All Questions
166
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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 ...
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-2
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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:
$$\displaystyle I = \int_{...
- 113
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votes
3
answers
130
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Is there a proof that a physical system with a *stationary* action principle cannot always be modelled by a *least* action principle?
I'm aware that with Lagrangian mechanics, the path of the system is one that makes the action stationary. I've also read that its possible to find a choice of Lagrangian such that minimization is ...
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4
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113
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Directly integrating the Lagrangian for a simple harmonic oscillator
I've just started studying Lagrangian mechanics and am wrestling with the concept of "action". In the case of a simple harmonic oscillator where $x(t)$ is the position of the mass, I ...
- 13
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Is there a Lorentz invariant action for a free multi-particle system?
I want to write down a Lorentz-invariant action of free multi-particle systems.
I know that a Lorentz-invariant action for each particle might be expressed as
$$
S[\vec{r}]=\int dt L(\vec{r}(t),\dot{\...
- 145
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1
answer
66
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Landau/Lifshitz action as a function of coordinates [duplicate]
In Landau/Lifshitz' "Mechanics", $\S43$, 3ed, the authors consider the action of a mechanical system as a function of its final time $t$ and its final position $q$. They consider paths ...
- 349
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1
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175
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Doubts about Noether's theorem derivation
Assume you have an action:
$S[q] = \int L(q, \dot q, t)$ (i.e $q$ is a function of time). (1) Then you do a transformation on $q(t)$ such as $\sigma(q(t), a)$ where $a$ is infinetisemal and this ...
-2
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2
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97
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On the physical meaning of functionals and the interpretation of their output numbers
I am studying about functionals, and while looking for some examples of functionals in physics, I have run into this handout .
Here are two questions of mine.
1- This handout starts as follows (the ...
user386360
2
votes
3
answers
148
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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
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1
answer
85
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Hamiltonian analysis of relational $N$-Particle Dynamics
I am following "A Shape Dynamics Tutorial, Flavio Mercati" (https://arxiv.org/abs/1409.0105), and have problems understanding the hamiltonian formulation of $N$-particle dynamics as sketched ...
- 513
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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 ...
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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
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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 ...
1
vote
1
answer
51
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Lagrange momentum for position change
After the tremendous help from @hft on my previous question, after thinking, new question popped up.
I want to compare how things behave when we do: $\frac{\partial S}{\partial t_2}$ and $\frac{\...
- 525
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2
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103
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Momentum $p = \nabla S$
My book mentions the following equation:
$$p = \nabla S\tag{1.2}$$ where $S$ is the action integral, nabla operator is gradient, $p$ is momentum.
After discussing it with @hft, on here, it turns out ...
- 525
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1
answer
156
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Differentiation of the on-shell action with respect to time
From the on-shell action, we derive the following two:
$\frac{\partial S}{\partial t_1} = H(t_1)$,
$\frac{\partial S}{\partial t_2} = -H(t_2)$,
where $H = vp - L$ is the energy function.
I have two ...
- 525
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0
answers
89
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Why is the action for a field a quadruple integral over spacetime? [duplicate]
I've been trying to get started on classical field theories. As I had been studying classical mechanics from Goldstein, I decided to start from there. Goldstein introduces the action $$S=\int \mathscr{...
- 873
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votes
2
answers
705
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Using the principle of inertia to motivate the principle of least action?
Can we motivate the principle of least action with the principle of inertia that causes a mass particle to resist changes in its momentum? After all, the principle of inertia is the starting point and ...
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1
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296
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Does a constant in the action always have unobservable consequences in classical mechanics?
Background
So in classical mechanics, my understanding is that for the action by using a the principle of least action one can get the equations of motion. Adding a constant to the action does not ...
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236
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How does nature know Hamilton's principle? [duplicate]
I have gone through some of the questions asked here re Hamilton's principle, but could not readily find an answer to the following:
Hamilton's principle states that paths particles follow extremizes ...
user228424
1
vote
1
answer
118
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Is it possible to built a variational principle for this first-order system?
Imagine there is a mechanical system described in unitary units by the equation:
$$\dot{x} = -\text{sgn}(x)\sqrt{|x|},\quad x(0)=1 \tag{Eq. 1}$$
such it has a finite duration solution:
$$x(t) = \frac{...
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2
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257
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Why is the action integral of relativity particles $S = -mc\int ds$? [duplicate]
In my classical mechanic course material, it states that
(In context of relativity) The path of a particle is called its "world line". Each world line can be noted mathematically using the ...
- 301
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1
answer
501
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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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1
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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:
$$
\sum_i \boldsymbol{\mathrm{F_i}} \; \cdot \delta \boldsymbol{\mathrm{...
- 1,525
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1
answer
140
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How to show that the Action has units Energy·time?
The Lagrangian, which has units of Energy, is defined as that which when summed over time gives the Action, the action being more fundamental.
But how does summing over units of Energy across time ...
- 381
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3
answers
170
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Is the Lagrangian formulation a mathematical inevitability? [duplicate]
An analogy with functions:
Say, we have a function $f(x)$ and we have an equation to solve, $f(x)=0$. We can always re-formulate the problem of solving $f(x)=0$ with the problem of extremising $F(x)$, ...
- 6,355
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votes
1
answer
115
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Derivation of the virial theorem from the action and boundary term
In this answer, it is said that the invariance of the action under the transformation $$ x \rightarrow (1+\epsilon)x\tag{0}$$
gives, up to some boundary terms the virial theorem.
I tried to interpret ...
- 1,168
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2
answers
378
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The action of a physical system
My knowledge in this topic is as follows, correct me if I'm wrong.:
the action $S$ of a physical system is quantity such that the system evolves so that it's extemized, maximized or minimized, usually ...
- 123
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1
answer
750
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Independence of position and momentum in action
Why are position and momentum independent with respect to the Hamiltonian Action $S_H$ given by
$$
S_H = \int_{t_1}^{t_2} (p \dot q - H) dt \ \ \ ? \tag{1}
$$
While deriving Hamilton's equations from ...
- 23
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2
answers
139
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Other infinitesimal variation of the action
I was reading this post about the virial theorem where the virial theorem comes from varying the action by the infinitesimal rescaling $x\rightarrow(1+\epsilon)x$ and asking that $\delta S=0$ under ...
- 1,168