All Questions
18
questions with no upvoted or accepted answers
6
votes
0
answers
413
views
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
2
votes
1
answer
333
views
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
1
answer
264
views
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
2
answers
66
views
Is there a $n$-dimensional system such that the minimal action from a path from $x$ to $y$ is the scalar product?
Suppose we work (with a particle) in $\mathbb{R}^n$.
Is there a Euler-Lagrange equation associated to the particle in question such that the minimal action of all path going from a position $x\in \...
- 121
2
votes
0
answers
270
views
Decoupling of generalized coordinates in lagrangian
Say you have a lagrangian $L$ for a system of 2 degrees of freedom. The action, S is:
$S[y,z] = \int_{t_1}^{t_2} L(t,y,y',z,z')\,dt \tag{1}$
If $y$ and $z$ are associated with two parts of the ...
- 1,222
1
vote
1
answer
51
views
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
1
vote
0
answers
70
views
Hamilton's principle for fields
According to Goldstein, Hamilton's principle can be summerized as follows:
The motion of the system from time $t_{1}$ to time $t_{2}$ is such that the line integral (called the action or the action ...
- 1,131
1
vote
0
answers
96
views
Equation of Motion from Action for a Scalar Field + Matter
In a review on quintessence, the equations of motion (EoM) for the action
$$
S=\int\!\mathrm{d}^4x\sqrt{-g}\left(\frac{M_p^2R}{2}-\frac{g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi }{2}-V\left(\phi\...
- 938
1
vote
0
answers
62
views
Lagrangian of free particle relativistic case
Why must the covariant Lagrangian of a free particle be a first-order differential?
1
vote
0
answers
445
views
What is the physical interpretation of the action integral, without the stationary action principle?
I'm still wondering about the physical interpretation of the action integral of some mechanical system (classical theory here, to simplify things):
\begin{equation}\tag{1}
A = \int_{t_1}^{t_2} L(q, \, ...
- 7,592
1
vote
0
answers
89
views
Cauchy problem for Hamilton-Jacobi equation
In Arnol'd V.I, "Mathematical methods of classical mechanics" p.257, I was asked to find a solution for the Cauchy problem
$$H=\frac{p^2}{2},\ \ \ S_0=\frac{q^2}{2}$$
of the Hamilton-Jacobi equation
...
- 627
1
vote
0
answers
81
views
When the equations of motion are not unique (eg. when they are given by eigenvectors), which will the free particle adhere to?
For this question I think it will be easier to express the usual equation describing the motion of a "free particle,"--viz. $g_{ij}\dot{x}^i\dot{x}^j$--in matrix form as follows:
$$g_{ij}\dot{x}^i\...
- 1,517
0
votes
0
answers
50
views
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
$$
S[\vec{r}]=\int dt L(\vec{r}(t),\dot{\...
- 145
0
votes
0
answers
82
views
Why does this method of deriving the classical free particle Lagrangian not work?
I was reading volume two in Landau and Lifshitz's Course of Theoretical Physics (The Classical Theory of Fields). In it, Dr. Landau develops the relativistic Lagrangian as follows: one has $$S=\alpha\...
- 3,967
0
votes
0
answers
87
views
Lagrangian of Charged Particle Evaluated On-Shell
I am trying to calculate the Lagrangian of a charged particle in background gauge field evaluaed on-shell.
Let $A^{\mu}(x)$ be a gauge field. The action of a charged particle in this background gauge ...
- 2,923