All Questions
Tagged with degrees-of-freedom classical-mechanics
79 questions
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How to determine the nr. of degrees of freedom (for a system), for calculating the Lagrangian?
I want to have an understanding as what constitutes a degree of freedom of a system that we consider. My understanding of it, I believe, is pretty naive. I associate it with how an object moves in ...
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What is the degrees of freedom (Lagrange equation) of two connected spool rolling down two inclines?
I'm quite confused as to how to use the Lagrange equation [second type] in a system which features a spool rolling down an incline. I think this particular example is quite representative of what is ...
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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 rod (or wire) that is rotating with a permanent angular velocity $\omega$. The whole system is under the influence of a uniform gravitational ...
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Degree of Freedom and Equations of Motion [duplicate]
I am a bit confused about the number of degrees of freedom of a simple pendulum. Is there
only angular displacement, or
angular displacement and angular velocity?
Also, I am a bit confused on the ...
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Locally accessible dimensions of configuration space
I am reading a book called "Structure and Interpretation of Classical Mechanics"
by MIT Press.While discussing configuration space and degrees of freedom,the authors remark the following:
Strictly ...
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Degrees of freedom for Constrained Motion
I'm starting to learn about Degrees of freedom, and the idea of 'constrained motion' seems strange to me, surely any particle with a predefined path is 'constrained' in its motion, We also had ...
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About Yo-yo motion and forces constraint
The purpose of the Euler-Lagrange equation, is supposed to enable us to describe a system with the fewest possible coordinates, using generalized coordinates instead of traditional ones.
However, in ...
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Do we consider a spring to be a constraint in classical mechanics. If yes/no why so?
I was brushing up on my DOF concepts before moving on to Lagrangian mechanics. One of my professors told me that a spring is not considered a constraint but his explanation was not satisfactory in my ...
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A rod is moving in space and an insect is on it. How many degrees of freedom does the insect have?
Is the answer 7? The number of degrees of freedom of a system can be viewed as the minimum number of coordinates required to specify a configuration. Applying this definition, we have:
For a single ...
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Precise Definition of Degrees of Freedom [duplicate]
I am taking Analytical Mechanics and while reading Goldstein's and LL something bothered me: can I say that a degree of freedom is an independent (generalized) coordinate?
What bothers me is that we ...
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Is the equation for degrees of freedom $f=3N-k$ valid for all cases?
Consider the example of a linear triatomic molecule. Now at low temperatures, where we can exclude vibration, quite clearly degrees of freedom, $f=5$, with 3 translational and 2 rotational degrees of ...
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How many degrees of freedom does a diatomic and triatomic molecule have at high temperatures?
I understand that a diatomic molecule has 3 translational and 2 rotational degrees of freedom. But since there is only 1 vibrational mode associated with a diatomic molecule and 1 vibrational mode is ...
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How to determine the number of constraints in this problem set?
I don't understand how to determine the number of constraints in this given problem set in classical mechanics (picture attached down below).
Now lets take a look at this problem for example where a ...
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Why can we assume independent variables when using Lagrange multipliers in non-holonomic systems?
I'm studying from Goldstein's Classical Mechanics, 3rd edition. In section 2.4, he discusses non-holonomic systems. We assume that the constraints can be put in the form
$$f_\alpha(q, \dot{q}, t) =0, ...
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How did the number of unknowns change in the Dirac equation?
So I haven't seen this argument addressed in any textbook which makes me doubt it's legitimacy. Here goes:
Since Newton's $F=ma$ is essentially a second-order differential equation. Any equation of ...
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Degree of freedom and Grubler formula
I am attempting to apply the Grubler formula (which can be found here: https://learnmech.com/how-to-calculate-degree-of-freedom-of/) to determine the number of degree of freedom, but it does not seem ...
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Doubt in Arnold's "Mathematical Methods of Classical Mechanics", Chapter 2
My question is about Arnold's book "Mathematical Methods of Classical Mechanics", chapter 2, section B (pg. 16).
He talks about systems with one degree of freedom, i.e. systems described by $...
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Elastic collisions and internal degrees of freedom
As I was considering elastic collisions today a question popped into my head. Do elastic collisions imply that there are no internal degrees of freedom in the colliding objects which couple ...
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Number degrees of freedom for Sphere on a inclined plane
How many generalized coordinates are required to describe the dynamics of a solid sphere is rolling without slipping on an inclined plane?
What I think is that there two translational degrees of ...
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Dynamics: why do physicists include derivatives like $\dot{\theta}$ in the state space for a system like a pendulum?
I come from statistics, so my experience with physics is spotty, especially on some simple stuff. I have been working on some applications related to control theory lately, and was looking at some ...
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How many DOF does this system have?
I saw the problem above and thought it would be fun to solve it using lagrangians. However, in order to do this, one has to know the DOF of the system. And this is where it gets confusing for me. ...
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Degree of freedom in Lagrange's formalism
Degrees of freedom $=3K-N$ where $K$ is number of particles and $N$ is number of constraints. How to find the number of degrees of freedom for a rigid body which has both translation and rotation, ...
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How much degree of freedom of a rigid body in $N$-dimensional space?
Well I have the answer it is
$\frac{N(N+1)}{2}$ but what the procedure to derive it .
I tried this.
1).I have $N$ number of translation freedom.
To calculate the number of rotational freedom I tried ...
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Why are $p$ and $q$ independent variables in Hamiltonian formalism?
Let's say we have $(q, \dot{q})$ as the generalised coordinate and generalised velocity. If we have a Lagrangian given by
$$L=Aq\dot{q}+Bq$$
where $A$ and $B$ are constants that give the right units ...
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Why should degrees of freedom be independent?
To define the position of a system of $N$ particles in space, it is necessary to specify $N$ radius vectors, i.e. $3N$ co-ordinates. The number of independent quantities which must be specified in ...
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What does it mean, when one says that system has $N$ constants of motion?
For example for an isolated system the energy $E$ is conserved. But then any function of energy, (like $E^2,\sin E,\frac{ln|E|}{E^{42}}$ e.t.c.)
is conserved too. Therefore one can make up infinitely ...
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Why not a $(q,\dot{q})$ space in Lagrangian Mechanics?
We know that the Lagrangian $\mathcal{L}(q,\dot{q},t)$ which is function of generalized co-ordinate, generalized velocity and time. We consider the dynamics of particle is in configuration space. But ...
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Rigorous definition of degrees of freedom
According to this Wikipedia article, the definition of degrees of freedom is:
The degree of freedom (DOF) of a mechanical system is the
number of independent parameters that define its ...
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Independent coordinates of a rigid body
This is a quote from Classical mechanics by Goldstein:
"To fix a point in a rigid body, it is not necessary to specify its distances to all other points in the body ; we need only state the ...
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Representation of Holonomic Constraints by independent generalized coordinates
Say we have a system with N particles described by N position vectors: $\{\vec{r_{i}}\};$ $i=1,...N$
Say we have a holonomic constraint: $$f(\{\vec{r_{i}}\},t)=0 \tag{1}$$
Since we have one holonomic ...
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Understanding the Degrees of freedom of a Ballbot
A Ball Balancing Robot is dynamically stable robot capable of omnidirectional motion. It possesses non-holonomic properties and is a special case of underactuated system, classified as a Shape-...
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Pendulum constrained by a spring and generalized forces [closed]
I've been going through some problem sets used in a classical mechanics course offered a few semesters ago as a way to prepare for when I have to take that course next semester and I've hit a snag ...
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Question about holonomic constraints
Goldstein says that when a system of $N$ particles is subject to $k$ holonomic constraints, the positions $\mathbf{r}_1, \dots, \mathbf{r}_N$ can be parameterized by $3N - k$ independent coordinates $...
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How to count degrees of freedom
I am able to visualize and see that $y=A \sin{x}$ has 1 degree of freedom, because $z=0$ and $y$ depends on $x$. However, its plot looks like a 2D plane, even though according to the DOF it is a 1D ...
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What are the degrees of freedom of a dumbbell?
Edit 1: May be I should modify my question after getting the answers. I see why $(X_c, Y_c, Z_c, \theta, \phi)$ are legitimate Dof's of the dumb-bell, I never had any problem with that.
Please ...
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How do the degrees of freedom change
I came across the following problem in an old exam:
How many degrees of freedom does a system of 4 mass points (A,B,C,D) have, if the distances AB, BC and CD are given?
So my attempt was to say ...
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How can one modify the Nambu-Goto action to include the longitudinal degrees of motion?
The Nambu-Goto action is given by
$$ S = -\frac{T_0}{c} \int_{-\infty}^{+\infty} d\tau \int_{0}^{\sigma} d\sigma \sqrt{ \Bigg(\frac{\partial X^\mu}{\partial \tau} \frac{\partial X_\mu}{\partial \...
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Holonomic constraints and degrees of freedom
Wikipedia and other sources define holonomic constraints as a function
$$ f(\vec{r}_1, \ldots, \vec{r}_N, t) \equiv 0, $$
and says the number of degrees of freedom in a system is reduced by the ...
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Understanding dependent/independent variables in physics
How does one determine the independent and dependent variables?
What do the terms mean?
Can they be derived from a formula?
For example I saw in a textbook $F = k\Delta l$, Hooke's Law, that $F$ is ...
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Number of degrees of freedom if independent coordinates are functions of time
If there are $n$ independent coordinates for a system of particles, but all of them are functions of time, is it correct to say that the degree of freedom is 1? If we know the instant of time we're ...
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How to determine the number of generalized coordinates in the following example?
Question:
In this exercise one needs to determine the generalised coordinates. We have a pendulum in a magnetic field $B$. The pendulum rotates around its axis with an angular velocity $ω$ on the ...
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Degrees of freedom for a bead on a parabolic wire?
How many degrees of freedom does a bead on a parabolic wire have? I think it must be two degrees of freedom since the bead is constrained to move on the wire (up, down motion and left/right motion). ...
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How do you find the degrees of freedom of a rigid body moving parallel to a fixed plane surface? [closed]
Find the degrees of freedom of a rigid body moving parallel to a fixed plane surface.
I know the definition of degrees of freedom which meant we need minimum number of coordinate to specify something....
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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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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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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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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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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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How to determine whether a set of coordinates are independent and sufficient to determine the system completely?
In Analytical mechanics, when we formulate our principles, in general, it is assumed that we start with a cartesian coordinate system, and then find some generalised coordinates $q_j$s they are all ...
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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 ...