All Questions
239
questions
-2
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2
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Are there any experiments that examine Hamilton's Principle directly?
Or can it be examined?
I 'd glad if you can share some ideas about "principles" in general.
- 109
3
votes
1
answer
82
views
Does quasi-symmetry preserve the solution of the equation of motion?
In some field theory textbooks, such as the CFT Yellow Book (P40), the authors claim that a theory has a certain symmetry, which means that the action of the theory does not change under the symmetry ...
- 51
1
vote
2
answers
111
views
Does Hamilton's principle allow a path to have both a process of time forward evolution and a process of time backward evolution?
This is from Analytical Mechanics by Louis Hand et al. The proof is about Maupertuis' principle. The author seems to say that Hamilton's principle allow a path to have both a process of time forward ...
- 353
2
votes
0
answers
73
views
Why can't we treat the Lagrangian as a function of the generalized positions and momenta and vary that? [duplicate]
Some background: In Lagrangian mechanics, to obtain the EL equations, one varies the action (I will be dropping the time dependence since I don't think it's relevant) $$S[q^i(t)] = \int dt \, L(q^i, \...
- 93
7
votes
3
answers
2k
views
Something fishy with canonical momentum fixed at boundary in classical action
There's something fishy that I don't get clearly with the action principle of classical mechanics, and the endpoints that need to be fixed (boundary conditions). Please, take note that I'm not ...
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0
votes
1
answer
76
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In Lagrangian mechanics, do we need to filter out impossible solutions after solving?
The principle behind Lagrangian mechanics is that the true path is one that makes the action stationary. Of course, there are many absurd paths that are not physically realizable as paths. For ...
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0
votes
0
answers
88
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Deeper explanation for Principle of Stationary Action [duplicate]
The Principle of Stationary Action (sometimes called Principle of Least Action and other names) is successfully applied in a wide variety of fields in physics. It can often can be be used to derive ...
- 16.9k
-2
votes
1
answer
108
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Why the choice of Configuration Space in Hamilton's Principle is $(q, \dot{q}, t)$? [closed]
In most physics books I've read, such as Goldstein's Classical Mechanics, the explanation of Hamilton's principle took into consideration the equation (1) known as Action:
$$\displaystyle I = \int_{...
- 113
3
votes
3
answers
130
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Is there a proof that a physical system with a *stationary* action principle cannot always be modelled by a *least* action principle?
I'm aware that with Lagrangian mechanics, the path of the system is one that makes the action stationary. I've also read that its possible to find a choice of Lagrangian such that minimization is ...
- 50.2k
1
vote
1
answer
33
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Regarding varying the coefficients of constraints in the pre-extended Hamiltonian in Dirac method
consider the following variational principle:
when we vary $p$ and $q$ independently to find the equations of motion, why aren't we explicitly varying the Coeff $u$ which are clearly functions of $p$ ...
5
votes
3
answers
630
views
Is Principle of Least Action a first principle? [closed]
It is on the basis of Principle of Least Action, that Lagrangian mechanics is built upon, and is responsible for light travelling in a straight line.
Is its the classical equivalent of Schrodinger's ...
0
votes
1
answer
58
views
Euler-Lagrange confusion
Consider the action $S = \int dt \sqrt{G_{ab}(q)\dot{q}^a\dot{q}^b}.$
Now for computing the Euler-Lagrange equations, we need the time derivative of $\frac{\partial L}{\partial \dot{q}^c} = \frac{1}{\...
- 298
2
votes
3
answers
148
views
Derivation of Hamiltonian by constraining $L(q, v, t)$ with $v = \dot{q}$
I am trying to reconstruct a derivation that I encountered a while ago somewhere on the internet, in order to build some intuition both for $H$ and $L$ in classical mechanics, and for the operation of ...
- 23
0
votes
1
answer
79
views
Is $F=-\nabla V$ a form of the least action principle? [closed]
Only for conservative systems, of course.
0
votes
1
answer
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 $$\frac{d}{d t}\left(\frac{\partial L}{\partial \...
- 1,270
3
votes
3
answers
788
views
Confusion of variable vs path in Euler-Lagrange equation, Hamiltonian mechanics, and Lagrangian mechanics
In Lagrangian mechanics we have the Euler-Lagrange equations, which are defined as
$$\frac{d}{dt}\Bigg(\frac{\partial L}{\partial \dot{q}_j}\Bigg) - \frac{\partial L}{\partial q_j} = 0,\quad j = 1, \...
- 3,350
2
votes
2
answers
154
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What is the most general transformation between Lagrangians which give the same equation of motion?
This question is made up from 5 (including the main titular question) very closely related questions, so I didn't bother to ask them as different/followup questions one after another. On trying to ...
- 785
0
votes
0
answers
34
views
Obtaining the critical solution for a functional
So, I was trying to calculate the critical solution $y = y(x)$ for the following functional:
$$J[y] = \int_{0}^{1} (y'^{\: 2} + y^2 + 2ye^x)dx$$
and a professor of mine said I should use conservation ...
1
vote
1
answer
94
views
Variational Principle for Free Particle Motion (Relativistic)
This is the same problem as someone asked before: Problem understanding something from the variational principle for free particle motion (James Hartle's book, chapter 5)
The question below:
Here ...
- 349
0
votes
3
answers
67
views
Least action principle and uniform motion
I'm trying to apply the principle of least action to the case of a uniform motion under no potential. Assume the object starts with initial velocity $v_0$, moving from point $A$ to point $B$. We know ...
- 1,513
1
vote
2
answers
54
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Number of classical paths linking $(x_1, t_1)$ and $(x_2, t_2)$ [duplicate]
In classical mechanics, the least action principle states that the real path linnking $(x_1, t_1)$ and $(x_2, t_2)$ is an extremal of the action functional.
The question is, how many such solutions ...
- 1,957
5
votes
1
answer
712
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Confusion in derivation of Euler-Lagrange equations
Here's a screenshot of derivation of Euler-Lagrange from feynman lecture https://www.feynmanlectures.caltech.edu/II_19.html
My doubt is in the last paragraph. I get that $\eta = 0$ at both ends, but ...
- 301
0
votes
1
answer
152
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How did Landau & Lifshitz (Mechanics) get Equation 2.5?
I understood everything in Landau & Lifshitz's mechanics book until Equation 2.4,but I'm not sure what he means when he says "effecting the variation" and gets Equation 2.5.
Can anyone ...
2
votes
1
answer
61
views
Confusion on variation of $\dot{q}$ while applying Hamilton Principle to Lagrangian Mechanics
We restrict that $$\delta q\mid _{t_{1}}= \delta q\mid _{t_{2}}=0$$ while applying Hamilton Principle ($\delta\int_{t_{1}}^{t_{2}}Ldt=0$) to get Euler-Lagrange’s Equations. Hence adding a $$\frac{d}{...
- 77
0
votes
2
answers
324
views
Euler-Lagrange intuition
We know from euler-lagrange, that $S$ should be minimized, which in turn means (KE-PE) should be minimized at each smallest interval along the path.
I'm not trying to understand the math here, it's ...
- 525
1
vote
1
answer
87
views
Is Hamilton’s principle valid for systems that are not monogenic?
I have read in Goldstein that Hamilton’s principle works only for monogenic systems. Is it true? I thought that the action principle is universal?
0
votes
4
answers
359
views
A step in the derivation of the Euler-Lagrange equations using Hamilton's Principle
I am going through the derivation of the Euler-Lagrange equations from Hamilton's principle following Landau and Lifshitz Volume 1. We start by writing the variation in the action as,
$$\delta S = \...
1
vote
3
answers
870
views
Deriving Hamilton's Principle from Lagrange's Equations
I'm trying to derive Hamilton's Principle from Lagrange's Equations, as I've heard they're logically equivalent statements, and am stuck on a final step. For simplicity, assume we're dealing with a ...
- 245
4
votes
2
answers
137
views
Gauge Symmetry of the Lagrangian
My teacher told the following statement to me during office hours. Is it correct and if so, how could one go about proving it?
Given a material system subject to holonomic and smooth constraints ...
- 402
2
votes
2
answers
705
views
Using the principle of inertia to motivate the principle of least action?
Can we motivate the principle of least action with the principle of inertia that causes a mass particle to resist changes in its momentum? After all, the principle of inertia is the starting point and ...
- 735
0
votes
0
answers
71
views
Deriving Euler-Lagrange equation [duplicate]
I have derive the Euler-Lagrange equation which is equation (2) for a condition in which generalised velocity is independent on the generalised coordinate but when generalised velocity is dependent on ...
2
votes
2
answers
64
views
In the context of field-theoretic constrained dynamics, do we have the freedom to choose the Lagrange multipliers to be time-independent?
Let us work in a box $(t,\overrightarrow{x}) \in [0,1] \times [0,1]^3$. For any function on this box, we impose some Dirichlet boundary condition on the temporal direction and periodic boundary ...
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0
votes
0
answers
30
views
Why is the action guranteed to have unique extrema in classical mechanics? [duplicate]
Reading any classical mechanics book which introduces the Lagrangian formalism of mechanics, a one particle system is introduced to show that we obtain the euler-lagrange equations from Newton's ...
- 460
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1
answer
140
views
Why is the action guranteed to have one unique extrema in classical mechanics? [duplicate]
Reading any classical mechanics book which introduces the Lagrangian formalism of mechanics, a one particle system is introduced to show that we obtain the euler-lagrange equations from Newton's ...
- 460
2
votes
1
answer
296
views
Does a constant in the action always have unobservable consequences in classical mechanics?
Background
So in classical mechanics, my understanding is that for the action by using a the principle of least action one can get the equations of motion. Adding a constant to the action does not ...
- 1,389
2
votes
0
answers
43
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Can Lagrangians model all possible dynamics? [duplicate]
We use Lagrangians and variational calculus for almost all of physics, from Newtonian mechanics to QFT.
Is there any theorem in mathematics that guarantees that all possible dynamics of objects (say ...
- 173
1
vote
2
answers
145
views
Solving 3D Kepler Problem substitution goes wrong
I'm trying to arrive at the effective potential equation in Kepler Problem using Routh reduction method. We can procede in two ways, either using polar coordinates in the plane where the orbit happens ...
- 221
2
votes
0
answers
48
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Why do we discount higher-order variations when applying variational methods in analytical mechanics? [duplicate]
In No-Nonsense Classical Mechanics, the calculus of variations is introduced with what I'm sure is a standard example. We try to find the minima of function $f(x) = x^2$ by evaluating it at $x + \...
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votes
2
answers
236
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How does nature know Hamilton's principle? [duplicate]
I have gone through some of the questions asked here re Hamilton's principle, but could not readily find an answer to the following:
Hamilton's principle states that paths particles follow extremizes ...
user228424
2
votes
3
answers
144
views
What is the intuitive implication behind $L' = L + \frac{df}{dt}$ not affecting equations of motion?
I am referring to another post on the same question as this post: Lagrangian $L' = L + \frac{df}{dt}$ gives the same equations of motion
In the first paragraph, the poster says:
"It is well ...
- 23
2
votes
2
answers
682
views
Proof of principle of stationary action when the Lagrangian is not $L=T-V$
The principle of stationary action claims that the action $S$ takes a stationary value in a real system, where:
$$S = \int_{t_1}^{t_2} L dt\tag{1}$$
and $L$ is the Lagrangian of the system. It can be ...
- 337
1
vote
1
answer
118
views
Is it possible to built a variational principle for this first-order system?
Imagine there is a mechanical system described in unitary units by the equation:
$$\dot{x} = -\text{sgn}(x)\sqrt{|x|},\quad x(0)=1 \tag{Eq. 1}$$
such it has a finite duration solution:
$$x(t) = \frac{...
- 93
1
vote
1
answer
502
views
What is the difference between variational principle, principle of stationary action and Hamilton's principle?
In advanced mechanics, we learn about the variational principle, the principle of stationary action, and the Hamilton's principle. I feel that the difference between them is not very clearly organized ...
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3
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1
answer
396
views
Why is d'Alembert's principle not as applicable in physics as the principle of stationary action?
Any textbook in classical mechanics will tell you that there are two different routes one can follow to derive the Euler-Lagrange equations:
Route 1: Write d'Alembert's principle in the form $\sum_{i=...
- 1,092
1
vote
1
answer
300
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D'Alembert Principle and Euler-Lagrange. Virtual displacement
I have a little trouble with d'Alembert Principle and with virtual displacement.
Imagine that with the d'Alembert Principle:
$$
\sum_i \boldsymbol{\mathrm{F_i}} \; \cdot \delta \boldsymbol{\mathrm{...
- 1,525
-3
votes
1
answer
284
views
Why principle of least action is true? [duplicate]
Recently I am studying lagrangian mechanics where I came across the topic "principle of least action" which states that a system always takes the path of least action or when the action is ...
0
votes
1
answer
58
views
Why do we maximize according to the values of Lagrange multipliers?
In some Lagrangian problems, when we use the lagrange multipliers to minimize a function $f(x)$ they write:
\begin{equation}
\max_{\lambda,\mu} \min_{x}\mathcal{L} = \max_{\lambda,\mu}\min_{x} \Big( ...
- 231
1
vote
1
answer
130
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Lagrangian mechanics and geodesics in configuration space? [duplicate]
In lagrangian mechanics Is the path that take a System in the configuration space between initial and final state is identical to the geodesic which connect the two points?
3
votes
5
answers
371
views
Axiomatising classical mechanics to arrive at the principle of stationary action - what are the fundamental definitions of momentum, etc.?
$\newcommand{\d}{\mathrm{d}}\newcommand{\l}{\mathcal{L}}$Throughout all my study of physics, it has never been clear what is a definition, what is an axiom, what is a law and what is a proof in ...
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4
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503
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Is there any physical significance to the non-uniqueness of the least action principle?
In classical mechanics we often define the action as the quantity
$$ \int_{0}^{T} \left[ T - V \right] dt$$
Which in many applications is some variant of
$$ \int_{0}^{T} \left[ \frac{1}{2}m \left( x' ...
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