All Questions
166
questions
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?
7
votes
1
answer
167
views
Modifying Feynman-Wheeler absorber theory to work with arbitrary potentials?
I'm trying to consider relativistic multi-body dynamics in special relativity. In classical mechanics, it's easy to write a simple $n$-body system with arbitrary potential $V$:
\begin{equation}
m \...
- 716
1
vote
3
answers
660
views
Is minimizing the action same as minimizing the energy?
When we differentiate the total energy with respect to the time and set it to zero (make it stationary), we get an expression as similar to what we get while we minimize action. Also putting the time ...
- 111
13
votes
3
answers
2k
views
In a Lagrangian, why can't we replace kinetic energy by total energy minus potential energy?
TL;DR: Why can't we write $\mathcal{L} = E - 2V$ where $E = T + V = $ Total Energy?
Let us consider the case of a particle in a gravitational field starting from rest.
Initially, Kinetic energy $T$ is ...
- 255
3
votes
1
answer
340
views
Is action maximized for a system in stable equilibrium?
Others have asked in general about cases in which the action integral is not minimized, but my question is specific: Can we show that the conventional action integral is always maximized for a system ...
- 33
1
vote
1
answer
440
views
Hamilton-Jacobi equation and Action Functional
Let the action functional $S[q]$ given by
\begin{equation}\label{eq16}
S[q]=\int\limits^{t_2}_{t_1}L\left(q^i(t),\dot{q}^i(t)\right)dt.\tag{1}
\end{equation}
Also, we know that using Legendre ...
- 403
1
vote
0
answers
44
views
Variation of classical action due to variation of position and time [duplicate]
I am reading about the Brown-York formalism in https://link.springer.com/article/10.12942/lrr-2004-4. The document says that if we consider a deformation of our end points $(q,t$) in position and time,...
- 700
1
vote
1
answer
242
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Do solutions to the Euler Lagrange equation for physical Lagrangians actually minimize the action? [duplicate]
Do solutions to the Euler Lagrange equation for physical Lagrangians actually minimize the action? In other words, is it known that for all Lagrangians used in application, that the unique solution to ...
- 1,036
3
votes
1
answer
894
views
Is there anything natural about the principle of "stationary action"?
In Taylor's classical mechanics, he derived Lagrange equations and showed that they are equivalent to Newton's second law. Then, it was obvious that Lagrange equations are similar to the Euler-...
- 381
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
0
votes
1
answer
123
views
Issue with a derivation of the Hamilton-Jacobi equation
I'm trying to derive the HJ the easiest way I can but some issues come up.
$$\mathrm{dS}=\dfrac{\partial S}{\partial q}\mathrm{d}q+\dfrac{\partial S}{\partial t}\mathrm{d}t\Rightarrow\displaystyle{S=\...
- 1,168
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 ...
4
votes
2
answers
448
views
Assumptions reg. Kinetic energy and Potential energy in the Lagrangian formulation
I have recently been introduced to Lagrangian mechanics. My previous exposure to Lagrangian math has been in the form of optimizing constrained functions using Lagrange multipliers.
I get the math ...
- 143
7
votes
2
answers
411
views
Is there a deep reason why action comes from a local lagrangian?
In both classical and quantum physics Lagrangians play a very important role. In classical physics, paths that extremize the action $S$ are the solutions of the Euler-Lagrange equations, and the ...
- 1,290
3
votes
1
answer
143
views
How does a falling rock minimize action?
Consider a single two dimensional system with a rock that is influenced by gravity. The Action of this system is defined as $\int_0^\infty [T(\dot x(t))-V(x(t))]dt$, where $T$ is the kinetic and $V$ ...
- 1,772