Linked Questions
19 questions linked to/from Adding a total time derivative term to the Lagrangian
8
votes
3
answers
4k
views
Non-uniqueness of the Lagrangian
Goldstein, 3rd ed
$$
\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_{j}}\right)-\frac{\partial L}{\partial q_{j}}=0\tag{1.57}
$$
expressions referred to as "Lagrange's equations."
...
- 1,340
12
votes
2
answers
2k
views
Can the Lagrangian be written as a function of ONLY time?
The lagrangian is always phrased as $L(t,q,\dot{q})$.
If you magically knew the equations $q(t)$ and $\dot{q}(t)$, could the Lagrangian ever be written only as a function of time?
Take freefall for ...
- 494
10
votes
2
answers
2k
views
Undefined Hamiltonian for this particular Lagrangian [duplicate]
So, this a question from a Phd qualifying examination. Given the following Lagrangan $$L=\frac{1}{2}\dot{q}\text{sin}^2q,$$ what is the Hamiltonian for this system? So, finding the canonical momentum $...
- 353
11
votes
3
answers
7k
views
Does a four-divergence extra term in a Lagrangian density matter to the field equations?
Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation:
${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu A^\...
- 1,181
5
votes
3
answers
2k
views
Lagrangian gauge invariance $L'=L+\frac{df(q,t)}{dt}$
So, I have to prove directly (e.g. by substitution) that if a path satisfies the Euler-Lagrange equations for the Lagrangian $L$ it does so for $$L'=L+\frac{df(q,t)}{dt}.$$ Let me tell you what I have ...
- 153
7
votes
2
answers
3k
views
Lagrangian $L' = L + \frac{df}{dt}$ gives the same equations of motion
It is well known that when a Lagrangian $L$ is incremented by the total time derivative of a function $f$ that does not depend on the time derivatives of the generalized coordinates, the same ...
- 9,444
8
votes
2
answers
1k
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How to determine the Lagrangian's "true" explicit dependence on time?
If your Lagrangian satisfies
$$
\frac{\partial \mathcal L}{\partial t} = 0
$$
then you're happy, energy is conserved, etc. However, if the above doesn't hold, that doesn't necessarily mean energy ...
- 343
2
votes
1
answer
2k
views
Lagrangian isn't unique [closed]
If $L$ is a Lagrangian for a system of n degrees of freedom satisfying Lagrange's equations, show by direct substitution that
$$L' = L + \frac{\mathrm{d}F(q_1,\dots,q_n,t)}{\mathrm{d}t}$$
also ...
- 135
-1
votes
2
answers
496
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
$$
S=m\int_a^b\frac{1}{2}(\dot{x})^2dt
$$
...
- 1,558
3
votes
2
answers
123
views
Why are you allowed to omit the $V^2$ term in the non-inertial frame?
I'm trying to find trying to find the Lagrangian and Hamiltonian for a particle in a non-inertial frame, but when I try to do so, I always get a quadratic term, which textbooks like Landau & ...
- 57
0
votes
1
answer
568
views
Uniqueness of Lagrangian and Hamiltonian [duplicate]
Is a Lagrangian unique in the same field? Is Hamiltonian unique?
If it is unique then please explain why is it so and if it is not then please explain why is it not so.
1
vote
1
answer
211
views
Lagrangian formalism (demonstration)
My question is about the multiplicity of the Lagrangian to a Physics system.
I pretend to demonstrate the following proposition:
For a system with $n$ degrees of freedom, written by the Lagrangian ...
- 1,092
1
vote
0
answers
241
views
Proof of Lagrangian
I'm having some trouble with some math on a problem for a physics class (looking for help with some partial derivatives, not an answer).
Let $$L'=L+\dfrac{dF}{dt},$$ where $L$ is a Lagrangian and $F$ ...
- 133
0
votes
2
answers
115
views
Symmetric part contributing time derivative implies that it does not appear in Equations of Motion (EoM)
Given the following Lagrangian,
$L=\frac{1}{2} g_{i j} \dot{q}^{i} \dot{q}^{j}+b_{i j} \dot{q}^{i} q^{j}-U(q)$
I am told that,
The eom depend only on the anti-symmetric part of $b_{i j} .$ The ...
- 139
0
votes
3
answers
141
views
How can I know a system described by a lagrangian?
I have a lagrangian
\begin{equation}
L=\frac{1}{2}\dot{q}^{2}-\frac{1}{2}q\dot{q}-aq^{2}
\end{equation}
$a>0$, and I can't realise what physical system it describes, since I guess potential energy $...
- 21