All Questions
166
questions
1
vote
1
answer
408
views
Invariance of action integral under point transformations
I am reading Goldstein classical mechanics chapter 2 p. 35.
Here the author states that the action integral
$$\int L(q,\dot q,t)dt$$
is invariant under change in generalized coordinates
$$q_i=q_i(s_1,...
- 31
1
vote
0
answers
95
views
Why does the action $S=\int L dt=\int (T-V) dt$ have to be minimised (or maximised) to produce Newton's Second Law? [duplicate]
We have recently covered the Lagrangian in our lectures, whereby it was shown that all equations of motion ($x(t)$) satisfying the Euler-Lagrange equation with Lagrangian $L=T-V$, where $T=\frac{1}{2}...
- 485
5
votes
2
answers
610
views
Principle of stationary action vs Euler-Lagrange Equation
I am a bit confused as to what I should use to derive the equations of motions from the lagrange equation.
Suppose I have a lagrange function:
$$L(x(t), \dot{x}(t)) = \frac{1}{2}m\dot{x}^2-\frac{1}{...
- 714
0
votes
1
answer
97
views
What systems can the Principle of Least Action be applied? [duplicate]
When reading about the Calculus of Variation and Hamilton's principle I come across quotes like this
Hamilton's principle states that the differential equations of motion for any physical system ...
- 50.2k
1
vote
1
answer
175
views
Fermat's principle in classical mechanics?
I do know the principle of least action, but is it possible to formulate classical mechanics based on the principle of least time? That is, if we know the initial state $(x_i,p_i)$ of the particle and ...
- 735
-1
votes
2
answers
488
views
Is the Lagrangian of a non-relativistic particle just $\dot{x}$?
Let
$$
S= m \int_a^b \dot{x}dt
$$
Using the relation $L\to L^2/2$, (see Geodesic Equation from variation: Is the squared lagrangian equivalent?)
I obtain
$$
S=m\int_a^b\frac{1}{2}(\dot{x})^2dt
$$
...
- 1,548
6
votes
1
answer
241
views
Solving free particles with Fourier series
Here's a silly idea : take the action of a free particle,
$$S = \int_{t_1}^{t_2} \dot{x}^2 dt$$
Our configuration space is the space of $C^1$ functions over $[t_1, t_2]$, which is spanned by the ...
- 16.5k
2
votes
2
answers
617
views
Does it make sense to say that the action is even or odd under time reversal?
The action of a system in mechanics is an integral over time defined as $$S[x(t)]=\int\limits_{t_1}^{t_2}L(x,\dot{x},t)dt.$$ Here, the time $t$ is integrated making the left hand side depend only on ...
- 11.8k
4
votes
1
answer
229
views
Why is there a Lagrangian? [duplicate]
In all discussions regarding the Lagrangian formulation it has always been said that $L = T - V $, only is a correct guess that when operated via through the Euler -Lagrange equation yields something ...
- 63
9
votes
2
answers
983
views
Why does the 'metric Lagrangian' approach appear to fail in Newtonian mechanics?
A well known derivation of the free-space Lagrangian in Special Relativity goes as follows:
The action $\mathcal{S}$ is a functional of the path taken through
configuration space, $\mathbf{q}(\lambda)...
- 91
1
vote
2
answers
298
views
Action principle and Functional derivative in CM
I want to extremize this well known action.
$$S[\phi]=\int \mathcal{L}(\phi(t),\dot{\phi}(t)) dt $$
The result is also well known. It turns out to be E-L equation.
The Action principle states that the ...
- 107
1
vote
0
answers
63
views
How principle of least action? [duplicate]
I had learned the principle of least action.But I didn't get the motive behind taking the least action. Or why should the particle follow a path where it have a least action?
- 1,348
3
votes
2
answers
302
views
Derivation of Hamilton-Jacobi equation
I am trying my own way of deriving the Hamilton Jacobi equation
$$\frac{\partial S}{\partial t} = -H \tag{1}$$
through direct variation. I think the difficulty of doing this is that the upper limit ...
- 121
0
votes
2
answers
285
views
Taylor expansion in derivation of Noether-theorem
In my classical mechanics lecture we derived the Noether-theorem for a coordinate transformation given by:
$$ q_i(t) \rightarrow q^{'}_i(t)=q_i(t) + \delta q_i(t) = q_i(t) + \lambda I_i(q,\dot q,t).$$...
- 13
4
votes
1
answer
1k
views
Show two Lagrangians are equivalent
I need to show that these two Lagrangians are equivalent:
\begin{align}
L(\dot{x},\dot{y},x,y)&=\dot x^2+\dot y + x^2-y ,\\
\tilde{L}(\dot x, \dot y, x, y)&=\dot x^2+\dot y -2y^3.
\end{align}
...
- 247