All Questions
156
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Generalized momentum
I am studying Hamiltonian Mechanics and I was questioning about some laws of conservation:
in an isolate system, the Lagrangian $\mathcal{L}=\mathcal{L}(q,\dot q)$ is a function of the generalized ...
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1
vote
1
answer
56
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Confusing Goldstein Statement about Magnitude of the Lagrangian
On page 345 of Goldstein's Classical Mechanics 3rd Ed., he writes:
...the Hamiltonian is dependent both in magnitude and in functional form upon the initial choice of generalized coordinates. For the ...
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4
votes
3
answers
152
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Analyzing uniform circular motion with Lagrangian mechanics
Consider swinging a ball around a center via uniform circular motion. The centripetal acceleration is provided by the tension of a rope. Now, is this force a constraint force? If it is, since it is ...
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votes
1
answer
76
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Derivation of lagrange equation in classical mechanics
I'm currently working on classical mechanics and I am stuck in a part of the derivation of the lagrange equation with generalized coordinates. I just cant figure it out and don't know if it's just ...
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votes
2
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330
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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 ...
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2
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91
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How to change generalised coordinates in a Lagrangian without inverting the coordinate transformation?
Given a Lagrangian using the standard cartesian coordinates.
$$ \mathcal{L} = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2) - \frac{1}{2}k(x^2 + y^2) $$
How to move to the hyperbolic coordinates given as
$$2 x ...
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2
votes
1
answer
122
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Independence of generalized coordinates in the derivation of Lagrange equations from d'Alembert's Principle
I am confused by this remark in the derivation of Lagrange equations from d'Alembert's principle in Goldstein:
I am not comfortable that I understand why, at this late stage of the derivation, they ...
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2
answers
296
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Lagrangian mechanics and generalized coordinates
In Lagrangian mechanics, we use what is called the generalized coordinates (gc's) as the variable of the machanics problem in hand. These gc's represent the degrees of freedom that the studied system ...
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1
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95
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Virtual work of constraints in Hamilton‘s principle
Goldstein 2ed pg 36
So in the case of holonomic constraints we can move back and forth between Hamilton's principle and Lagrange equations given as $$\frac{d}{d t}\left(\frac{\partial L}{\partial \...
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1
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48
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How to determine which coordinates to use for calculating the Hamiltonian? [closed]
In my classical mechanics course, I was tasked with finding the Hamiltonian of a pendulum of variable length $l$, where $\frac{dl}{dt} = -\alpha$ ($\alpha$ is a constant, so $l = c - \alpha t$.).
I ...
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answers
77
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Constraint force using Lagrangian Multipliers
Consider the following setup
where the bead can glide along the rod without friction, and the rod rotates with a constant angular velocity $\omega$, and we want to find the constraint force using ...
- 193
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47
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Can anyone in here explain to me what exactly is 'Quasi-Generalised Co-ordinates'?
This comes straight up from a certain text that I was going through, which of course is in the form of a question which asks 'A solid cylinder is rolling without slipping and how many generalized co-...
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1
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72
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Why are constraint forces and gradient of constraint functions perpendicular?
My question is about the general relationship between the constraint functions and the constraint forces, but I found it easier to explain my problem over the example of a double pendulum:
Consider a ...
- 193
2
votes
6
answers
238
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Lagrangian - How can we differentiate with respect to time if $v$ not a function of time?
In the Lagrangian itself, we know that $v$ and $q$ don't depend on $t$ (i.e - they are not functions of $t$ - i.e., $L(q,v,t)$ is a state function.)
Imagine $L = \frac{1}{2}mv^2 - mgq$
Euler-Lagrange ...
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1
answer
72
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Requirement of Holonomic Constraints for Deriving Lagrange Equations
While deriving the Lagrange equations from d'Alembert's principle, we get from $$\displaystyle\sum_i(m\ddot x_i-F_i)\delta x_i=0\tag{1}$$ to $$\displaystyle\sum_k (\frac {\partial\mathcal L}{\partial\ ...
- 193
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2
answers
108
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How to explain the independence of coordinates from physics aspect and mathmetics aspect?
When I was studying Classical Mechanics, particularly Lagrangian formulation and Hamiltonian formulation. I always wondering how to understand the meaning of independence of parameters used of ...
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0
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51
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How can you immediately check if a lagrangian contains a cyclic coordinate, regardless of coordinate system choice? [duplicate]
If we look at a simple cannonsball that gets shot out we quickly see the cyclic coordinate in the Lagrangian:
$$L=\frac{1}{2}m{\dot{x}}^2+\frac{1}{2}m{\dot{y}}^2-mgy$$
Since the coordinate $x$ isn't ...
- 2,035
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1
answer
255
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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 ...
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votes
2
answers
396
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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
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1
answer
232
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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 ...
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0
answers
117
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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 ...
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23
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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
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0
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204
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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
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6
answers
641
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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 ...
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1
answer
45
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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$ ...
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1
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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 \...
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votes
1
answer
918
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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
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2
answers
291
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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 ...
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0
answers
130
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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{...
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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}\...
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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 ...
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1
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167
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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 ...
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4
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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
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1
answer
981
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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
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1
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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
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1
answer
350
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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. ...
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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
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votes
1
answer
425
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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
$$...
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2
answers
620
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Conservation theorem for cyclic coordinates in the Lagrangian
Suppose $q_1,q_2,...,q_j,..,q_n$ are the generalized coordinates of a system.
$q_j$ is not there in the Lagrangian (it is cyclic).
Then $\frac{\partial L}{\partial\dot q_j}=constant$
In Goldstein, it ...
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1
answer
68
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Are generalised coordinates necessarily independent of one another?
I'm solving the equations of motion for a spring attached to a wall with a mass $m$ on the other end that is subject to Earth's gravitational field, $\vec{g}$. An obvious set of coordinates is the ...
- 1,775
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2
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86
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Contradicting Changes in a Lagrangian under Transformation
The change in a Lagrangian with no explicit time dependence $L(\mathbf{q},\mathbf{\dot q})$ can be written using the chain rule:
$$δL = \frac{\partial L}{\partial \mathbf{q}}\cdot δ\mathbf{q} + \frac{...
- 545
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2
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388
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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
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3
answers
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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:
...
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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}}{\...
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1
answer
56
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Energy change under point transformation
How do the energy and generalized momenta change under the following
coordinate
transformation $$q= f(Q,t).$$
The new momenta: $$P = \partial L / \partial \dot Q = \partial L / \partial \dot q\times ...
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781
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Degrees of freedom for Constrained Motion
I'm starting to learn about Degrees of freedom, and the idea of 'constrained motion' seems strange to me, surely any particle with a predefined path is 'constrained' in its motion, We also had ...
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61
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Can we take any physical quantity as a generalized co-ordinate in Lagrangian function?
Lagrangian function is a function of generalized co-ordinates $q_1,q_2,....$ & possibly of time $t$.
i.e. $L=L(q_1,q_2,....;t)$
Consider a simple pendulum.
Can I take
$q_1$ = kinetic energy of ...
- 1,034
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1
answer
98
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Why is this the requirement for invertibility within the context of canonical transformations in mechanics?
I'm reading "Analytical Mechanics", by Hand and Finch. In page 210, there's the following statement, regarding the generating function $F$ for some lagrangian such that $L'=\lambda L-\frac{\...
- 1,775
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1
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212
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I'm studying analytical mechanics and it states that it always true that generalized coordinates times generalized forces have the dimension of energy
Since the terms $q$ "generalized coordinates" are not necessarily ‘lengths’, the quantities $Q$ "generalized forces" also do not necessarily have the dimension of a ‘force’. ...
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