All Questions
26
questions
5
votes
1
answer
456
views
How can one modify the Nambu-Goto action to include the longitudinal degrees of motion?
The Nambu-Goto action is given by
$$ S = -\frac{T_0}{c} \int_{-\infty}^{+\infty} d\tau \int_{0}^{\sigma} d\sigma \sqrt{ \Bigg(\frac{\partial X^\mu}{\partial \tau} \frac{\partial X_\mu}{\partial \...
- 73
5
votes
2
answers
1k
views
Why can we assume independent variables when using Lagrange multipliers in non-holonomic systems?
I'm studying from Goldstein's Classical Mechanics, 3rd edition. In section 2.4, he discusses non-holonomic systems. We assume that the constraints can be put in the form
$$f_\alpha(q, \dot{q}, t) =0, ...
- 28.3k
4
votes
1
answer
413
views
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
4
votes
1
answer
637
views
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 ...
4
votes
0
answers
281
views
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 ...
3
votes
1
answer
788
views
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 ...
- 163
3
votes
3
answers
4k
views
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
votes
1
answer
674
views
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
3
votes
2
answers
243
views
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
2
votes
1
answer
214
views
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
2
votes
3
answers
17k
views
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 ...
2
votes
1
answer
1k
views
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 ...
- 366
2
votes
1
answer
661
views
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
2
votes
0
answers
141
views
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
1
vote
3
answers
184
views
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