Linked Questions

3 votes
2 answers
93 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 & ...
17 votes
1 answer
3k views

On a trick to derive the Noether current

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$. The trick to get the Noether current consists in making the ...
0 votes
2 answers
142 views

Zero Lagrangian

If $L(q,\dot{q},t)$ is a lagrangian of a system, then $L' = L + \frac{dF(q,t)}{dt}$ is also a valid lagrangian and both lagrangians will lead to the same equation of motion. But, what if I choose $F(q,...
2 votes
2 answers
1k views

When can one omit a total time derivative in the Lagrangian formulation?

I am studying Lagrangian and Hamiltonian mechanics and i am using Landau & Lifshitz and Goldstein books. Both of them state that a modified lagrangian $$L'=L+\frac{df}{dt}$$ gives the same ...
53 votes
7 answers
8k views

Why isn't the Euler-Lagrange equation trivial?

The Euler-Lagrange equation gives the equations of motion of a system with Lagrangian $L$. Let $q^\alpha$ represent the generalized coordinates of a configuration manifold, $t$ represent time. The ...
4 votes
1 answer
176 views

How to check $\mathbf v' · \mathbf V$ and $\mathbf v'^2$ are time derivatives of some other functions?

From Landau, Lifshitz Mechanics p.127 $$ \renewcommand{\vec}[1]{\mathbf{#1}} L'=\frac{1}{2}m(\vec{v}'^2+\vec{v'}\cdot\vec{V}+\vec{V}^2)-U $$ He states that "$\vec{V}^2(t)$ can be written as the ...
3 votes
2 answers
168 views

Confusion about Noether's theorem

In my field theory class we recently derived Noether's theorem: We consider a infinitessimal transformation $\phi \to \phi + \epsilon \,\delta\phi$ of our field which preserves action i. e. $\delta S =...
7 votes
1 answer
1k views

What variables does the action $S$ depend on?

Action is defined as, $$S ~=~ \int L(q, q', t) dt,$$ but my question is what variables does $S$ depend on? Is $S = S(q, t)$ or $S = S(q, q', t)$ where $q' := \frac{dq}{dt}$? In Wikipedia I've ...
1 vote
1 answer
1k views

Proving invariance of Lagrange's equations under transformation of the form $L' = L + \dfrac{d}{dt}g(\mathbf{q},t)$

A problem in my classical mechanics textbook is stated as follows: Show that if the Lagrangian $L(\mathbf{q},\dot{\mathbf{q}},t)$ is modified to $L'$ by any transformation of the form $$ L' = L + ...
9 votes
2 answers
3k views

When can we add a total time derivative of $f(q, \dot{q}, t)$ to a Lagrangian?

The other day, I was listening to this lecture on the Lagrangian for a charged particle in an electromagnetic field, and at one point in the video, the lecturer mentions that we can add any total time ...
8 votes
2 answers
853 views

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 ...
3 votes
1 answer
161 views

Prove an action expansion's even-indexed terms have to be integrated, where the odd-indexed terms are only derivatives of the potential (WKB)

After assuming a wavefunction of a form: $$ \psi \approx A \exp{\left(i \frac{S(x)}{\hbar}\right)}$$ and letting $$S = \hbar^0 S_0 + \hbar^1 S_1 + \hbar^2 S_2 +...$$ The odd-indexed terms of the ...
2 votes
3 answers
1k views

Is the path of stationary action unique? What are the physical implications of $L_{\dot{x}}=L_x$

Below, for any function $Q$ the notation $Q_x$ means $\frac{\partial Q}{\partial x}$, and $Q_{xx}$ means $\frac{\partial^2 Q}{\partial x^2}$. In physics, the trajectory of a particle is given by the ...