All Questions
156
questions
1
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114
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Is the order of ordinary derivatives interchangeable in classical mechanics?
I am having trouble with a term that arises in a physics equation (deriving the Lagrange equation for one particle in one generalized coordinate, $q$, dimension from one Cartesian direction, $x$).
My ...
- 111
1
vote
1
answer
167
views
Why are there $2s -1$ independent integrals of motion?
I was reading Mechanics by Landau and Lifshitz and I am confused when it is stated in chapter 2 section 6 that one of the integrals of motion is not independent and can be considered an additive ...
- 219
2
votes
4
answers
220
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A doubt regarding $L=T-V$ and explicit time dependence
Edit: After having some clarity, I chose to write an answer instead of editing the question itself. Scroll down to read it after reading the problem that follows.
Let's say $\vec{r}=\vec{r}(q_1,q_2 ......
user292464
3
votes
1
answer
979
views
Lagrangian Mechanics: semi-holonomic constraints
By switching to a different set of coordinates, can you make problem with semi holonomic constraints into a problem with holonomic constraints? If so, then when can you do this? I wold like to know if ...
- 169
1
vote
1
answer
77
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Independence and ambiguity of holonomic constraints
I've got a couple of questions concerning constraint equations:
Suppose I've got $n$ holonomic constraint equations for a particle, how can I be sure those are all the ones there are and I didn't ...
- 468
1
vote
3
answers
146
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Partial derivatives of canonical momenta in Poisson brackets
I will simply give an example for a general doubt about the Hamiltonian formulation. So, consider the spherical pendulum of length $l$ as an example of my perhaps more general question.
The Lagrangian ...
- 387
0
votes
1
answer
350
views
2DOF robot arm dynamic model (Double Compound Pendulum - Modeling without Lagrangian)
Consider 2DOF robotic arm. No gravity. Instead of modeling it with two torque inputs at joints, I want to model it as two forces $F_1$ and $F_2$ applied at distance r (motor radius) from joints. ...
- 111
2
votes
1
answer
615
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Proof that the Euler-Lagrange equations hold in any set of coordinates if they hold in one
This is a question about a specific proof presented in the book Introduction to Classical Mechanics by David Morin. I have highlighted the relevant portion in the picture below.
In the remark, he ...
anon
0
votes
1
answer
425
views
Covariance of Euler-Lagrange equations under arbitrary change of coordinates
I'm trying to prove that the Euler-Lagrange equation
$$\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}_i})-\frac{\partial L}{ \partial q_i}=0$$
is invariant under an arbitrary change of coordinates
$$...
- 4,235
-1
votes
2
answers
620
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Conservation theorem for cyclic coordinates in the Lagrangian
Suppose $q_1,q_2,...,q_j,..,q_n$ are the generalized coordinates of a system.
$q_j$ is not there in the Lagrangian (it is cyclic).
Then $\frac{\partial L}{\partial\dot q_j}=constant$
In Goldstein, it ...
- 293
0
votes
1
answer
68
views
Are generalised coordinates necessarily independent of one another?
I'm solving the equations of motion for a spring attached to a wall with a mass $m$ on the other end that is subject to Earth's gravitational field, $\vec{g}$. An obvious set of coordinates is the ...
- 1,775
1
vote
2
answers
86
views
Contradicting Changes in a Lagrangian under Transformation
The change in a Lagrangian with no explicit time dependence $L(\mathbf{q},\mathbf{\dot q})$ can be written using the chain rule:
$$δL = \frac{\partial L}{\partial \mathbf{q}}\cdot δ\mathbf{q} + \frac{...
- 545
1
vote
2
answers
388
views
Lagrangian Dynamics of an inverted Spherical Cart Pendulum
Introduction
I have to come up with a PD-controller for an inverted Spherical Cart Pendulum, therefore I tried to compute the Dynamics of such a Pendulum.
The Spherical Cart Pendulum is a hybrid ...
22
votes
3
answers
2k
views
Equations of motion only have a solution for very specific initial conditions
An exercise made me consider the following Lagrangian
$$L = \dot{x}_1^2+\dot{x}_2^2+2 \dot{x}_1 \dot{x}_2 + x_1^2+x_2^2.\tag{1}$$
If I didn't make a mistake the equations of motion should be given by:
...
- 531
0
votes
1
answer
88
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Step in derivation of Lagrangian mechanics
There is a step in expressing the momentum in terms of general coordinates that confuses me (Link)
\begin{equation}
\left(\sum_{i}^{n} m_{i} \ddot{\mathbf{r}}_{i} \cdot \frac{\partial \mathbf{r}_{i}}{\...
- 335