All Questions
156
questions
2
votes
2
answers
108
views
How to explain the independence of coordinates from physics aspect and mathmetics aspect?
When I was studying Classical Mechanics, particularly Lagrangian formulation and Hamiltonian formulation. I always wondering how to understand the meaning of independence of parameters used of ...
- 529
1
vote
0
answers
51
views
How can you immediately check if a lagrangian contains a cyclic coordinate, regardless of coordinate system choice? [duplicate]
If we look at a simple cannonsball that gets shot out we quickly see the cyclic coordinate in the Lagrangian:
$$L=\frac{1}{2}m{\dot{x}}^2+\frac{1}{2}m{\dot{y}}^2-mgy$$
Since the coordinate $x$ isn't ...
- 2,035
1
vote
1
answer
255
views
Canonical equations of motion
The Hamiltonian is obtained as the Legendre transform of the Lagrangian:
\begin{equation}
H(q,p,t)=\sum_i p_i \dot{q_i} - L(q,\dot{q},t)\tag{1}
\end{equation}
If the Hamiltonian is expressed in ...
- 860
3
votes
2
answers
396
views
Hamiltonian conservation in different sets of generalized coordinates
In Goldstein, it says that the Hamiltonian is dependent, in functional form and magnitude, on the chosen set of generalized coordinates. In one set it might be conserved, but in another it might not. ...
- 860
1
vote
1
answer
232
views
Forces of constraint and Lagrangian in a half Atwood Machine with a real pulley
I was thinking about this problem and had some trouble about the constraint equation.It's just a pulley with mass and moment of inercia $I$ that is atached to two blocks, just like in the picture. And ...
- 95
2
votes
0
answers
117
views
Choosing coordinates to solve problems using Lagrangian mechanics
I am trying to obtain the equations of motion using the euler-lagrange equation.
First, let $x$ be the distance of disc R from the wall. Let $y_p$ and $y_q$ be the distance of disc P and disc Q from ...
- 21
0
votes
0
answers
23
views
What is the derivative of $z$ with respect to $\dot z$? [duplicate]
Let's say the lagrange function of my system is $L = T(z,\dot z) - m g z$ and I want to determine the equations of motion.
Why is $\frac{\partial L}{\partial \dot z} = \frac{\partial T(z, \dot z) }{\...
1
vote
0
answers
204
views
Why introduce Lagrange multipliers? [duplicate]
For a non-relativistic particle of mass $m$ with a conservative force with potential $U$ acting on the particle and a holonomic constraint given by $f(\mathbf{r},t)=0$, the system can be incorporated ...
- 1,092
1
vote
6
answers
641
views
How to define $p$ and $q$ in Hamiltonian system?
In Lagrangian mechanics, once we define $q$ which is about the position, then we automatically get $\dot q$ such that the data $(q,\dot q)$ uniquely determines the state of the system.
But in ...
- 121
1
vote
1
answer
45
views
Lagrangian energy equation with a nonholonomic constraint?
Problem 6.8 on p. 39 in David Morin's The Lagrangian Method gives a stick pivoted at the origin and rotating around the pivot with constant angular velocity $\dot{\alpha}$ (which is given as $\omega$ ...
- 51
0
votes
1
answer
396
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Calculating the Generalized force with and without the lagrangian
In my mechanics class, I learned that the components of the generalized force, $Q_i$, could be calculated using:
$$\begin{equation}\tag{1}Q_i = \sum_j \frac{\partial \mathbf{r}_j}{\partial q_i}\cdot \...
0
votes
1
answer
918
views
Deriving Lagrange equation with constraint
I'm having a hard time understanding the derivation of Lagrange equation from Newton's law when there is constraint (I'm ok with the basic case where there is only kinetic energy and potential ...
- 159
1
vote
2
answers
291
views
Choosing coordinates in Lagrangian Mechanics
Consider the problem of a hoop rolling down an inclined plane, with the plane sliding (frictionless) in a horizontal motion.
I don't know how to choose the generalized coordinates for this system. In ...
- 487
3
votes
0
answers
130
views
What is the geometric interpretation of a general 'state space' in classical mechanics?
Let $\pmb{q}\in\mathbb{R}^n$ be some n generalized coordinates for the system (say, a double pendulum). Then the 'state space' is often examined using either the 'Lagrangian variables', $(\pmb{q},\dot{...
- 477
1
vote
0
answers
44
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Is there always a transformation between canonical variables?
Let us suppose that a for a given monogenic and holonomic system we can construct two different collection of canonical variables $\{\underline{q}, \underline{p}\}$ and $\{\underline{Q}, \underline{P}\...
- 402