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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.

Definition: The minimum number of independent coordinates required to simplify the system completely along with the constraints is called the degree of freedom of a dynamical system.

For the $N$ number of particles moving freely in $d$-dimensional space, the degrees of freedom is represented by the equation $f=Nd$. If there are constraints then $f=Nd-k$,where $k$ is the number of constraints.

For example, when a single particle moves in space, it has three degree of freedom, but if it is constrained to move along a certain space curve, it has only one.

Note: Imposing constraints is a way of simplifying the problems mathematically in that the number of equations of motion are reduced to the same number as the number of degrees of freedom.

References: https://en.wikipedia.org/wiki/Degrees_of_freedom_(mechanics)

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Do Maxwell's Equations overdetermine the electric and magnetic fields?

Maxwell's equations specify two vector and two scalar (differential) equations. That implies 8 components in the equations. But between vector fields $\vec{E}=(E_x,E_y,E_z)$ and $\vec{B}=(B_x,B_y,B_z)$, there are only 6 unknowns. So we have 8…
Warrick
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In counting degrees of freedom of a linear molecule, why is rotation about the axis not counted?

I was reading about the equipartition theorem and I got the following quotations from my books: A diatomic molecule like oxygen can rotate about two different axes. But rotation about the axis down the length of the molecule doesn't count. - Daniel…
user36790
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What is the definition of how to count degrees of freedom?

This question resulted, rather as by-product, the discussion on how to count degrees of freedom (DOF). I extend that question here: Are necessary1 derivatives such as velocities counted as individual DOFs or together with the respective…
Tobias Kienzler
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Counting Degrees of Freedom in Field Theories

I'm somewhat unsure about how we go about counting degrees of freedom in classical field theory (CFT), and in QFT. Often people talk about field theories as having 'infinite degrees of freedom'. My understanding of this is that we start with…
Jojo
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Gauge-fixing of an arbitrary field: off-shell & on-shell degrees of freedom

How to count the number of degrees of freedom of an arbitrary field (vector or tensor)? In other words, what is the mathematical procedure of gauge fixing?
Moumita
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Conservation of Mathematical Constraints when deriving Energy and Momentum from $F=ma$

Background: Starting from $F = ma$, integrating with respect to time, and using basic calc, one can derive $\int Fdt = m (v_f - v_i)$ Starting from $F = ma$, integrating with respect to distance, and substituting $a\ ds = v\ dv$ (from calculus),…
user1476176
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Gauge fixing and degrees of freedom

Today, my friend (@Will) posed a very intriguing question - Consider a complex scalar field theory with a $U(1)$ gauge field $(A_\mu, \phi, \phi^*)$. The idea of gauge freedom is that two solutions related by a gauge transformation are identified…
Prahar
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Why are there only $1+3+3=7$ Additive Integrals of Motion?

1. I was reading Landau & Lifschitz's book on Mechanics, and came across this sentence on p.19: "There are no other additive integrals of the motion. Thus every closed system has seven such integrals: energy, three components of momentum, and three…
Chill2Macht
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Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics

How do I definitively show that there are only two propagating degrees of freedom in the Lorenz Gauge $\partial_\mu A^\mu=0$ in classical electrodynamics. I need an clear argument that involves the equations of motion for just the potentials $A^0$…
QuantumDot
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How to deduce $E=(3/2)kT$?

It says in my course notes for undergraduate environmental physics that a particle has so-called "kinetic energy" $$E=\frac{3}{2}kT=\frac{1}{2}mv^{2}$$ Where does this formula come from? What is $k$?
Niklas Rosencrantz
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Transverse Magnetic (TM) and Transverse Electric (TE) modes

I'm reading and working my way through "Plasmonics Fundamentals" by Stefan Maier and I've come across a step in the workings that I'm struggling to understand when working out the electromagnetic field equations at a dielectric-conductor interface.…
Tom
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Degree of freedom paradox for a rigid body

Suppose we consider a rigid body, which has $N$ particles. Then the number of degrees of freedom is $3N - (\mbox{# of constraints})$. As the distance between any two points in a rigid body is fixed, we have $N\choose{2}$ constraints giving…
yayu
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Counting degrees of freedom for gravitational waves as a gauge field

Sean Carroll has a new popularization about the Higgs, The Particle at the End of the Universe. Carroll is a relativist, and I enjoyed seeing how he presented the four forces of nature synoptically, without a lot of math. One thing I'm having…
user4552
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How do we know from representation theory that a massless spin-1 particle has only two polarizations?

In chapter 8.2.3 of Schwartz' textbook "Quantum Field Theory and the Standard Model", the author states the following, Finally, we expect from representation theory that there should only be two polarizations for a massless spin-1 particle, so the…
ersbygre1
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Number $g(T)$ of relativistic degrees of freedom as a function of temperature $T$

Let us consider the total number of relativistic degrees of freedom $g(T)$ for particle species in our universe: $$g(T)=\left(\sum_Bg_B\right)+\frac{7}{8}\left(\sum_Fg_F\right)$$ Where the sums are over the degrees of freedom for bosons ($B$) and…
Kagaratsch
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