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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.
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,
$$
k\ L(q,\dot q)=q\frac{\p …
9
votes
Accepted
Is the Lagrangian density a functional or a function?
The Lagrangian density is a function.
Consider the following examples:
$$
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] …
6
votes
Accepted
Variation of Lagrangian density $\mathcal{L}$ w.r.t $x^{\mu}$
There are two kinds of derivatives we should distinguish:
$$
\frac{\mathrm d\mathcal L}{\mathrm dx}=\lim_{h\to 0}\frac{1}{h}\big[\mathcal L(\phi(x+h),\phi'(x+h),x+h)-\mathcal L(\phi(x),\phi'(x),x)\big …
2
votes
Accepted
Derivation of the Cartan Field equation
Note that the spin tensor is skew-symmetric in its lower indices,
$$
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
$$
-\frac14\varepsilon^{\mu\alpha\beta}\nabla_\alpha(\delta^\nu_\beta R)\delta g_{\mu\nu}=-\frac14\varepsilon^{\mu\alpha\nu}\nabla_\alpha R\,\delta g_{\mu\nu}
$$
…