All Questions
Tagged with classical-mechanics lagrangian-formalism
251
questions with no upvoted or accepted answers
8
votes
0
answers
336
views
Euler-Lagrange equations for chain fountain
Most of us are familiar with chain fountains.
I was wondering how this phenomenon is explained in the Lagrangian mechanics. I mean do we know how the Euler-Lagrange equations look like for this system?...
- 3,464
7
votes
0
answers
135
views
Variational principle with $\delta I \neq 0$
In Covariant Phase Space with Boundaries D. Harlow allows boundary terms in the variation of the action. If we have some action $I[\Phi]$ on some spacetime $M$ with boundary $\partial M = \Gamma \cup \...
- 36.4k
6
votes
0
answers
413
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Is there a modified Least Action Principle for nonholonomic systems?
We know that one can treat nonholonimic (but differential) constraints in the same manner as holonimic constraints. With a given Lagrange Function $L$, the equations of motion for a holonomic ...
- 6,763
5
votes
0
answers
60
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Group theoretical approach to conservation laws in classical mechanics
I'm doing some major procrastination instead of studying for my exam, but I wanted to share my thought just to confirm if I'm right.
Suppose that the action, $S(\mathcal{L})$ forms the basis of a ...
- 372
5
votes
0
answers
339
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Lagrangian of the Euler equations - why are Lin constraints required?
The following equation describes the motion of a rigid body rotation, such as a gyroscope:
$$
\frac{d\textbf{L}}{dt} ={\bf{\tau}}= \textbf{r}\times m\textbf{g}= {\omega}\times \textbf{L}$$
where $...
- 185
5
votes
0
answers
950
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Intuition behind the principle of virtual work
To derive Lagrange's Equations we need the principle of virtual work first. This principle states that whenever a system of $K$ particles is constrained to a submanifold $\mathcal{M}\subset \mathbb{R}^...
- 36.4k
5
votes
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470
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Naive questions on the classical equations of motion from the Chern-Simons Lagrangian
Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial _0a_i-...
- 3,744
4
votes
0
answers
170
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Noether's Theorem in non-conservative systems
In most books on classical mechanics, Noether's Theorem is only formulated in conservative systems with an action principle. Therefore I was wondering if it is possible to also do that in non-...
- 470
4
votes
0
answers
72
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Classical "bird flocking" Hamiltonian with velocity-velocity interaction
Consider the following classical Lagrangian with an interaction between velocities:
$$\mathcal{L} = \sum_{i} \frac{1}{2}m \mathbf{v}_{i}^{2} + \sum_{i < j} J(r_{ij}) \hat{\mathbf{v}}_{i} \cdot \hat{...
- 1,330
4
votes
0
answers
81
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Cases of various time symmetries
Is it possible to cook up three physically relevant examples where the Lagrangian has explicit time dependence but the system still has one of the following?
time-reversal invariance,
time ...
- 11.8k
4
votes
0
answers
178
views
Deriving the Lagrangian of a set of interacting particles only from symmetry
In section 5 of Landau and Lifshitz's Mechanics book, they show that the Lagrangian of a free particle must be proportional to its velocity squared, $\mathcal{L} = \alpha\mathbf{v}^2$ using only ...
- 2,157
4
votes
0
answers
721
views
Equilibrium points of three masses on a rigid spring ring with gravity
I'm trying to find the equilibrium points of a given system using Lagrangian mechanics (the system is still not rotating at the beginning).
should I find the diagonal matrix for the characteristic ...
4
votes
0
answers
230
views
Closed trajectories for Kepler problem with classical spin-orbit corrections?
Kepler problem explains closed elliptic trajectories for planetary systems or in Bohr's classical atomic model - let say two approximately point objects, the central one has practically fixed position,...
- 960
3
votes
0
answers
101
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Meaning of equations associated with Legendre transform
In the famous paper about semiclassical Bloch theory https://arxiv.org/abs/cond-mat/9511014, the Lagrangian
\begin{eqnarray}
L (\mathbf{k},\dot{\mathbf{k}}) = -e \delta \mathbf{A}(r,t)\cdot\dot{\...
3
votes
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{...
- 477
3
votes
0
answers
138
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Particle in "external potential" VS particle on "curved surface": equivalence?
Let's consider a non-relativistic particle - its position is $x(t)\in \mathbb{R}^n$ - in an external potential $\phi$, with Lagrangian $$L=\dot{x}^i \eta_{ij}\dot{x}^j/2 - \phi(x),\tag{1}$$
where $\...
- 5,078
3
votes
0
answers
121
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Intuitive explanation on why velocity = 0 for a inverted pendulum on a wheel system
I believe I have solved below problem. I am not looking for help on problem-solving per se. I am just looking for an intuitive explanation.
Problem statement: wheel mass = $m_1$, even mass rod BC mass ...
- 131
3
votes
0
answers
77
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Is there a unique accepted Lagrangian formulation of Nambu mechanics?
In section 5 of their 2000 paper "Nambu Mechanics in the Lagrangian Formalism", Ogawa & Sagae critique previous attempts by Bayen & Flato and by Takhtajan to formulate the theory ...
- 3,684
3
votes
0
answers
122
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Physical interpretation of the definition of angular momentum in classical mechanics
To what I understand, the following is a valid way to introduce the angular momentum $\mathbf L$ in the Lagrangian system of a rigid body. We can consider the extended configuration space to be $M\...
3
votes
0
answers
396
views
Are all canonical transformations either a point transformation, gauge transformation or a combination of them?
It's regularly argued that in the Hamiltonian formalism, we have more freedom to choose our coordinates and that this is arguably its most important advantage.
To quote from two popular textbooks:
[S]...
- 10.1k
3
votes
0
answers
830
views
Gauge freedom in Lagrangian corresponds to canonical transformation of Hamiltonian
I want to show that the gauge transformation
$$L(q,\dot{q},t)\mapsto L^\prime(q,\dot{q},t):=L(q,\dot{q},t)+\frac{d}{dt}f(q, t)$$
corresponds to a canonical transformation of the Hamiltonian $H(p, q, ...
- 1,187
3
votes
0
answers
141
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Relativistic configuration space in classical mechanics
Okay so a couple of questions. Firstly I realise that in order to study the dynamics of one particle (classically), we define the Lagrangian and Hamiltonian to be the maps from the tangent and ...
- 504
3
votes
0
answers
276
views
Lagrangian formulation of a themodynamics problem
I was wondering whether it is possible to derive the model of a thermodynamical system by combining thermodynamic equations and Lagrangian mechanics.
Let's consider the following closed system.
A ...
- 43
3
votes
0
answers
714
views
Non-canonical transformation
I would like to know any method to transform a known non-canonical set of variables to a canonical set for a given system. The Lagrangian and Hamiltonian are known in the non-canonical variables. I ...
- 81
3
votes
0
answers
222
views
Does the additivity property of Integrals of motion and Lagrangians valid in all situations?
I would like to know if the additivity property of an integral (constant) of motion valid in all situations ? It works for energy but does it work for all other integrals of motion in all kinds of ...
- 345
3
votes
0
answers
606
views
A question on Lagrangian dynamics an the velocity phase space
I've struggled in the past with understanding why we can treat position and velocity as independent variables in the Lagrangian, but I think I may have finally become a bit more enlightened on the ...
- 3,063
3
votes
2
answers
243
views
Locally accessible dimensions of configuration space
I am reading a book called "Structure and Interpretation of Classical Mechanics"
by MIT Press.While discussing configuration space and degrees of freedom,the authors remark the following:
Strictly ...
- 1,487
2
votes
0
answers
37
views
Physical model described by modify Helmholtz equation
The wave equation $\partial_t^2u=c\Delta u$ is usually handled through a time-harmonic ansatz, which reduces it to Helmholtz equation $\Delta u+\omega^2u=0$.
I'm interested in the following modified ...
- 121
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 ...
- 31
2
votes
1
answer
72
views
Some doubts about action symmetry
We know that Symmetry of the Lagrangian ($\delta L = 0$) always yields some conservation law.
Now, if $\delta L \neq 0$, that doesn't mean we won't have conservation law, because if we can show action ...
- 525
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
2
votes
0
answers
57
views
What are the extra terms in the generalized momentum regarding the Lagrangian formalism?
In the lectures, we have defined the generalized momentum in the Lagrangian to be:
$$p_i=\frac{\partial L}{\partial\dot q_i}.$$
But with this definition, if we do not make any assumptions about the ...
- 193
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
2
votes
1
answer
104
views
Derivatives of the multipliers appearing in the Euler-Lagrange equation
(This is a crossed post where physical considerations can be helpful.)
Let $\Omega\subset \mathbb{R}^2$ be some simply connected domain and consider the functional
\begin{align}
V\left[u(x,y),v(x,y),w(...
2
votes
1
answer
117
views
Do Legendre transformation form a group?
In my classical mechanics class, my professor asked if Legendre transformations form a group, and in my little knowledge about groups, I know that a transformation group consists of a set of ...
- 21
2
votes
0
answers
44
views
How are conjugate variables in mechanics and stat mech related to duality in convex optimization?
I recently studied duality in optimization where a primal optimization problem can be casted as a dual problem which provides meaningful lower bounds on the primal. There is also a notion of conjugate ...
- 131
2
votes
0
answers
141
views
Understanding the Degrees of freedom of a Ballbot
A Ball Balancing Robot is dynamically stable robot capable of omnidirectional motion. It possesses non-holonomic properties and is a special case of underactuated system, classified as a Shape-...
- 71
2
votes
1
answer
209
views
Non-Holonomic constraint in rigid body dynamics
I have solved many problems on Holonomic constraint using Lagrange multiplier method but I don't know how to tackle problems on non-Holonomic constraint.
Can anyone help me with the following problem ...
- 33
2
votes
1
answer
803
views
A particle constrained to always move on a surface whose equation is $\sigma (\textbf{r},t)=0$. Show that the particle energy is not conserved
In Goldstein's Classical mechanics question 2.22
Suppose a particle moves in space subject to a conservative potential $V(\textbf{r})$ but is constrained to always move on a surface whose equation is ...
- 173
2
votes
0
answers
210
views
Galileo's Principle of Relativity in Lagrangian Mechanics
Q. Does Galileo's principle of relativity I imply Galileo's principle of relativity II?
Galileo's principle of relativity I: Newton’s equations $\ddot{x} = F(x, \dot{x}, t)$ are invariant under the ...
- 41
2
votes
1
answer
245
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How to calculate generalized force $Q_\phi$ with d'Alembert's principle?
The related post was found here Lagrangian formalism application on a particle falling system with air resistance and also Wikipedia's definition on generalized force. Essential
$$\frac{d}{dt}\frac{\...
- 3,256
2
votes
0
answers
161
views
Problem understanding something from the variational principle for free particle motion (James Hartle's book, chapter 5)
I am currently studying general relativity from James Hartle's book and I have trouble understanding how he goes to equation (5.60) from equation (5.58). It's about the variational principle for free ...
2
votes
0
answers
186
views
Lagrangian and convexity
Is it possible to model any physical system with a Lagrangian convex in its velocity variables ?
I am aware that many Lagrangian can model the same system and maybe not all of them are partially ...
2
votes
1
answer
109
views
How does derivation of Lagrange equation from d’Alembert principle differ from the derivation of it from principle of least action?
Using d’Alembert principle, one doesn't require any assumption like the one made in other case where particle has to follow the path of least action.
- 21
2
votes
1
answer
333
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How do we get Maupertuis Principle from Hamilton's Principle?
Maupertuis principle says that if we know the initial and final coordinates but not time, the total energy and the fact that energy is conserved, we can choose the "right" path from all mathematically ...
2
votes
0
answers
76
views
Conflict of domain and endpoints in Noether's theorem and energy conservation
In the derivation of energy conservation, there is the transformation $q(t)\rightarrow q'(t)=q(t+\epsilon)$, whose end points are kind of fuzzy. The original path $q(t)$ is only defined from $t_1$ to $...
2
votes
0
answers
186
views
Conserved quantities of a simple Lagrangian
Suppose the following Lagrangian with 2 degrees of freedom:
$$L = \frac{3}{2}\dot{q}^2_{1} \ + \frac{3}{2}\dot{q}^2_{2} \ + \dot{q}_{1}\dot{q}_{2}$$
My aim is to find all the conserved quantities of ...
2
votes
0
answers
226
views
Are first Integrals the same in Lagrangian and Hamiltonian formalism?
A first integral in Lagrangian formalism is defined as a function which is constant along the solutions $(q,\dot{q})$ where $q$ are the generalized coordinates; while a first integral in Hamiltonian ...
- 423
2
votes
1
answer
264
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Clarifications regarding the Maupertuis/Jacobi principle
I'm slightly confused regarding the Maupertuis' Principle. I have read the Wikipedia page but the confusion is even in that derivation. So, say we have a Lagrangian described by $\textbf{q}=(q^1,...q^...
- 353
2
votes
0
answers
140
views
Deriving Hamilton's equations independently
The usual way to derive Hamilton's equations is to perform Legendre transformation of the Lagrangian and then use the stationarity principle. However, this procedure seems a little artificial to me ...
- 21