Tagged Questions

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Why does the arc-length formula have form $\int_a^b\left|\left|\frac{d\vec{f}(t)}{dt}\right|\right|_2dt$ for C1 curves?

This discussion focuses on $\mathcal{C}^1$ curve on $\mathbb{R}^n$. But feel free to talk about the case where we only have a continuous curve or the scenario with a manifold with a metric in general. ...
68 views

Shallow tent like soap film

A soap film circle in $x-y$ plane with center at origin can be carefully pricked with a blunt soapy pin at center and drawn out a little bit on $z$-axis forming a surface of revolution somewhat like a ...
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Let $\varphi:(M,g,\nabla)\to\mathbb{R}^n$ be a smooth embedding of a convex hypersurface. I want to explicitly calculate $$\langle \varphi,\varphi_{\ast}(\nabla\|\varphi\|^2)\rangle.$$ In particular, ...
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Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$\mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx}$$ ...
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In the paper "The existence of minimal immersions of 2-spheres" by Sacks and Uhlenbeck the authors claim that the Euler Lagrange equation for the modified functional $E_\alpha(s) = \int_M (1 + ... 1answer 50 views Calculation mistake in variation of length functional? This should be pretty simple to check if you know the basics of variational calculus. I feel like I am making an obvious mistake somewhere like not using chain rule somewhere. Let$g : \mathbb{R}^n ...
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In the article you can find at http://www.cc.gatech.edu/~turk/my_papers/schange.pdf, precisely at page 2 of the .pdf, there is a functional E which is said to be a measure of the aggregate squared ...
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geodesic metric

I'm trying to prove that the line segment is the minimizer of the distance $$d(x,y)=\inf l(\gamma),$$ where $x,y\in X$, $X$ is a Banach space, $\gamma$ is a path from $x$ to $y$ and ...
443 views

Geodesic and Euler - Lagrange equation

If we have a metric $g_{\mu \nu}$, defined in a Riemannian manifold we can write the equation of the geodesic: $$\frac{d^2x^\mu}{dt^2}+\Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{dt}\frac{dx^\beta}{dt}$$ ...
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Vector Field Generating Variation Along Curve

I'm learning a proof of the fact that length extremising curves are geodesics of the Levi-Civita connection, and have found something I don't understand. The argument states the following. Suppose ...
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Levi-Civita Connection for 2-dimensional Riemannian manifold

I'm trying to show the following. Suppose $(M, g)$ is a $2$-dimensional Riemannian manifold with connection $\nabla$. Suppose also that $\nabla$ is metric compatible, and that length extremizing ...
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Convert line integral between different metrics?

If I have $$\int\limits_0^T \frac{\sqrt{\dot{x}(t)^2+\dot{y}(t)^2}}{\sqrt{2 y(t)}}dt$$ I can convert this problem of finding the solution to the brachistochrone problem to a geometric problem by ...
292 views

Geometric proof of the geodesics of a sphere?

I have seen the standard variational proof that great circles are the geodesics on the $2$-sphere. Do you know a purely geometric proof of this fact, not involving calculus of variations or ...
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Checking my understanding of $T^*M$ as a symplectic manifold and the links between the classical description of Lagrangians vs this invariant way.

I am working through a book titled "An introduction to mechanics and symmetry" by Marsden and Ratiu. I have written up a brief summary trying to solidify my understanding of the general principles. ...
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Sufficient conditions for Hessian definiteness for critical points of functionals

Let $C$ be the set of smooth curves from the unit interval into $\mathbb{R}^n$. Let $f : C \rightarrow \mathbb{R}$ be a functional on these curves given by $f(x) := \int_0^1 L(x,\dot{x}) dt$. Define ...
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Geodesic First Variation

I'm trying to prove that if the first variation of length vanishes then the curve $\gamma$ must be an affinely parameterised geodesic. In the following $T=\dot{\gamma}$. So I've attacked the ...
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Is there an action functional whose critical points are the geodesics of a arbitrary connection on TM?

Geodesics of the Levi-Civita connection may be defined as the critical points of the action functional $S[\gamma]=\int \lvert\dot{\gamma}\rvert\,dt$ (or square it, if you like). The Euler-Lagrange ...
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Geodesics in a manifold M diffeomorphic to $\mathbb S^2$

I am now reading the book Calculus of Variations written by Jost and I encountered the following problem (in Theorem 2.3.3.): Let $M$ be a differentiable submanifold of $\mathbb R^d$ diffeomorphic to ...
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Divergence Theorem/ partial integration of higher order

I'm in the middle of a proof and i'm trying to understand a step of the proof which does give me a hard time. The proof is about minimal surfaces and at the moment I'm trying to understand why the ...
173 views

variation problem of constrained area and minimized distance

$$c=\int_{x_1}^{x_2}f_{gr}(x)\;dx$$ The integral is a time-like curve between $x_1$ and $x_2$ and at imagine fgf(x1) is a lower left corner of the rectangle and fgf(x2) is the upper right corner and ...
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Closed Geodesics as minimisers of action functional

Suppose I have a Riemannian surface $(M,g)$. It's clear that closed geodesics are critical points of the length functional $l(\gamma)=\int\left|\gamma(t)^{\prime}\right|_{g(\gamma(t))}dt$ over the ...
458 views

minimal surface of revolution when endpoints on x-axis?

What is the formula for the planar curve through $(\pm a,0)$ of fixed length $l$ which has minimal-area surface of revolution when rotated about the x-axis? I get the area of the surface to be ...
113 views

I have some trouble understanding the following question: Suppose we have 1st fundamental form $E \, dx^2+2F \, dx \, dy+G \, dy^2$ and we are given that for any $u,v$, the curve given by $x=u, y=v$ ...
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Is there a fundamental misunderstanding here or have I made an algebraic slip?

Is there a fundamental misunderstanding here or have I made an algebraic slip? I have a Riemannian metric of the form $ds^2={du^2+dv^2\over 1-u^2-v^2}$ on an open disc and I want to prove that radial ...