## Archive of posts with tag “mathematical physics”

### The duality between two plane trajectories related by a conformal map

The conformal map $\fc wz$ transforms the trajectory with energy $-B$ in potential $\fc Uz\ceq A\v{\d w/\d z}^2$ into the trajectory with energy $-A$ in potential $\fc Vw\ceq B\v{\d z/\d w}^2$. I will prove this beautiful result and show some implications of it.

- Categories: physics
- Tags: complex, classical mechanics, canonical transformation, kepler problem, mathematical physics, vector analysis, long paper

### Rotational symmetry of plane lattices as a simple example of algebraic number theory

For a plane lattice, there is only a finite number of different rotational symmetries that are compatible with the discrete translational symmetry. For example, the 5-fold rotational symmetry is not one of them. Why is that? It turns out that whether an $m$-fold symmetry is compatible with translational symmetry is the same as whether $\varphi(m)\le2$.

- Categories: math
- Tags: complex, condensed matter physics, algebraic number theory, lattice, mathematical physics

### A measure-theoretic formulation of statistical ensembles (part 2)

For sake of rigor and generalizability, I feel it necessary to try to have a mathematical formulation for statistical ensembles. I chose measure spaces as the underlying mathematical structure of thermal systems and tried to justify the method of statistical ensembles by deducing them from some axioms.

- Categories: physics
- Tags: mathematical physics, statistical mechanics, functional analysis, measure theory, probability, long paper

### A measure-theoretic formulation of statistical ensembles (part 1)

For sake of rigor and generalizability, I feel it necessary to try to have a mathematical formulation for statistical ensembles. I chose measure spaces as the underlying mathematical structure of thermal systems and tried to justify the method of statistical ensembles by deducing them from some axioms.

- Categories: physics
- Tags: mathematical physics, statistical mechanics, functional analysis, measure theory, probability, long paper