All Questions
239
questions
3
votes
3
answers
788
views
Confusion of variable vs path in Euler-Lagrange equation, Hamiltonian mechanics, and Lagrangian mechanics
In Lagrangian mechanics we have the Euler-Lagrange equations, which are defined as
$$\frac{d}{dt}\Bigg(\frac{\partial L}{\partial \dot{q}_j}\Bigg) - \frac{\partial L}{\partial q_j} = 0,\quad j = 1, \...
- 3,350
2
votes
2
answers
154
views
What is the most general transformation between Lagrangians which give the same equation of motion?
This question is made up from 5 (including the main titular question) very closely related questions, so I didn't bother to ask them as different/followup questions one after another. On trying to ...
- 785
0
votes
0
answers
34
views
Obtaining the critical solution for a functional
So, I was trying to calculate the critical solution $y = y(x)$ for the following functional:
$$J[y] = \int_{0}^{1} (y'^{\: 2} + y^2 + 2ye^x)dx$$
and a professor of mine said I should use conservation ...
1
vote
1
answer
94
views
Variational Principle for Free Particle Motion (Relativistic)
This is the same problem as someone asked before: Problem understanding something from the variational principle for free particle motion (James Hartle's book, chapter 5)
The question below:
Here ...
- 349
0
votes
3
answers
67
views
Least action principle and uniform motion
I'm trying to apply the principle of least action to the case of a uniform motion under no potential. Assume the object starts with initial velocity $v_0$, moving from point $A$ to point $B$. We know ...
- 1,513
1
vote
2
answers
54
views
Number of classical paths linking $(x_1, t_1)$ and $(x_2, t_2)$ [duplicate]
In classical mechanics, the least action principle states that the real path linnking $(x_1, t_1)$ and $(x_2, t_2)$ is an extremal of the action functional.
The question is, how many such solutions ...
- 1,957
5
votes
1
answer
712
views
Confusion in derivation of Euler-Lagrange equations
Here's a screenshot of derivation of Euler-Lagrange from feynman lecture https://www.feynmanlectures.caltech.edu/II_19.html
My doubt is in the last paragraph. I get that $\eta = 0$ at both ends, but ...
- 301
0
votes
1
answer
152
views
How did Landau & Lifshitz (Mechanics) get Equation 2.5?
I understood everything in Landau & Lifshitz's mechanics book until Equation 2.4,but I'm not sure what he means when he says "effecting the variation" and gets Equation 2.5.
Can anyone ...
2
votes
1
answer
61
views
Confusion on variation of $\dot{q}$ while applying Hamilton Principle to Lagrangian Mechanics
We restrict that $$\delta q\mid _{t_{1}}= \delta q\mid _{t_{2}}=0$$ while applying Hamilton Principle ($\delta\int_{t_{1}}^{t_{2}}Ldt=0$) to get Euler-Lagrange’s Equations. Hence adding a $$\frac{d}{...
- 77
0
votes
2
answers
324
views
Euler-Lagrange intuition
We know from euler-lagrange, that $S$ should be minimized, which in turn means (KE-PE) should be minimized at each smallest interval along the path.
I'm not trying to understand the math here, it's ...
- 525
1
vote
1
answer
87
views
Is Hamilton’s principle valid for systems that are not monogenic?
I have read in Goldstein that Hamilton’s principle works only for monogenic systems. Is it true? I thought that the action principle is universal?
0
votes
4
answers
359
views
A step in the derivation of the Euler-Lagrange equations using Hamilton's Principle
I am going through the derivation of the Euler-Lagrange equations from Hamilton's principle following Landau and Lifshitz Volume 1. We start by writing the variation in the action as,
$$\delta S = \...
1
vote
3
answers
870
views
Deriving Hamilton's Principle from Lagrange's Equations
I'm trying to derive Hamilton's Principle from Lagrange's Equations, as I've heard they're logically equivalent statements, and am stuck on a final step. For simplicity, assume we're dealing with a ...
- 245
4
votes
2
answers
137
views
Gauge Symmetry of the Lagrangian
My teacher told the following statement to me during office hours. Is it correct and if so, how could one go about proving it?
Given a material system subject to holonomic and smooth constraints ...
- 402
2
votes
2
answers
705
views
Using the principle of inertia to motivate the principle of least action?
Can we motivate the principle of least action with the principle of inertia that causes a mass particle to resist changes in its momentum? After all, the principle of inertia is the starting point and ...
- 735