Questions tagged [geodesic]

Question 1

Motivation: I am trying to find and plot the Geodesics on the Torus, and obtained the following ODE: $$ \dot{\theta}(\varphi)=f_P(\theta) \qquad f_P(\theta)=\frac{1}{Pr}(R+r\cos\theta)\sqrt{(R+r\cos\...

Question 2

Let us consider a Lagrangian system for which the equations of motion come from Hamilton's principle and are such that the control variable $\tau$ equals the equations of motion of a free system, ...

Question 3

Suppose I have the ellipsoid $$ \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1 $$ And I have two points on this surface, $P_1 = (x_1, y_1, z_1)$, and $P_2= (x_2, y_2, z_2) $. I am ...

Question 4

Let $g_{ij}$ be the components of a smooth 2D metric tensor with the quadratic line element $$ds_0^2 = g_{ij}\, dx^i \, dx^j = e^0 \cdot g_{ij}\, dx^i \, dx^j.$$ Perturb the metric into $$ds^2_\...

Question 5

I'm tying to find where I went Wrong in my efforts to solve problem 6.1 in Taylor’s book on classical mechanics. Using spherical polar coordinates $(r,\theta,\phi)$, show that the length of a path ...

Question 6

I am fairly new to the theory of CAT spaces, and I am trying to understand which familiar matrix spaces are CAT($\kappa$) for some $\kappa$. For instance, the space of real symmetric positive definite ...

Question 7

We have a curve in a Riemannian manifold $\gamma: [0,1] \to \mathcal{M}$. We know the Riemannian metric at each point $\gamma(t)$. Is this sufficient to do parallel transport of a vector along the ...

Question 8

Let $(M,g)$ be a Cartan-Hadamard manifold (simply connected, complete and with sectional curvature K≤0 ) and let $K \subset M$ be a closed convex body with $C^{1,1}$ boundary. Fix an interior point $p ...

Question 9

There are several notions of strict convexity for geodesic metric spaces. I showed that if $X$ is strictly convex by distances, then $X$ is strictly convex by balls. Proof. Let us fix $x,a,b \in X$ ...

Question 10

In the manifold $$ M=\{x\in\mathbb{R}^2:\ |x|>a\}\quad(a>0), $$ you can write $M\cong (a,\infty)\times S^1$ with polar coordinates $(s,\theta)$ and take a conformal rescaling of the Euclidean ...

Question 11

How do you find the parametrization of the shortest curve between two points on the following surface: I am primarily interested in the parametrization r(t), more so than the length. Although ...

Question 12

I am calculating the length of an arc between two points on a sphere labeled in both Cartesian coordinates and spherical coordinates to confirm that my conversions and formulas are correct. I'm ...

Question 13

Given the triangle in the Poincare model, would most mathematicians notate a side of the triangle $\overline{\text{BC}}$ or ? Basically, is there a standard convention based on function/visual ...

Question 14

I was discussing with a friend and he said that the Riemannian exponencial is Locally bi-Lipschitz. But it is not clear to me why and I coundn't find this fact in any textbook. I tryied working it out ...

Question 15

I'm working on this exercise for my riemannian geometry course, but I’m a bit stuck. It's about the flat torus $T^2(v_1,v_2)=\mathbb R^2/\Lambda$ where $\Lambda=\mathbb Zv_1\oplus\mathbb Zv_2$. ...

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Questions tagged [geodesic]

Is there a way to rewrite a differential equation to obtain uniqueness? Lipschitz

Are the motion equations of an optimal control problem geodesics on a manifold?

Finding the minimum distance path on the surface of an ellipsoid [duplicate]

How do geodesics in 2D change under conformal change of the metric?

Geodesics on spheres (from Taylor’s classical mechanics) [duplicate]

Are $SO(n)$ and/or $SE(n)$ CAT spaces?

If the Riemannian metric is known for each point along a curve, can you compute the parallel transport of a vector along the curve?

Do geodesics from an interior point intersect the boundary of a convex body only once?

Space that is strictly convex by balls but not strictly convex by distances

In $\Bbb{R}^2\setminus\Bbb{D}^2$ with warped metric write the limit of θ-coordinate as the geodesic reaches the boundary in terms of initial data

Geodesic on crescent-shaped 3-d uv-surface

calculation discrepancy measuring arclength between points in Cartesian vs. spherical coordinates

What is the convention for notating geodesics in hyperbolic geometry?

Is the Riemannian exponential locally bi-Lipschitz?

Flat torus defined by a lattice and minimal length of closed geodesics

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