# Geodesics and potential function

MathOverflow Asked by Bruno Peixoto on October 29, 2020

I try to assemble concepts of differential geometry for my own comprehension of the subject. I understand a manifold is a higher dimensional surface. It has a metric which perform inner product in the tangent space. A curve on the respective manifold has a covariant derivative, which remains on the tangent bundle. A geodesic is a manner to comprehend straight lines on a manifold. They might be closed like on a sphere. They are the main tool of physicists to comprehend the universe through the lagrangian and hamiltonian framework. On the lagrangian framework, there is a kinetic $$K = g_{ij} dot{x}_i dot{x}_j$$ and potential $$U = V(x)$$ energies which computes the lagrangian $$L = K – U$$. The ausence of potential energy coincide with the geodesic equation. The Einstein notation is in force here.

In mathematical terms, I do not comprehend the role of the potential or dissipative term on the geodesic equation and further explanations on similar manner as I will explain shortly.

As far I comprehend, the geodesic statement is: given two points A and B, the geodesic which binds both points on a simply connected non-compact smooth manifold is the solution to the boundary value problem of former equation below.

The same statement but for the latter equation entertains other interpretation, no longer a geodesic at strict sense. I comprehend from physical perspective that given the manifold endowed by a metric, the second and third are relative to the actuation of forces on the motion particle. But for this, one defines the particle, which is merely an abstract conception, a trick to better comprehend intuitively.

$$begin{equation} ddot{x}^j + Gamma^j_{i k} dot{x}^i dot{x}^k = 0 end{equation}$$

$$begin{equation} ddot{x}^j + Gamma^j_{i k} dot{x}^i dot{x}^k + g^{ji} frac{partial V}{partial x^i} = 0 end{equation}$$

$$begin{equation} ddot{x}^j + Gamma^j_{i k} dot{x}^i dot{x}^k + g^{ji} frac{partial V}{partial x^i} + g^{ji} frac{partial R}{partial dot{x}^i} = 0 end{equation}$$

## Related Questions

### Does the following sum converge?

1  Asked on December 21, 2021 by ryan-chen

### Is a homotopy sphere with maximum Morse perfection actually diffeomorphic to a standard sphere?

0  Asked on December 21, 2021 by fredy

### Differential birational equivalence

0  Asked on December 20, 2021

### Calculating $n$-dimensional hypervolumes ($n sim 50$), for example

1  Asked on December 20, 2021 by luka-klini

### Definition of a system of recurrent events

1  Asked on December 20, 2021 by rob-arthan

### Regularity of a conformal map

2  Asked on December 20, 2021 by amorfati

### Is there a standard definition of weak form of a nonlinear PDE?

1  Asked on December 20, 2021

### Is there an algebraic version of Darboux’s theorem?

1  Asked on December 20, 2021

### The strength of “There are no $Pi^1_1$-pseudofinite sets”

1  Asked on December 20, 2021

### Are groups with the Haagerup property hyperlinear?

1  Asked on December 18, 2021 by maowao

### Proof of second incompleteness theorem for Set theory without Arithmetization of Syntax

0  Asked on December 18, 2021

### Characterization of effective descent morphism

0  Asked on December 18, 2021 by kind-bubble

### Analogue of decay of Fourier coefficients of a smooth function on $mathbb{S}^1$

2  Asked on December 18, 2021

### Strong Data Processing Inequality for capped channels

1  Asked on December 18, 2021 by thomas-dybdahl-ahle

### lattice suprema vs pointwise suprema

2  Asked on December 18, 2021 by giuliosky

### In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?

2  Asked on December 18, 2021

### Are those distributional solutions that are functions, the same as weak solutions?

0  Asked on December 18, 2021

### Ability to have function sequence converging to zero at some points

1  Asked on December 16, 2021 by mathcounterexamples-net

### Curious anti-commutative ring

1  Asked on December 16, 2021 by robert-bruner

### Minkowski (box-counting) dimension of generalized Cantor set

0  Asked on December 16, 2021

### Ask a Question

Get help from others!