All Questions
15
questions
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1
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76
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In Lagrangian mechanics, do we need to filter out impossible solutions after solving?
The principle behind Lagrangian mechanics is that the true path is one that makes the action stationary. Of course, there are many absurd paths that are not physically realizable as paths. For ...
- 50.2k
4
votes
4
answers
263
views
Variation of a function
I'm studying calculus of variations and Lagrangian mechanics and i don't understand something about the variational operator
Let's say for example that i got a Lagrangian $L [x(t), \dot{x}(t), t] $ ...
- 309
0
votes
0
answers
73
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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 ...
1
vote
2
answers
131
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Questions regarding the derivation of Euler-Lagrange Equation from Taylor's Classical Mechanics
I'm self-studying classical mechanics from Taylor's book. I saw his derivation of the Euler-Lagrange Equation and I'm confused about something, he created a 'wrong' function $$Y(x) = y(x)+\eta(x)\tag{...
- 115
3
votes
2
answers
148
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How to prove that $ \delta \frac{dq_i}{dt} = \frac{d \delta q_i}{dt} $? [duplicate]
During the proof of least action principle my prof used the equation $ \delta \frac{dx}{dt} = \frac{d \delta x}{dt} $. We were not proved this equality. I was curious to know why this is true so I ...
1
vote
1
answer
123
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Reasoning behind $\delta \dot q = \frac{d}{dt} \delta q$ in deriving E-L equations [duplicate]
Consider a Lagrangian $L(q, \dot{q}, t)$ for a single particle. The variation of the Lagrangian is given by:
$$\delta L= \frac{\partial L}{\partial q}\delta q + \frac{\partial L}{\partial \dot q}\...
0
votes
1
answer
83
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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
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
0
answers
45
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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
2
votes
2
answers
952
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Confusing Total Derivative and Partial Derivative in Classical Field Theory - Noether Theorem
I'm really confused about total derivatives and partial derivatives.
My multivariable calculus book (Guidorizzi vol 2 Um Curso de Calculo) says that if I have a function like $f(a(u,v),b(u,v))$ then ...
0
votes
2
answers
285
views
Taylor expansion in derivation of Noether-theorem
In my classical mechanics lecture we derived the Noether-theorem for a coordinate transformation given by:
$$ q_i(t) \rightarrow q^{'}_i(t)=q_i(t) + \delta q_i(t) = q_i(t) + \lambda I_i(q,\dot q,t).$$...
- 13
1
vote
0
answers
50
views
Functional derivative of a symmetrized field
I'm confused whether a symmetrisation/antisymmetrization of a function with respect to its arguments, i.e., $$F(x_1,x_2,...,x_n)=\frac{1}{\sqrt{N!}}\sum_{\pi}\textrm{sgn}(\pi)~f(x_{\pi(1)},...,x_{\pi(...
- 1,323
0
votes
1
answer
452
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Least action principle : is $ \delta \frac{dx}{dt} = \frac{d \delta x}{dt} $ always true?
(Just some recalls)
We have an action on which we want to apply Least action principle.
$$ S=\int_{t_i}^{t_f} L(q,\dot{q},t)dt$$
We assume that $t \mapsto q(t)$ is the function that will extremise ...
- 1,450
25
votes
2
answers
2k
views
Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $
I am reading about Lagrangian mechanics.
At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed.
The fact that the two are ...
- 281
148
votes
9
answers
18k
views
Calculus of variations -- how does it make sense to vary the position and the velocity independently?
In the calculus of variations, particularly Lagrangian mechanics, people often say we vary the position and the velocity independently. But velocity is the derivative of position, so how can you treat ...
- 2,155