All Questions
84
questions
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
13
votes
2
answers
408
views
Anticommutation of variation $\delta$ and differential $d$
In Quantum Fields and Strings: A Course for Mathematicians, it is said that variation $\delta$ and differential $d$ anticommute (this is only classical mechanics), which is very strange to me. This is ...
1
vote
2
answers
345
views
Jerk mechanics - Lagrangian
I have a Lagrangian with the form
$$L = L[q(t,\alpha), \dot{q}(t,\alpha), \ddot{q}(t,\alpha), t],$$
to which I am applying the calculus of variations. The problem is that when I apply the calculus, I ...
- 3,122
2
votes
1
answer
244
views
Field theory Euler-Lagrange problem term
Consider the following Lagrangian (density)
$$
\mathcal{L} = (\mu/2) (\partial_t q)^2 - (Y/2) (\partial_x q)^2 -\alpha(\partial_x{}^2 q)^2
$$
$\mu, Y, \alpha, q$ are respectively mass/unit length, ...
- 1,031
1
vote
1
answer
123
views
Reasoning behind $\delta \dot q = \frac{d}{dt} \delta q$ in deriving E-L equations [duplicate]
Consider a Lagrangian $L(q, \dot{q}, t)$ for a single particle. The variation of the Lagrangian is given by:
$$\delta L= \frac{\partial L}{\partial q}\delta q + \frac{\partial L}{\partial \dot q}\...
1
vote
0
answers
185
views
Euler-Lagrange equations when Lagrangian becomes unbounded at the limit of boundary conditions after change of variables
Consider an action integral $$I = \int_{t_0}^{t_f}L(\eta(\theta(t)),\dot{\eta}(\theta(t)),t)\,dt\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(0)$$ with boundary conditions $\dot{\theta}(t_0) = \dot{\theta}(t_f)=0$ ...
- 11
2
votes
1
answer
183
views
Symmetry Condition in Noether's Theorem
Suppose $q = \{q_1,\cdots, q_i\}$ is a coordinate system for Lagrangian $L(q,\dot{q},t)$. In this text by David Morin, on page 16 in chapter 6, it states that a symmetry is a transformation of the ...
- 106
0
votes
1
answer
42
views
Change in potential energy after infinitesimal variation in position
The a particle with the potential $V(x^2+y^2)$ undergoes an active transformation where
$x\rightarrow x+y\delta$
$y\rightarrow y-x\delta$
The exercise was to prove that the Lagrangian of the system ...
- 62
1
vote
2
answers
173
views
Euler-Lagrange partial with respect to $r$ of a dot product involing velocity and a vector potential [closed]
I will outline what I believe to a correct way to go from a Lagrangian of a charged particle in a EM field to Lorentz force via the Euler-Lagrange equations. At the very beginning when I use the EL ...
- 1,031
0
votes
1
answer
83
views
The use of $x_\varepsilon (t) = x(t) + \varepsilon (t)$ and $x_\varepsilon (t) = x(t) + \varepsilon \eta (t)$ in proving Hamilton's principle
The following Wikipedia page uses $x_\varepsilon (t) = x(t) + \varepsilon (t)$ in the proof.
https://en.wikipedia.org/wiki/Hamilton%27s_principle#Mathematical_formulation
But in my mechanics book (by ...
- 485
2
votes
1
answer
212
views
D'Alembert derivation of Lagrange Equation - why can it use both virtual and normal differentials?
In "Classical Mechanics" by Goldstein and "A Students Guide to Lagrangians and Hamiltonians" by Hamill I noticed that both the virtual displacement derivatives and the normal displacement derivatives ...
user248988
2
votes
1
answer
252
views
Partial time derivative of the on-shell action
I have a few questions about differentiating the on-shell action.
Here is what I currently understand (or think I do!):
Given that a system with Lagrangian $\mathcal{L}(\mathbf{q}, \dot{\mathbf{q}}, ...
- 2,157
2
votes
1
answer
100
views
Confusion regarding the time derivative term in Lagrange's equation
I am solving a pendulum attached to a cart problem. Without going into unnecessary details, the generalised coordinates are chosen to be $x$ and $\theta$. The kinetic energy of the system contains a ...
- 192
0
votes
1
answer
216
views
Is the action (a functional) a functional of both the Lagrangian (a function) and the trajectory (a function) or only the Lagrangian?
In Lagrangian Mechanics, would it be correct for me to say that the action (which is a functional) is a functional of two functions, namely the Lagrangian (a function) and the trajectory in ...
- 483
0
votes
0
answers
45
views
About Lagrange equation [duplicate]
$$\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j}.$$
I don't understand partial derivative by "function" (e.g. $q_j$).
$q$ ...
- 37