All Questions
30
questions
37
votes
6
answers
66k
views
What are holonomic and non-holonomic constraints?
I was reading Herbert Goldstein's Classical Mechanics. Its first chapter explains holonomic and non-holonomic constraints, but I still don’t understand the underlying concept. Can anyone explain it to ...
- 381
7
votes
2
answers
3k
views
Covariance of Euler-Lagrange equations under change of generalized coordinates
Suppose I have an inertial frame with coordinate $\{q\}$. Now I define another reference frame with coordinate $\{q'(q,\dot q,t)\}$. I obtain the equation of motion in $\{q'\}$ in two different ways:
...
- 4,014
9
votes
5
answers
2k
views
Why are coordinates and velocities sufficient to completely determine the state and determine the subsequent motion of a mechanical system?
I am a Physics undergraduate, so provide references with your responses.
Landau & Lifshitz write in page one of their mechanics textbook:
If all the co-ordinates and velocities are ...
- 840
7
votes
2
answers
5k
views
Centrifugal Force and Polar Coordinates
In Classical Mechanics, both Goldstein and Taylor (authors of different books with the same title) talk about the centrifugal force term when solving the Euler-Lagrange equation for the two body ...
- 115
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 ...
1
vote
1
answer
472
views
Why is it important that there is no variation of time $\delta t=0$ in the definition of virtual displacement?
In Goldstein's Classical mechanics I found a proposition that I don't understand:
Similarly, the arbitrary virtual displacement $\delta \mathbf{r}_i$ can be connected with the virtual displacement $...
12
votes
2
answers
878
views
Why do we use the Lagrangian and Hamiltonian instead of other related functions?
There are 4 main functions in mechanics. $L(q,\dot{q},t)$, $H(p,q,t)$, $K(\dot{p},\dot{q},t)$, $G(p,\dot{p},t)$. First two are Lagrangian and Hamiltonian. Second two are some kind of analogical to ...
- 353
57
votes
7
answers
9k
views
Why isn't the Euler-Lagrange equation trivial?
The Euler-Lagrange equation gives the equations of motion of a system with Lagrangian $L$. Let $q^\alpha$ represent the generalized coordinates of a configuration manifold, $t$ represent time. The ...
- 1,826
7
votes
2
answers
3k
views
How do you derive Lagrange's equation of motion from a Routhian?
Given a Routhian $R(r,\dot{r},\phi,p_{\phi})$, how do you derive Lagrange's equation for $r$? Do you just solve the following for $r$?
$$\frac{d}{dt}\frac{∂R}{∂\dot{r}}-\frac{∂R}{∂r}=0.$$
And as a ...
user27771
3
votes
1
answer
3k
views
Lagrangian Mechanics: When to Use Lagrange Multipliers?
I've seen a few other threads on here inquiring about what is the point of Lagrange Multipliers, or the like. My main question though is, how can I tell by looking at a system in a problem that ...
- 181
9
votes
1
answer
3k
views
Point of Lagrange multipliers
I am trying to understand how for a constrained system the introduction of Lagrange multipliers facilitates the incorporation of the holonomic constraints. I am using Classical Mechanics by John ...
- 187
1
vote
2
answers
328
views
Geometrical picture of change of coordinates in case of Lagrangian
I was reading the part that Euler-Lagrange equation holds even on changing the coordinates. In the book by David Morin, the author talks of geometrical picture of the change of coordinates. He makes ...
- 658
27
votes
3
answers
3k
views
Lagrange's equation is form invariant under EVERY coordinate transformation. Hamilton's equations are not under EVERY phase space transformation. Why?
When we make an arbitrary invertible, differentiable coordinate transformation $$s_i=s_i(q_1,q_2,...q_n,t),\forall i,$$ the Lagrange's equation in terms of old coordinates $$\frac{d}{dt}\left(\frac{\...
- 11.8k
2
votes
1
answer
442
views
What is the function type of the generalized momentum?
Let
$$L:{\mathbb R}^n\times {\mathbb R}^n\times {\mathbb R}\to {\mathbb R}$$
denote the Lagrangian (it should be differentiable) of a classical system with $n$ spatial coordinates. In the action
$...
- 8,523
1
vote
2
answers
398
views
Total time derivatives and partial coordinate derivatives in classical mechanics
This may be more of a math question, but I am trying to prove that for a function $f(q,\dot{q},t)$
$$\frac{d}{dt}\frac{∂f}{∂\dot{q}}=\frac{∂}{∂\dot{q}}\frac{df}{dt}−\frac{∂f}{∂q}.\tag{1}$$
As part of ...
