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Results tagged with variational-calculus
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Variational calculus is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find extrema of functionals: mappings from a set of functions to the real numbers. The archetype application in physics is Lagrangian mechanics, seeking extrema of action functionals.
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Variational calculus and KKR method for band structure calculation
However, my understanding of variational techniques is that you are supposed to minimize (or extremize) an action integral of a lagrangean.
You can minimize whatever you want.
See, for example, …
- 23.3k
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Questions regarding the derivation of Euler-Lagrange Equation from Taylor's Classical Mechanics
How does that work?
A minimum is a stationary point (a point where the first derivative is zero). If you want to see whether it is a maximum or a minimum (or a saddle point) look at the second deriv …
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Evaluating functional derivatives
I am having difficulty evaluating the following derivative:
$$I = \frac{\delta}{\delta x(t)}\frac{\delta}{\delta x(t')}\int_{u_i}^{u_f}\frac{du}{2}\left(\frac{dx}{du}\right)^2~.$$
... How can this de …
- 23.3k
3
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Four-divergence term in Lagrangian
It is known that adding a four divergence term, $\partial_\mu A^\mu$ does not affect the equations of motion. I am trying to reason this based on the Euler-Lagrange equation. But I want to show this …
- 23.3k
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When computing the Euler–Lagrange equations, why do we assume the coordinates do not depend ...
Consider a function $F$ of two variables $a$ and $b$:
$$
F(a,b)\;,
$$
where the usual notation of parentheses and a comma indicates that $F$ is a function of two independent variables, $a$ and $b$.
Co …
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Variational derivative of a Lagrangian
I would like to know how exactly to calculate the variational derivative of:
$$L = \oint dx \: \frac{m}{a} \dot{u}(x,t) - ca(u')^2$$
with respect to $u$, ($\delta L / \delta u$).
Following my lecture …
- 23.3k
1
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How can I show that applying Hamiltonian dynamics recovers the original wave equation?
$$ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}\tag{1}$$
with $ u = u(t, x)$ over domain $x \in [0, l] = \Omega$. This can be represented as a Hamiltonian system with …
- 23.3k
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Steps taken to differentiate action in wave equation
Which then leads to the wave equation. But, how was that result (i) obtained?
It is basic multi-variable differential calculus.
See, for example, this answer, which explains something quite similar. …
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Inquiry about applying stationary action to field lagrangian
My questions are
how does he get to the second line?
Integration by parts.
Why does the last term vanish as I quoted he said above.
Presumably because his $\delta \phi$ vanishes at the boundar …
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Understanding this Lagrangian calculation
I was trying to understand this section of a Wikipedia article:
$$0 = \delta \int \sqrt{2T} d\tau =
\int \frac{\delta T}{\sqrt{2T}} d\tau =
\frac{1}{c} \delta \int T d\tau$$
For the life of m …
- 23.3k
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Why partial derivative in Lagrange is partial?
Then we differentiate action with respect to $\epsilon$ to see how action changes when $\epsilon$ changes:
$$\frac{dS}{d\epsilon} = \int \frac{\partial L}{\partial x} \frac{\partial x}{\partial \epsi …
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Dummy index question
The Maxwell's Lagrangian density is given by the equation, $$\mathcal L = -\frac{1}{4} \space F_{\mu\nu} \space F^{\mu\nu},\tag{A}$$ where $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$.
.. …
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Equations of motion for Lagrangian of scalar QED
I have to determine the equations of motion for both the complex scalar field $\varphi$ and the electromagnetic field $A_\mu$ by using the Euler-Lagrange equations.
Now I know, that because the sca …
- 23.3k
1
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Taking functional derivatives of generating functional
My main confusion is about how to take derivatives of function like $J(y_1)$ wrt to $J(x)$. For example what is the value of $$\frac{\delta J(y_1)}{\delta J(x)}$$. Is that equal some kind of delta fu …
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3
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Variation of a function
Let's say for example that i got a Lagrangian $L [x(t), \dot{x}(t), t] $ which is a functional
No, $L$ is not a functional. In physics we reserve the term "functional" to refer to a function that ta …
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