All Questions
84
questions
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
1
answer
104
views
Derivatives of the multipliers appearing in the Euler-Lagrange equation
(This is a crossed post where physical considerations can be helpful.)
Let $\Omega\subset \mathbb{R}^2$ be some simply connected domain and consider the functional
\begin{align}
V\left[u(x,y),v(x,y),w(...
0
votes
0
answers
25
views
Help with deriving euler-lagrange equation (evaluating at $ \varepsilon = 0$ before solving partials) [duplicate]
I am using wiki here to help me understand the deriving of the euler-lagrage equations
How do we get from:
\begin{equation}
\left.\frac{dJ_\varepsilon}{d\varepsilon}\right|_{\varepsilon = 0} = \int_a^...
- 283
0
votes
2
answers
90
views
Help with deriving the Euler-Lagrange equation (evaluating at $ \varepsilon = 0$ before solving partials)
I am using wiki here to help me understand the deriving of the euler-lagrage equations
How do we get from:
\begin{equation}
\left.\frac{dJ_\varepsilon}{d\varepsilon}\right|_{\varepsilon = 0} = \int_a^...
- 283
0
votes
0
answers
71
views
Deriving Euler-Lagrange equation [duplicate]
I have derive the Euler-Lagrange equation which is equation (2) for a condition in which generalised velocity is independent on the generalised coordinate but when generalised velocity is dependent on ...
2
votes
0
answers
48
views
Why do we discount higher-order variations when applying variational methods in analytical mechanics? [duplicate]
In No-Nonsense Classical Mechanics, the calculus of variations is introduced with what I'm sure is a standard example. We try to find the minima of function $f(x) = x^2$ by evaluating it at $x + \...
- 31
0
votes
1
answer
102
views
For virtual displacement in the Lagrangian, why is $\delta \dot{x_i} = \delta \frac{dx_i}{dt} = \frac{d}{dt}\delta x_i \equiv 0$?
I am having trouble understanding why $$\delta \dot{x_i} = \delta \frac{dx_i}{dt} = \frac{d}{dt}\delta x_i \equiv 0.\tag{7.132}$$
you can see my explanation leading up to it below.
I would greatly ...
- 283
1
vote
1
answer
300
views
D'Alembert Principle and Euler-Lagrange. Virtual displacement
I have a little trouble with d'Alembert Principle and with virtual displacement.
Imagine that with the d'Alembert Principle:
$$
\sum_i \boldsymbol{\mathrm{F_i}} \; \cdot \delta \boldsymbol{\mathrm{...
- 1,525
1
vote
0
answers
68
views
Action in Lagrangian Mechanics [duplicate]
I editted this question since it was closed because it is a duplicate. However, answers in the referenced question didn't solve my question, so I am writing it again.
Lagrangian mechanics is built ...
- 515
3
votes
2
answers
148
views
How to prove that $ \delta \frac{dq_i}{dt} = \frac{d \delta q_i}{dt} $? [duplicate]
During the proof of least action principle my prof used the equation $ \delta \frac{dx}{dt} = \frac{d \delta x}{dt} $. We were not proved this equality. I was curious to know why this is true so I ...
0
votes
1
answer
579
views
Independent generalized coordinates are dependent
(This is not about independence of $q$,$\dot q$)
A system has some holonomic constraints. Using them we can have a set of coordinates ${q_i}$. Since any values for these coordinates is possible we say ...
- 1,270
1
vote
1
answer
27
views
Representing infinite paths from one point to other using parameterization
Below the picture is from the standard Goldstein textbook of Classical Mechanics. In Hamilton's principle part, there is an equation that represents the set of all possible paths from $x1$ to $x2$ (as ...
- 31
1
vote
2
answers
304
views
About virtual displacement
Thornton Marion
The varied path represented by $\delta y$ can be thought of physically as a virtual displacement from the actual path consistent with all the forces and constraints (see Figure above)....
- 1,270
3
votes
2
answers
275
views
How do we know that solving Euler-Lagrange equation gives us the correct equation of motion in any coordinate?
As far as I understand it: If we defined a quantity called the Lagrangian as the difference between the kinetic energy and potential energy for a particle in one dimension in Cartesian coordinates, ...
- 195
1
vote
1
answer
134
views
From the initial condition problem of the Euler-Lagrange equation to the principle of least action
I browsed through many similar questions about the Initial Condition Problem (ICP) and Boundary Value Problem (BVP) for Euler-Lagrange equations, here some interesting but (in my opinion) incomplete ...
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