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Results tagged with variational-calculus
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user 84967
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.
9
votes
Accepted
Is the Lagrangian density a functional or a function?
The Lagrangian density is a function.
Consider the following examples:
$$
A[f]=\int_0^1\mathrm dx\ f(x)
$$
and
$$
B(f(x))=f(x)
$$
It is clear that $A$ is a functional, because for example
$$
A[\sin] …
2
votes
Accepted
Derivation of the Cartan Field equation
Note that the spin tensor is skew-symmetric in its lower indices,
$$
s_{ij}{}^k=-s_{ji}{}^k
$$
Therefore, we have $s_{ij}{}^k=s_{[ij]}{}^k$. From this, its easy to see that
$$
A_{ij}{}^k=s_{ij}{}^k \ …
1
vote
Accepted
Equation of motion from $D=3$ Lorentz Chern-Simons action
Note that the term that is missing is
$$
-\frac14\varepsilon^{\mu\alpha\beta}\nabla_\alpha(\delta^\nu_\beta R)\delta g_{\mu\nu}=-\frac14\varepsilon^{\mu\alpha\nu}\nabla_\alpha R\,\delta g_{\mu\nu}
$$
…
6
votes
Accepted
Variation of Lagrangian density $\mathcal{L}$ w.r.t $x^{\mu}$
There are two kinds of derivatives we should distinguish:
$$
\frac{\mathrm d\mathcal L}{\mathrm dx}=\lim_{h\to 0}\frac{1}{h}\big[\mathcal L(\phi(x+h),\phi'(x+h),x+h)-\mathcal L(\phi(x),\phi'(x),x)\big …
13
votes
Accepted
When is numerical value of Lagrangian evaluated on-shell a full differential?
Theorem: let $L$ be a homogeneous function of degree $k$; then the on-shell lagrangian is a total derivative.
Proof: according to the Euler's homogeneous function theorem,
$$
k\ L(q,\dot q)=q\frac{\p …