All Questions
156
questions
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
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
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 $$\frac{d}{dt}\left(\frac{\...
- 11.8k
22
votes
3
answers
2k
views
Equations of motion only have a solution for very specific initial conditions
An exercise made me consider the following Lagrangian
$$L = \dot{x}_1^2+\dot{x}_2^2+2 \dot{x}_1 \dot{x}_2 + x_1^2+x_2^2.\tag{1}$$
If I didn't make a mistake the equations of motion should be given by:
...
- 531
13
votes
1
answer
1k
views
Finding generalized coordinates when the implicit function theorem fails
Given some coordinates $(x_1,\dots, x_N)$ and $h$ holonomic constraints, it should always be possible to reduce the coordinates to $n=N-h$ generalized coordinates $(q_1,\dots, q_n)$. This is ...
- 261
12
votes
2
answers
878
views
Why do we use the Lagrangian and Hamiltonian instead of other related functions?
There are 4 main functions in mechanics. $L(q,\dot{q},t)$, $H(p,q,t)$, $K(\dot{p},\dot{q},t)$, $G(p,\dot{p},t)$. First two are Lagrangian and Hamiltonian. Second two are some kind of analogical to ...
- 353
9
votes
5
answers
2k
views
Why are coordinates and velocities sufficient to completely determine the state and determine the subsequent motion of a mechanical system?
I am a Physics undergraduate, so provide references with your responses.
Landau & Lifshitz write in page one of their mechanics textbook:
If all the co-ordinates and velocities are ...
- 840
9
votes
1
answer
3k
views
Point of Lagrange multipliers
I am trying to understand how for a constrained system the introduction of Lagrange multipliers facilitates the incorporation of the holonomic constraints. I am using Classical Mechanics by John ...
- 187
7
votes
2
answers
3k
views
Covariance of Euler-Lagrange equations under change of generalized coordinates
Suppose I have an inertial frame with coordinate $\{q\}$. Now I define another reference frame with coordinate $\{q'(q,\dot q,t)\}$. I obtain the equation of motion in $\{q'\}$ in two different ways:
...
- 4,014
7
votes
2
answers
2k
views
Why is this a non-holonomic constraint?
Wikipedia states:
holonomic constraints are relations between the position variables (and possibly time1) which can be expressed in the following form:
$$f(q_{1},q_{2},q_{3},\ldots ,q_{n},t)=0$$
...
- 1,772
7
votes
2
answers
5k
views
Centrifugal Force and Polar Coordinates
In Classical Mechanics, both Goldstein and Taylor (authors of different books with the same title) talk about the centrifugal force term when solving the Euler-Lagrange equation for the two body ...
- 115
7
votes
3
answers
928
views
Do generalized coordinates have to be orthogonal?
I've been picking up concepts on mechanics from online posts and Wikipedia pages, so please forgive my limited understanding. I'm currently trying to figure out whether it's okay to choose an ...
- 196
7
votes
2
answers
3k
views
How do you derive Lagrange's equation of motion from a Routhian?
Given a Routhian $R(r,\dot{r},\phi,p_{\phi})$, how do you derive Lagrange's equation for $r$? Do you just solve the following for $r$?
$$\frac{d}{dt}\frac{∂R}{∂\dot{r}}-\frac{∂R}{∂r}=0.$$
And as a ...
user27771
6
votes
2
answers
330
views
Generalized vs curvilinear coordinates
I am taking the course "Analytical Mechanics" (from on will be called "AM") this semester. In our first lecture, my professor introduced the notion of generalized coordinates. As ...
- 150
6
votes
1
answer
3k
views
Lagrangian of a 2D double pendulum system with a spring
In the figure above (please excuse my Picasso drawing skills), we have the general 2D double pendulum system with a slight modification, there's a spring connecting the masses instead of the usual ...
- 133