All Questions
166
questions
2
votes
1
answer
156
views
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
4
votes
0
answers
89
views
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 $$S=\int \mathscr{...
- 873
2
votes
2
answers
705
views
Using the principle of inertia to motivate the principle of least action?
Can we motivate the principle of least action with the principle of inertia that causes a mass particle to resist changes in its momentum? After all, the principle of inertia is the starting point and ...
- 735
2
votes
1
answer
296
views
Does a constant in the action always have unobservable consequences in classical mechanics?
Background
So in classical mechanics, my understanding is that for the action by using a the principle of least action one can get the equations of motion. Adding a constant to the action does not ...
- 1,389
0
votes
2
answers
236
views
How does nature know Hamilton's principle? [duplicate]
I have gone through some of the questions asked here re Hamilton's principle, but could not readily find an answer to the following:
Hamilton's principle states that paths particles follow extremizes ...
user228424
1
vote
1
answer
118
views
Is it possible to built a variational principle for this first-order system?
Imagine there is a mechanical system described in unitary units by the equation:
$$\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
2
answers
257
views
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
1
vote
1
answer
501
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 ...
- 11.8k
1
vote
1
answer
300
views
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:
$$
\sum_i \boldsymbol{\mathrm{F_i}} \; \cdot \delta \boldsymbol{\mathrm{...
- 1,525
0
votes
1
answer
140
views
How to show that the Action has units Energy·time?
The Lagrangian, which has units of Energy, is defined as that which when summed over time gives the Action, the action being more fundamental.
But how does summing over units of Energy across time ...
- 381
1
vote
3
answers
170
views
Is the Lagrangian formulation a mathematical inevitability? [duplicate]
An analogy with functions:
Say, we have a function $f(x)$ and we have an equation to solve, $f(x)=0$. We can always re-formulate the problem of solving $f(x)=0$ with the problem of extremising $F(x)$, ...
- 6,355
2
votes
1
answer
115
views
Derivation of the virial theorem from the action and boundary term
In this answer, it is said that the invariance of the action under the transformation $$ x \rightarrow (1+\epsilon)x\tag{0}$$
gives, up to some boundary terms the virial theorem.
I tried to interpret ...
- 1,168
2
votes
2
answers
378
views
The action of a physical system
My knowledge in this topic is as follows, correct me if I'm wrong.:
the action $S$ of a physical system is quantity such that the system evolves so that it's extemized, maximized or minimized, usually ...
- 123
2
votes
1
answer
750
views
Independence of position and momentum in action
Why are position and momentum independent with respect to the Hamiltonian Action $S_H$ given by
$$
S_H = \int_{t_1}^{t_2} (p \dot q - H) dt \ \ \ ? \tag{1}
$$
While deriving Hamilton's equations from ...
- 23
3
votes
2
answers
139
views
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