Questions tagged [lie-derivative]

The Lie derivative gives a way to define the derivative of a tensor field in the direction of a vector field.

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Lie derivative of a function w.r.t. vector field and Lie algebra

A note for self-reference: this post continues but differs from another post: Lie derivative of a function (of a point) with respect to a vector field Lie derivative $L_XY$ of vector field $Y$ at a ...
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Component functions of the exterior derivative

Let $M$ be a smooth manifold, $(E_i)$ a smooth local frame for $M$, and ($\varepsilon^i$) the dual coframe. For each $i$, let $b^{i}_{jk}$ denote the component functions of the exterior derivative of (...
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Lie derivative from covariant derivative

The directional derivative is $\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \mathbf{v}$ The directional derivative on a manifold is the covariant derivative $\begin{align} \nabla_\...
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A basic question on Lie derivative

In the definition of Lie derivative $$ \mathcal{L}_VX^\mu=[V,X]^\mu=V^\mu\partial_\mu X^\nu\partial_\nu-X^\mu\partial_\mu V^\nu\partial_\nu=(V^\mu\partial_\mu X^\nu-X^\mu\partial_\mu V^\nu)\partial_\...
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In $SO(3)$, what is the derivative at time $t$ of $R(t) = \exp([\omega]_\times t)$?

I cannot wrap my head around something that seems pretty basic. In "A micro Lie theory for state estimation in robotics", the authors derive the structure of the Lie algebra of $SO(3)$ by ...
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Lie derivatives of tensor product

Given a smooth vector field $X = \sum X^i \frac{\partial}{\partial x_i}$ on $\mathbb{R}^n$, show that for all $i,j = 1,...,n$. $$\mathcal{L}_X(d x_i \otimes d x_j) = \sum_{r=1}^{n}(\frac{\partial X^j}{...
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Lie derivative of a metric

This question has been asked before but the answers were not clear to me so I am asking again. We have $g = \sum_{i,j} g_{ij} dx_i \otimes dx_j$ as a smooth $(0,2)$-tensor and asked to show that given ...
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Does a Lie derivative of a vector field involve subtracting vectors from different spaces?

Addition (and subtraction) is not by default defined for vectors in different spaces, even if those vector spaces are isomorphic (it is possible to define addition, but there are many ways to define ...
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lie bracket is vanish iff translation is commute

I want to prove following claim: Given two vector fields $X$ and $Y$ on smooth manifold $M$, $[X,Y]=0$ if and only if $\Phi^X_t \circ \Phi^Y_s \circ \Phi^X_{-t} \circ \Phi^Y_{-s}(q)=q$ for $\forall ...
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What's the advantages of writing standard calculus into lie differentiation?

Say $\dot{x} = f(x)$, $x\in\mathcal{M}$, $\phi: \mathcal{M} \mapsto \mathcal{M}$. Then $\dot{\phi} = f \cdot \nabla_x \phi $. However, one can define a lie differentiation and write $f \cdot \...
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tangent field is closed under lie bracket operation.

I want to prove this claim : If $[X,Y]=0$, then for any tangent field $Z$, $[X,[Y,Z]]=0$ I try to do this by using local coordinate. $$ X=\sum_{n=1}^{m}a_i\frac{\partial}{\partial x_i}\space\space\...
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How to use the Lie derivative to “perform” a parallel transport along a curve

Setup Consider a metric, for example that of a sphere with fixed radius $R$, i.e. $$ds^2 = R^2 d\theta^2 + R^2\sin^2\theta^2d\varphi^2,$$ and a curve on that sphere $\gamma = (\theta_0, \varphi)$, ...
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Commuting solenoidal fields

Given a pair of smooth divergence free vector fields $X,Y$ on $\mathbb{R}^n$, is there some known characterization whch tells when they do commute? My approach to solve the problem is to start from ...
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Taylor series using Lie derivatives

Suppose I have a (simple, compact) Lie group $G$ with generators $T_i$. I would like to Taylor expand a (smooth) function $f:U\to\mathbb{R}$. I can always write a normal expansion like \begin{align} x ...
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Derivative of the residue logarithm of a formal pseudo-differential series

In my lecture notes in the proof of Adler's theorem they use $$ D_t(res \log A)= res (D_t(A) \circ A^{-1}). $$ Where $D_t$ is derivation of the differential field $F$ and $A$ is a formal pseudo-...
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Confusion in calculating Lie derivative of lifted symplectical vector fields

I am a bit confused about a calculation I did and I do not see my mistake. First of I took any vector field $X \in \mathfrak{X}(M)$ and lifted it as some $\hat{X}\in \mathfrak{X}(T^*M)$ (s.t. $T\pi\...
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Pushfoward and Lie bracket

Consider two smooth vector fields $X,Y$, denote $[X,Y]$ their Lie bracket, $e^{tX}$ (resp. $e^{tY}$) the flow of $X$ (resp. $Y$) and $\varphi * X$ is the pushforward of $X$ by a diffeomorphism $\...
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Commuting vector fields with common first integrals

Let $M$ be a $n-$dimensional smooth manifold and $X\in\chi(M)$ be a smooth vector field defined on it. Let $f_1,...,f_{n-2}:M\rightarrow \mathbb{R}$ be functionally independent first integrals of $X$, ...
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Identities for Lie derivatives and Casimir operators

Suppose I have (simple, compact) Lie group with Lie derivatives $\partial_i$. These do not commute, but instead it is $[\partial_i,\partial_j]=\partial_i\partial_j-\partial_j\partial_i=-f_{ijk}\...
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Lie bracket of left-invariant vector fields, wrong reasoning

I'm trying to understand left-invariant vector fields, but I come to a contradiction. Can you tell me where I'm wrong? Here's the definition. Let $G$ be a Lie group. Denote by $L_g:G\to G$ the ...
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Commuting covariant Derivatives of a variation

I am trying to understand the proof of the First and Second Variation of Arclength formulas for Riemannian Manifolds. I want some verifaction that the following covariant derivaties commute. I find it ...
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What is the applications of Lie derivations?

Let $K$ be a commutative ring with unity. Let $A$ be a unital algebra over $K$. We write $[x,y]= xy-yx$ for every $x,y \in A$ and we call it Lie product (or Lie bracket). A linear map $L: A \to A$ is ...
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Confusion Regarding Definition of Lie Derivative

I am having some trouble reconciling two definitions for the Lie derivative. Let $X$ be a vector field on a smooth manifold, $M$, and let $\varphi_t(x)$ be the local flow through the point $x \in M$. ...
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Higher covariant derivatives and the exterior derivative

Let me start with the following tl;dr version of my question What is a higher-order derivative, in general? How does it relate to the exterior derivative and to differential forms? Suppose we ...
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If g is a solvable Lie algebra then its form of Killing is zero? [duplicate]

I know this is not true but I would like to know a counterexample. This is contrary implication of Cartan's criterion. I would also like someone to tell me what other conditions the soluble lie ...
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Covariant derivative of Killing field invariant under flow

Let $X$ and $Y$ be Killing fields on a Riemannian manifold, with $[X,Y]=0$. Is then the total covariant derivative $\nabla Y$ invariant under the flow of $X$? If so, how do you prove it? I've seen a ...
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What are vector fields given by the partial derivatives of coordinate functions?

I am using Sean Carroll's book "Spacetime and Geometry" to learn about differential topology from a physics point of view. After introducing vectors on a manifold, he defines the commutator of two ...
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Prove Taylor expansion $(\mathrm{exp}\,tX)(\mathrm{exp}\,tY)=\mathrm{exp}(t(X+Y)+\frac{1}{2}t^2[X,Y]+o(t^2))$

This is an exercise from John Lee’s book Introduction to Smooth Manifolds, GTM218, chapter 20. It says For Lie group $G$ and two vector fields $X,Y\in \mathrm{Lie}(G)$, we have $$(\mathrm{exp}\,...
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Prove a function is a first integral iff its Lie derivative vanishes

Definition. $X$ smooth vector field over a manifold $M$, the Lie derivative of $f\in C^{\infty}(M)$ along the flow of $X$ is: $\mathcal L_X f(m) = \langle df, X \rangle (m)$. Definition. $M$ manifold,...
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The Lie derivative of a vector field with respect to another is the Lie bracket. Where is this useful?

Maybe this is a weird question... It's known that the Lie derivative of one vector field with respect to another equals their Lie bracket. The proofs in the literature rely on viewing vector fields ...
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Lie derivative of vector field and Lie bracket

This is my problem: Let $X,Y$ be a couple of vector field over a smooth manifold $M$ and $p\in M$, we define Lie derivative of $Y$ along $X$ in $p$, the vector field $$\mathcal L_X(Y)_p=\lim_{h\to0}\...
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Why is this identity about commutators of Lie derivatives true?

I am reading the paper "On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations" by Lubich. On page 2147 the author claims $$[T,V](\psi) = T'(\psi) V(\psi) - V'(\psi) T(...
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Lie derivative of vector fields and flows of vector fields: Is this formula right?

In the lecture we have defined the Lie derivative as $$\mathcal{L}_{X}Y:=\frac{\mathrm{d}}{\mathrm{d}t}\bigg\vert_{t=0}\Phi_{t}^{\ast}Y$$ where $X,Y\in\mathfrak{X}(\mathcal{M})$ are vector fields on ...
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Definition of Lie derivative by Hitchlin

I am following Hitchlin notes on differentiable manifolds: Now suppose $Y$ is a vector field, considered as a map $Y : M → TM$. With a diffeomorphism $F : M → M$, its derivative $DF_x : T_x → T_{...
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interchanging limit and exterior derivative to prove $L_X(df) = d(L_Xf)$

Let $M$ be a smooth manifold, $f: M \to \Bbb{R}$ a smooth function and let $X$ be a smooth vector field on $M$, with flow $(t,p) \mapsto\phi(t,p) \equiv \phi_t(p)$. I'm trying to prove that the Lie ...
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vertical component of Lie bracket

Let $f:X\to Y$ be a submersion equipped with a connection given by a vertical projection $\mathrm V$. Let $\vec v_1,\vec v_2$ be vector fields on $Y$ with unique horizontal lifts $\vec u_1,\vec u_2$. ...
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Evaluating the Lie derivative of a tensor

Let $X$ be a vector field, $L_X$ the Lie derivative and $T$ a tensor of rank $(1,2)$. I would like to find $(L_X T)^a{}_{bc}$. What I have tried so far $$\begin{align*}(L_X T)^a{}_{bc}&=(L_XT)(dx^...
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Is there a coordinate-free proof of this Lie derivative identity?

Wikipedia mentions (here and here) that the Lie derivative has the following appealing commutator: $$[\mathcal{L}_X,\iota_Y]=\iota_{[X,Y]}$$ The only way I know to demonstrate this identity relies on ...
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The radial vector field $x_i \partial/\partial x_i$

In the Wikipedia article on Lie derivative we read that $\mathcal{L}_X \omega$ (where $X$ is a vector field and $\omega$ is a 1-form) evaluates the change of $\omega$ along the flow defined by the ...
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Vector fields such that their Lie derivatives differ

I'm trying to find some vectors fields $X$, $Y$, $Z$ on $\mathbb R^2$ such that the Lie derivatives $L_XZ\ne L_YZ$ at the origin, and $X=Y=\frac{\partial}{\partial x}$ on the x-axis. Now, we have ...
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Eulerian Variation of Four Velocity ??

Problem is Given in the Image, Iam unable to understand the way $\delta u^{\alpha}$ is calculated here, please help in this regard
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Dimensions of forms and vectors in Lie and interior derivatives

I read D. Bachman's as well as parts of H. M. Edwards' textbook on forms. Now trying to understand Lie and interior derivatives. Suppose $\omega$ is 1-form defined on $\mathbb{R}^n$. What is $\omega(...
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Lie Derivative in coordinates

Lie Derivative is defined through its action on vector fields, as: $\mathcal{L}_XY=[X,Y]$ This is linear in its arguments due to the linearity of Lie Bracket. While, for functions: $\mathcal{L}_Xf=...
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Leibniz-rule for Lie-derivative: special case for 1-forms

I have to prove the following statement for a 1-form $\alpha\in\Omega^{1}(\mathcal{M})$ and two vector fields $X,Y\in\mathfrak{X}(\mathcal{M})$: $$\mathrm{d}\alpha (X,Y) = X(\alpha(Y))-Y(X(\alpha))-\...
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A “vector” in the interior differentiation

I am trying to understand the interior derivative $\cal{i}$$_X \omega$... $\omega$ is a 1-form; $X$ is a "vector". I have been looking for worked examples here at Math.SE. What confuses me is that, ...
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Lie derivative of a functional

Suppose $F$ is a functional of tensor fields $g$ and $h$ in a manifold $M$ and $X$ a vector field. Does the lie derivative act on the functional as it act on a function , that is ,is this formula $$\...
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How to prove that $\frac{d}{dt}|_{t=t_0}(\theta_t^*A)_p=(\theta_{t_0}^*(\mathscr{L}_VA))_p$, 12.36 in Lee's Introduction to Smooth Manifolds

I'm interested in the proof of the following proposition, given in 12.36 of John M. Lee's Introduction to Smooth Manifolds: Suppose $M$ is a smooth manifold with or without boundary and $V\in \...
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Errant Minus Sign, Lie Derivative Formula…

I feel really silly asking this question, but.... I'm trying to prove the identity $$ \mathscr{L}_{fX}\omega \;\; =\;\; f\mathscr{L}_X\omega - \omega(X)df $$ for $f \in C^\infty(M), \; X \in \...
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Derivative of the pull-back on flow of a vector field

I've just started reading differential geometry, however i was trying to solve one of the examples from Nonholonomic mechanics and control by Bloch. Chapter 2, problem 2.6-1. Consider a two-form $\...
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Repetitive Lie Derivative

I'm currently diving into non linear control systems and I have some problems with the understanding of the repetitive Lie Derivative. Considering the following system (note: vector fields are bold ...

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