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Since the Euler-Lagrange equation for an autonomous Lagrangian $L(q,\dot q)$ is $$ \frac{\partial L}{\partial q} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot q}\right) = L_q - \dot q L_{q\dot q …

Here's a counterexample to the OP's literal question: Consider the following four symmetric (in fact, diagonal) $3$-by-$3$ matrices: $A_1 = \mathrm{diag}(0,0,0)$, $A_2 = \mathrm{diag}(\frac12,3,3)$, …

There is a counterexample with $d=1$ and $m=n=2$. Here is one way to construct such an example: Let $g:\mathbb{R}\to\mathbb{R}$ be a smooth function such that $g'(t)>0$ for $t\not=0$ and $g^{(k)}(0) …

I wasn't able to find a source to cite for the correct formulae, but it turns out that it's not that hard to work out the answer directly using differential geometry. First, a little notation: Let $u …

Remark: I had a little time to write a draft of my notes on the proofs of the claims I make below and have posted it on my home webpage here. (It would have made a very long post on MO, so I decided …

Well, here are a few comments that might be helpful: First, if one sets $q = a/R>0$, then the equation the OP wants to study can be written in the form $$ \dot\theta^2 +\omega^2\,\sin^2\theta - q^2\,t …

The answer is 'yes' there do exist such functions that are non-constant with singularities only along surfaces $\Sigma\subset\mathbb{C}^2$, and here is how one can understand them: First, it helps to …

The answer is that the only solutions have the form $$ f = (f_1,f_2) = \bigl(c, h(\,\overline{z}_1, z_2)\bigr) $$ where $h:\mathbb{C}^2\to\mathbb{C}$ is holomorphic and $c$ is a constant, which must e …

An approach that should work is to derive the differential equation that any minimizer would have to satisfy and check that its solutions are the known ones for which equality holds. To fill in the d …

This ODE has some very interesting properties. If one clears fractions and writes it out as $$ x(x+2y)(x-2y+1)\,y'' = (4x^2-8y^2+3x+4y)\,y' + x(4y-1)\,(y')^2, \tag1 $$ one recognizes this as the equa …

YangMills' answer shows that it is not always possible to represent a real $(1,1)$-form $\phi$ in the desired form globally on a compact complex manifold but doesn't answer the question of how to tell …

You are asking about the subject of orthogonal (coordinate) systems. There is an extensive literature on this subject, in particular by Darboux when $n=3$, and if you search on "triply orthogonal sys …

Edited on May 2, 2020: The OP pointed out that I had not addressed a special case (namely $C=1$ below), so I am amending my answer to address this and reorganizing so that the $C=1$ case gets addresse …

Your problem does not have a maximizing solution. Here is why: Assuming that $f$ is piecewise smooth and not identically constant, we can solve the Euler-Lagrange equations in an interval where $f …

There is a straightforward way to deduce necessary conditions for a space curve to lie on an ellipsoid, and it's really a matter of calculation to make these conditions explicit in terms of the curvat …