1
vote
2answers
37 views

Is it possible to write the curl in terms of the infinitesimal rotation tensor?

Is it possible to write the curl in terms of the infinitesimal rotation tensor? Basically, we can write the curl as a matrix operator $$ curl=\begin{bmatrix} 0 & -\partial z & \partial ...
3
votes
2answers
86 views

How is partial time derivative $\frac{\partial}{\partial t}$ defined for vector flows?

This question emerged when I was thinking about Liouville's theorem of classical mechanics. As far as I understand, the change of any function along the integral curves of Hamiltonian vector field is ...
1
vote
0answers
101 views

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
1
vote
1answer
41 views

Euler-Lagrange Eqn to find eqn of a straight line

I'm trying to see how we use the E-L equation \begin{equation} L_x(t,q(t),q'(t))-\dfrac{d}{dt}L_v(t,q(t),q'(t))=0 \end{equation} to find the shortest distance between two points in the Euclidean ...
0
votes
1answer
78 views

Cartesian vector field to vector field

Ok so I have a given vector field in Cartesian coordinates, say \begin{align*} \textbf{v}(x,y)=\frac{dx}{dt}\hat{\textbf{e}}_{1}+\frac{dy}{dt}\hat{\textbf{e}}_{2} \end{align*} Where $dx/dt$ and ...
3
votes
1answer
84 views

motivating the conservation of symplectic area by way of general (coordinate) covariance

I'm trying to motivate why a symplectic structure captures exactly the right structure one needs to do classical mechanics. The easiest part of this story goes like this: we need a procedure for ...
0
votes
1answer
40 views

Completely integrable geodesic flows without any degenerate point

Are there many examples of completely integrable geodesic flows (in the sense of Liouville), with say n integrals $f_1,\cdots, f_n$ such that everywhere, the differentials $(df_1,\cdots,df_n)$ are ...
4
votes
2answers
138 views

Distance metric for (infinite) lines in 3D

I would like a metric $d(\;)$ between pairs of (infinite) lines in $\mathbb{R}^3$, with these properties: If two lines $L_1$ and $L_2$ are parallel and separated by distance $x$, then $d(L_1,L_2) = ...
2
votes
0answers
450 views

Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
10
votes
1answer
310 views

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. ...
2
votes
1answer
90 views

Confusion over notation in a book on the mathematics of QFT by Faria-Melo

While formulating this question, I arrived at a likely interpretation provided in an answer to my own question below. My problem appears to be one of inexperience in working with ambient coordinates, ...
11
votes
2answers
621 views

How to introduce stress tensor on manifolds?

I want to understand the type of stress tensor $\mathbf{P}$ in classical physics. Usually in physics it is said that the force $\text d \boldsymbol F$ (vector) acting on an infinitesimal area $\text ...
1
vote
0answers
195 views

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 ...
4
votes
0answers
95 views

The Nambu bracket

Does anybody know how to show the Jacobi identity for the Nambu bracket in $\mathbb{R}^3$? The Nambu bracket with respect to $c \in \mathcal{F}(\mathbb{R}^3)$ is defined as $$\{F,G\}_c = \langle\nabla ...
12
votes
1answer
322 views

Is it possible to formulate variational calculus geometrically?

In textbooks I've seen differential geometry is done with finite-dimensional manifolds. Is it possible to generalise to banach manifolds so as to formulate the calculus of variations within it, or ...
2
votes
0answers
75 views

A lower positive bound on the number of closed orbits with given energy for a mechanical system

Let be given a mechanical system with configuration manifold $M,$ potential energy $V$ and kinetic energy $K$ corresponding to a riemannian metric on $M.$ Its dynamics is determined by the ...
0
votes
1answer
259 views

Rigid motion in curvilinear coordinates

I would like someone to clarify this since it has bedazzled me and can't seem to get a grip on it. Consider a 3D real space and Euclidean coordinates ($x_1,x_2,x_3$), with an associated standard basis ...
3
votes
0answers
72 views

what is the domain of the Lagrangian of a surface embedding?

If we view our Lagrangian particle mechanics geometrically, then we describe a particle trajectory as a map from R to a manifold, and the Lagrangian $L(x,\dot{x})$ as a function on the tangent bundle ...
14
votes
1answer
454 views

Coordinate-free techniques in Lagrangian mechanics

Consider the following Lagrangian (Exercise 3.6B from Abraham and Marsden's Foundations of Mechanics): $$ L(\upsilon)=\frac12g(\upsilon,\upsilon)+V(\tau_Q\upsilon)+g(\upsilon,Y(\tau_Q\upsilon)) $$ ...
12
votes
1answer
680 views

Car movement - differential geometry interpretation

The problem presented below is from my differential geometry course. The initial reference is Nelson, Tensor Analysis 1967. The car is modelled as follows: Denote by $C(x,y)$ the center of the ...