# geodesics, surfaces and Euler-Lagrange

can i  blatantly showed off to you what are -some friends and i- doing? There are also another beauties like this…

esquematización aproximada para “ver” donde “anda”  la curva geodésica

… let me tell that is important the next play:

$l=\int_I\|C'(u)\|du$

$=\int_0^t\|C'(u)\|du$

$=\int_0^sdu=s$

when $C$ is arc – parameterized.

### 6 responses to “geodesics, surfaces and Euler-Lagrange”

1. en un cono

“Geodesics on a Cone”

$z=\sqrt{x^2+y^2}$

http://www.maths.bris.ac.uk/~maxmr/mech2/mech2/conenotes.pdf

2. Frenet-Serret dynamics

http://arxiv.org/pdf/hep-th/0105040.pdf

… We consider the motion of a particle described by an action that is a functional of the Frenet-Serret [FS] curvatures associated with the embedding of its worldline in Minkowski space. We develop a theory of deformations tailored to the FS frame. Both the Euler-Lagrange equations and the physical invariants of the motion associated with the Poincar ́e symmetry of Minkowski space, the mass and the spin of the particle, are expressed in a simple way in terms of these curvatures. The simplest non-trivial model of this form, with the Lagrangian depending on the first FS (or geodesic) curvature, is integrable. We show how this integrability can be deduced from the Poincar ́e invariants of the motion. We go on to explore the structure of these invariants in higher-order models. In particular, the integrability of the model described by a Lagrangian that is a function of the second FS curvature (or torsion) is established in a three dimensional ambient spacetime… “

3. traducción de un trabajo original de L. Euler:

“A more accurate treatment of the problem of drawing the shortest line on a surface”

http://arxiv.org/pdf/0801.0897.pdf

4. c-kit

also

“Calculus of Variations 4: Several Functions of a Single Variable”

http://www.math.unl.edu/~scohn1/8423/cvar4.pdf

5. Cf:

“Calculus of Variations 5: Geodesics on Surfaces in R3”

http://www.math.unl.edu/~scohn1/8423/cvar5.pdf