All Questions
26
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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 ...
user400821
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69
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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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1
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1
answer
403
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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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1
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781
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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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1
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413
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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 ...
- 4,235
1
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3
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184
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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 ...
- 78
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93
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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 ...
- 409
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141
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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-...
- 71
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281
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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 ...
5
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456
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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 \...
- 73
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3
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221
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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 ...
- 11.9k
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182
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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 ...
- 440
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1
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1k
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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 ...
- 63
2
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1
answer
1k
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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 ...
- 356
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3
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87
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Constrained Curve in 3 Dimensions [closed]
I have a particle in a 3D space that moves on a curve of the function $$r(x)=\begin{bmatrix}x \\ x\sin(x) \\ \exp(x^2)\end{bmatrix}$$
I know that there must be 1 degree of freedom left thus $S = 3N-P$...
2
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1
answer
661
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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 ...
- 2,283
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3
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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, ...
- 31
3
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1
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674
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Generalised Coordinates in 3D Rotation
If you have N particles on a surface of a rigid body and the rigid body is rotating about some axis, we say there are six generalised coordinates for the system (N particles on the surface) and set up ...
- 31
2
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1
answer
214
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What is my state in the context of Hamiltonian mechanics?
I'm only beginning to learn the Lagrangian and Hamiltonian formulations (currently in chapter 9 of Goldstein), so please bear with me if my problem is too elementary.
I can see the point of going ...
- 545
1
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1
answer
317
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Rolling ball and number of generalized co-ordinates
Consider a sphere constrained to roll on a rough surface.
Book says it requires 5 generalized co-ordinates to specify sphere's configuration: 2 for its centre of mass and 3 for its orientation.
I ...
- 1,034
4
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1
answer
637
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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, $$
and says the number of degrees of freedom in a system is reduced by the ...
3
votes
1
answer
788
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Degrees of freedom of a point mass sliding on a rigid curved wire without friction
I am very new to the subject and am going through Structure and Interpretation of Classical Mechanics.
One exercise asks to find the degrees of freedom of a number of systems, one of which is a ...
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3
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9k
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Holonomic constraints and degrees of freedom?
Can we say that a constraint decreases the degrees of freedom of a system if and only if it is holonomic. Either way please can you explain why?
2
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3
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17k
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Definition of generalised coordinates?
I think the definition of generalised coordinates is something along the following lines:
A set of parameters that discribe the configuration of a system with respect to some refrence ...
3
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2
answers
243
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Locally accessible dimensions of configuration space
I am reading a book called "Structure and Interpretation of Classical Mechanics"
by MIT Press.While discussing configuration space and degrees of freedom,the authors remark the following:
Strictly ...
- 1,487
5
votes
2
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1k
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Why can we assume independent variables when using Lagrange multipliers in non-holonomic systems?
I'm studying from Goldstein's Classical Mechanics, 3rd edition. In section 2.4, he discusses non-holonomic systems. We assume that the constraints can be put in the form
$$f_\alpha(q, \dot{q}, t) =0, ...
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