confused by the fact that the freedom to choose a gauge for a theory controls the physical degrees of freedom. Please correct me if i am wrong ...
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What is the origin of the irreducible error $\propto e^{-N}$?

In this (timestamped) lecture, the professor says that any measurement includes an irreducible error that scales as $e^{-N}$ where $N$ is the number of degrees of freedom in the measurement apparatus ...
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Effective degrees of freedom of relativistic gases: Intuition for fermion factor 7/8?

When calculating the number density of a gas of identical massless particles you get the following integral $$ I_{n,\,\pm} \equiv \int_0^{\infty} \frac{u^2}{e^u\pm1} \,\text{d}u $$ with (+) for ...
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Reviewing an old argument about the number of DOFs of EM field

For free electromagnetic fields, it is possible to choose a gauge such that the scalar potential $\phi(t,{\bf x})=0$ and the vector potential ${\bf A}(t,{\bf x})$ satisfies the Coulomb gauge condition ...
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How do I calculate the bosonic/fermionic degrees of freedom from the helicity content?

After constructing a physical state and discovering the particle content, how can one find the fermionic and bosonic degrees of freedom? Eg. Constructing the physical states of an $\mathcal{N} = 2$ ...
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How do Maxwell's equations uniquely determine ${\bf E}$ and ${\bf B}$ despite no. of equations exceeding no. of unknowns?

Maxwell's equations in free space are given by $${\bf\nabla}\cdot\textbf{E}=0,~~{\bf\nabla}\cdot\textbf{B}=0$$ and $${\bf\nabla}\times\textbf{E}=-\frac{\partial\textbf{B}}{\partial t},~~{\bf\nabla}\...
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Lagrangian Mechanics - Bead sliding on a rotating rod

Say i have a bead of mass $m$ sliding on a friction-less rode (or wire) that is rotating with a permanent angular velocity $ω$. The whole system is under the influence of a uniform gravitational field ...
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Theory on domain walls

In Baryons in Quantum Chromodynamics, Zohar Komargodski have slide: I wanna understand: Why domein wall can have nontrivial worldvolume theory? When such solitonic objects have interior degrees of ...
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How many degrees of freedom does the air have?

Very simple question that I am overthinking... But how many degrees of freedom does the air have? Assuming let's say the air is confined in a rigid box.
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Regarding number of degrees of freedom of a dynamical system (as well as it's relation to number of equations of motion)

I would like to know why in the context of vibrating systems, we define degrees of freedom in terms of number of independent coordinates (by coordinates I mean the numbers which specify the components ...
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Does no-level-crossing theorem (aka avoided crossing) always hold in perturbation theory?

In perturbation, J.J. Sakurai Modern Quantum Mechanics Second Edition page 310 stated a no-level-crossing theorem stated that "a pair of energy levels connected by perturbation do not cross as ...
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About Degrees of Freedom and Energy

We know that the degree of freedom of oxygen is 5. For this the total kinetic energy of oxygen must be (5/2)nRT. But maximum books say that it should be (3/2) nRT where as our college teacher said it ...
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Degrees of freedom of constrained rigid body

A rigid body constrained at a distance $r$ from its center of mass with a ball-and-socket type constraint is considered to only have three (rotational) degrees of freedom. But isn't rigid body's ...
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Question about gauge symmetry confronted in Schwartz‘s book

This picture is from Schwartz book on QFT on page 131. I cannot understand that: What does the orange underlined sentences mean? How is Equation 8.108 derived? Could anyone kindly make some further ...
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According to equipartition theorem, how a harmonic oscillator has two degree of freedom?

I was reading a book on kinetic theory. In the part of equipartition theorem, I found a line as follows, " The average energy corresponding to one vibration is equal to twice the K.E. corresponding to ...
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Why a molecule has $3N$ degrees of freedom?

An atom has $3$ degrees of freedom because it can possess energy in $3$ independent ways- Kinetic Energy in three independent directions. Therefore, a collection of $N$ atoms will have $3N$ degrees of ...
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Difference between finite and infinitesimal motion

I am studying Arnold Sommerfeld mechanics. Here they talk about finite and infinitesimal motion. Quoted from the text: The simplest example of a non-holonomic condition is furnished by a sharp ...
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Do physical fields always have dimension of the Tangent Space at a point?

Say we only consider classical fields in 3 dimensions. In a 3 dimensional space you have scalar or vector fields, where scalar fields can be understood as vectors of one dimension. The other physical ...
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Validity of equipartition theorem and choice of coordinate axis

While reading through the basic derivation of how kinetic energy is related to temperature, I stumbled upon equipartition theorem where $\frac{1}{2}mv^2 = \frac{1}{2}kT$ thus $\frac{3}{2}kT$ in 3-...
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Wilsonion Renormalization Group in Asymptotically Free Theories

Consider some correlation function computed at some renormalization scale $\mu_0$ in an asymptotically free theory $$ \langle M(z; \mu_o) \rangle. $$ From what I understand of renormalization-group ...
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Degree of freedom of diatomic molecule [duplicate]

Is there any justification for degree of freedom of diatomic molecules to be two? I believe that degree of freedom are the number of coordinates needed to specify the position of a molecule so how ...
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What are degrees of freedom in this context?

For translational motion, $$H_\text{trans} = \frac{p_x^2}{2m} + \frac{p_y^2}{2m} + \frac{p_z^2}{2m}$$ For rotational motion, $$H_\text{rot} = \frac{1}{2} \frac{L_x^2}{I_x} + \frac{1}{2} \frac{...
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Degrees of freedom of large molecules

I know how to calculate degrees of freedom for small molecules, but how do I calculate degrees of freedom for larger molecules? For example, Benzene.
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graph relativistic degrees of freedom

I'm trying to graph the relativistic degrees of freedom, which should look like the figure And I am trying to guide me with this Phys.SE answer: Number $g(T)$ of relativistic degrees of freedom as a ...
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Number of independent components of the electromagnetic field

I'm taking a course on electrodynamics and I'm confused when we start talking about potentials. The electromagnetic field seems to have 6 independent components to me. It's described by six dynamical ...
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Spin of the particle and degrees of freedom

Wigner showed that irreducible representations of the Poincare group can be listed, depending of the mass being zero or larger then zero, as $2J+1$ dimensional representations where $J$ is half-...
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How many degrees of freedom does a system have if we only eliminate some of the constraints?

At the very bottom of page 11 in his The Variational Principles of Mechanics, Lanczos says the following: We sometimes prefer to eliminate only some of the kinematical conditions, and to leave the ...
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How constraining are conservation laws and continuity principles?

Suppose there are $N$ particles with masses $m_1, m_2, ..., m_n$. Consider the $3N$-dimensional classical configuration space of such particles. Consider some arbitrary physically possible trajectory $...
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Degrees of freedom of a molecule

We all know that if we consider a mono-atomic molecule, it has $3$ translational degrees of freedom only, along the $3$ principal coordinates of the Cartesian coordinate system. In case of a ...
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How to count degrees of freedom of metric in Newmann-Penrose formalism?

Usually a metric has 10 degrees of freedom. How to show the same in Newmann-Penrose formalism?
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Why do the degrees of freedom of a density operator not match up with the degrees of freedom of a state vector? [duplicate]

The density operator $\rho$ of a mixed 2-qubit system has $4^2-1=15$ degrees of freedom. We can require Tr[$\rho^2$] $ =1$ so that the system is in a pure state. Now we have 14 degrees of freedom. If ...
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What is the energy associated with the degrees of freedom of a nucleus?

I am following R.M. Martin in his book Electronic Structure: Basic Theory and Practical Methods. On page 11 he explains that we discriminate between properties of matter due to the electric ground ...
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Law of equipartition of energy, shouldn't kinetic energy per molecule by $(3/2)kT/f$

I study that according to the law of equipartition of energy the average kinetic energy associated with each degree of freedom is equal to $(1/2)kT$. But shouldn't it be $\frac{(3/2)kT}{f}$ where $f$ ...
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Is this a gauge symmetry?

Imagine a hypothetical action: $$S=\int \left(\frac{\partial}{\partial t}\phi(x,t)\right)^2 d^3x dt$$ Now we have a symmetry of the action: $$\phi(x,t)\rightarrow \phi(x,t)+\chi(x).$$ At time $t$, $\...
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Counting independent components of the Riemann curvature tensor

In 4D spacetime, we may choose a locally inertial frame at point P, that is we always have a transformation such that $g_{{\mu'}{\nu'}}(P) = \eta_{{\mu'}{\nu'}}$ and its first derivatives vanish. ...
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Non-holonomic constraints, degree of freedom and generalized coordinates

If a system has $N$ coordinates and $M$ number of holonomic constraints then number of degree of freedom $=N-M$ and generalized coordinates $=N-M$ too. But if there are $k$ non-holonomic constraints ...
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Count degrees of freedom of gauge tensors

For degrees of freedom (dof) it is said that spin-1 massless boson like photon has 2 dof in 4d, like U(1) gauge theory. it is said that spin-2 massless boson like photon has 2 dof in 4d, like ...
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Does a “gimbal hinge” have the same degrees of freedom as a ball-and-socket joint?

I am trying to design a joint to 3D print and considering different models. One model was a traditional spherical ball-and-socket joint. However, due to production issues, I am considering other ...
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Degree of freedom in ideal and real case

what will be the degree of freedom of a massless rod, moving freely in space with a particle which is constrained to move on it? What is meaning of massless rod?
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Infinite number of degrees of freedom

In a system with a finite number of degrees of freedom $\eta_i$, $i=1,\ldots, N$ , the partition function depends on the N external fields that may couple linearly to the $\eta_i$ in the Hamiltonian $...
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Some counting of field degrees of freedom for a classical spin-1/2 Dirac field

A classical real scalar field admits a decomposition $$\phi(x)\sim a_pe^{-ip\cdot x}+a_p^*e^{+ip\cdot x}$$ which tells that at each $x$, there exists a real number i.e., one degree of freedom at each ...
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Einstein solid degree of freedom

I was studying from Schroeder's thermal physics book. When it talks about Einstein solids it says that they have 2 degrees of freedom thus $U=NkT$ However, I thought when we talk about Einstein ...
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Physical degree of freedom and gauge fixing?

I'm confused with the gauge fixing in the Higgs mechanism. So if we have an action like $$S=\int |D\phi|-\frac{1}{4}F^2 -V(\phi) ~ ,\tag{1}$$ then expand around some non-trivial vacuum, then we have ...
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Degeneracy in Landau Levels

A subsection from "Landau Levels" from pg 21 from Lectures on Quantum Hall effect by David Tong. He shows and derives the energy of a charged particle in a planar surface under the action of a ...
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How many degrees of freedom does a spring pendulum have? [closed]

I've been looking at a spring pendulum system, but I'm not sure how many degrees of freedom it has.
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Rigorously define degrees of freedom

I want to understand if there is truly a rigorous definition for the degrees of freedom in a system. Say all of a system's physical states are contained in some set $S$. A seemingly acceptable (and I ...
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Why do we have redundant degrees of freedom?

Preliminaries: Consider the homogenous Maxwell's equations $$\partial_\mu F^{\mu\nu}=0.$$ and $$\partial_{\sigma} F_{\mu \nu}+\partial_{\mu} F_{\nu \sigma}+\partial_{\nu} F_{\sigma \mu}=0$$ Since ...
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How to know the number of constants of a free particle?

Landau-Lifshitz Mechanics says that there are $2s-1$ constants of a system with $s$ degrees of freedom (beginning of the second chapter on Conservation Laws). If this is true, for a single free ...

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