Ciro Santilli @cirosantilli  40

Like everything else in Lie group theory, you should first look at the matrix version of this operation: the matrix exponential.

The exponential map links small transformations around the origin (infinitely small) back to larger finite transformations, and small transformations around the origin are something we can deal with a Lie algebra, so this map links the two worlds.

The idea is that we can decompose a finite transformation into infinitely arbitrarily small around the origin, and proceed just like the product definition of the exponential function.

The definition of the exponential map is simply the same as that of the regular exponential function as given at Taylor expansion definition of the exponential function, except that the argument $x$ can now be an operator instead of just a number.

Examples:

The differential equation that is solved by the exponential function:with initial condition:

TODO find better name for it, "linear homogenous differential equation of degree one" almost fully constrainst it except for the exponent constant and initial value.

Is the solution to a system of linear ordinary differential equations, the exponential function is just a 1-dimensional subcase.

Note that more generally, the matrix exponential can be defined on any ring.

The matrix exponential is of particular interest in the study of Lie groups, because in the case of the Lie algebra of a matrix Lie group, it provides the correct exponential map.

This is how you transform the Lagrangian into the Hamiltonian.

Subtle is the Lord by Abraham Pais (1982) chapter 4 "Entropy and Probability" mentions well how Boltzmann first thought that the second law was an actual base physical law of the universe while he was calculating numerical stuff for it, including as late as 1872.

But then he saw an argument by Johann Joseph Loschmidt that given the time reversibility of classical mechanics, and because they were thinking of atoms as classical balls as in the kinetic theory of gases, then there always exist a valid physical state where entropy decreases, by just reversing the direction of time and all particle speeds.

So from this he understood that the second law can only be probabilistic, and not a fundamental law of physics, which he published clearly in 1877.

Musk is a truly ambivalent personality. Some points are very good. Some are very bad.

Respect on the technical side by Ciro Santilli.

But the way he treated his first wife Justine Musk, is very very weird, incomprehensible: www.marieclaire.com/sex-love/a5380/millionaire-starter-wife/

Positive Cirocoins for possibly going to reverse Twitter's unfair Trump ban if his Twitter acquisition goes through:

Negative Cirocoins for according to Wikipedia: