Let's say 10 kg block is sliding on a frictionless surface at a constant velocity, thus its acceleration is 0. According to Newton's second law of motion, the force acting on the block is 0: $a = 0$ $F = ma$ $F=0$ So let's say that block slid into a motionless block on the same surface, the...

In E Poisson "A relativist tool kit" p.75 it says that the Gauss-Weingarten relation is: $$e^{\alpha}_{a;\beta}e^{\beta}_{b}=\Gamma^{c}_{ab}e^{\alpha}_{c}-\epsilon K_{ab}n^{\alpha}$$ We have the definitions: $$A_{a|b}=A_{a,b}-\Gamma^{c}_{ab}A_c$$ and the relation $$A^{\alpha}_{;\beta}e^{\beta}_...

It should be stressed that Noether's theorem is a statement about consequences of symmetries of an action functional (as opposed to, e.g., symmetries of equations of motion, or solutions thereof, cf. this Phys.SE post). So to use Noether's theorem, we first of all need an action formulation. How ...

I'm trying to show that there is a canonical isomorphism between a finite-dimensional vector space $V$ (regarded as a $C^\infty$ manifold) and its tangent space $T_vV, v\in V$, without using a basis, in the following sense: Regard $\omega\in V^*$ as a linear function $\tilde\omega:V\to\mathbb{R}$...