/sci/ - Science & Math

>>11491730
Just the Cauchy criterion is enough. Given [math]u = (\dot{v},v)[/math] the Cauchy problem is well posed as long as [math]u \in L^2 \oplus H^1[/math] and namely the initial condition boundary intersects the characteristics transversally.
>>11503466
Analytic geometry.

>>11452301
The Lie algebra of the entire Euclidean group is 6D, yes. Without time we don't know what "boosts" are so your question depends on what you mean by "homogeneous". We have [math]{\bf E}(3) \cong \mathbb{R}^3\rtimes O(3)[/math] anyway and if by "homogeneous" you mean orientation preserving then it's [math]SO(3)[/math]; if by homogeneous you mean that [math]\mathbb{R}^3[/math] is a homogeneous space under it then it's the entire group.
>>11452306
>I'm not sure jet budles will ever make it into the applied realm
Putting a contact structure on [math]J^\infty\Omega[/math] lets you define the Lagrangian density as [math]\Omega^n(J^\infty\Omega)[/math], which lets you do quantum/classical field theory as if you're doing quantum/classical mechanics. I don't know what you mean by "applied" but it's the backbone of theoretical physics. Sardanashvily has a few nice articles on this.
>>11452314
>What about speeding up time tho?
In the non-relativistic case "time" is just the parameter on a physical trajectory, and time evolution is just an automorphism [math]\alpha_t[/math] on [math]C^\infty[/math] generated by the Poisson bracket [math]\{H,\cdot\}[/math] with the Hamiltonian. Again, "boosts" don't exist here and speeding up time just scales this parameter. It's not actually part of the isometries of space.