Credit: Me :D

From mathematics to physics and back again:

research at IRF


Daniel Kastinen

Credit: Me :D

Who am I?

  • Living in Kiruna since 2017
  • Finished PhD Nov. 2022
    From Meteors to Space Safety: Dynamical Models and Radar Measurements of Space Objects
  • Heired as permanent staff scientist February 2023
  • Task: Conduct research on meteors and space objects!

Solar Terrestrial and Atmospheric Research

Meteors and space objects

From mathematics to physics and back again?

  • Observe phenomena
  • Describe using mathematics
  • Make predictions / conclusions / extrapolations / ...
  • (Manipulate system)
  • Observe phenomena

Meteors and space objects

  • Meteors, meteoroids, dust
  • Space debris
  • Asteroids and comets
  • Solar system dynamics
  • Mesosphere and Lower Thermosphere

Meteors and space objects

  • Meteors, meteoroids, dust
  • Space debris
  • Asteroids and comets
  • Solar system dynamics
  • Mesosphere and Lower Thermosphere

First topic:

Meteors

Credit: Nathan Myhrvold

First topic:

Meteors

Where do meteoroids come from?

Rosetta OSIRIS - 67P/Churyumov-Gerasimenko

  • Comet sublimation
  • Asteroid collision
  • Catastrophic disruption
  • Interstellar?
  • Impact ejecta
  • ...

Meteors

Meteoroid dynamics in the solar system

Meteors

Meteoroid to a meteor

AllSky7

AllSky7

Perseid Meteor Shower 2023 ( youtube @MagicCarpetMedia )

EISCAT 3D

Photo: EISCAT

Meteors

My work: connecting the big picture

My primary math tools:

  • Differential equations
  • Numerical integration
  • Linear algebra & geometric algebra
  • Statistics
  • (broad base)

My primary math tools:

  • Differential equations
    • Coupled partial differential equations
    • Hamiltonian mechanics
    • ...
  • Numerical integration
    • Differential equation solvers
    • Root finding & integral estimation
    • ...
  • Linear algebra & geometric algebra
    • Quaternions
    • Matrix equation solving
    • ...
  • Statistics
    • Bayes' theorem
    • Markov-Chain Monte-Carlo methods
    • ...
  • (broad base)

First 2 years of uni speed-run! Lets'a go!

Lets dig into the math!

Hamiltonian mechanics

(ordinary) differential equations

A function and its derivatives

$$ f(t) = t^2 + 4 \\ \frac{\mathrm{d}}{\mathrm{d}t} f(t) = \frac{\mathrm{d}f}{\mathrm{d}t}(t) = \dot{f}(t) = 2 t \\ \frac{\mathrm{d}^2}{\mathrm{d}t^2} f(t) = \ddot{f}(t) = 2 $$

An equation of a function and its derivatives

$$ \dot{f}(t) = c f(t) $$ $$ f(t) = A e^{ct} $$ $$ f(0) = 420,\;\; \dot{f}(0) = \frac{420}{42} \Rightarrow \\ \Rightarrow f(t) = 420 e^{\frac{t}{42}} $$

Lets dig into the math!

Hamiltonian mechanics

Partial differential equations

A function of two variables!

$$ f(x, y) = x^2 + y^2 + x y^2 \\ \frac{\partial}{\partial x} f(x, y) = \partial_x f(x, y) = 2x + y^2 \\ \partial_y f(x, y) = 2y + 2yx \\ $$ $$ \partial_y \partial_x f(x, y) = \partial_y (2x + y^2) = 2y \\ \partial_x \partial_y f(x, y) = \partial_x (2y + 2yx) = 2y \\ $$

A pattern! (check out Lie algebra) If

$$ \frac{\partial}{\partial x} y = 0 \;\;\text{and}\;\; \frac{\partial}{\partial y} x = 0 \Leftrightarrow \\ \Leftrightarrow \partial_y \partial_x = \partial_x \partial_y $$

"Total" derivative (chain rule rules!)

$$ h(t, a), a(t) \\ \frac{\mathrm{d}h}{\mathrm{d}t} = \frac{\partial h}{\partial t} + \frac{\partial h}{\partial a} \frac{\partial a}{\partial t} $$

Lets dig into the math!

Hamiltonian mechanics

Vector notation and spaces

An element of a space: $\bar{x} \in \mathbb{R}^3$

$$ \bar{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \end{bmatrix} \\ $$

Functions between spaces

$$ \bar{f} : \mathbb{R} \mapsto \mathbb{R}^2 \\ \bar{f}(t) = \begin{bmatrix} t + 2 \\ t^2 \\ \end{bmatrix} $$ $$ g : \mathbb{R}^2 \mapsto \mathbb{R} \\ g(\bar{x}) = g(x_1, x_2) = x_1 x_2 $$

Derivatives of these functions

$$ \partial_t \bar{f} (t) = \begin{bmatrix} \partial_t ( t + 2 ) \\ \partial_t ( t^2 ) \\ \end{bmatrix} = \begin{bmatrix} 1 \\ 2t \\ \end{bmatrix} $$
$$ \partial_{\bar{x}} g (\bar{x}) = \nabla g (\bar{x}) = \\ = \begin{bmatrix} \partial_{x_1} \\ \partial_{x_1} \\ \end{bmatrix} g (\bar{x}) = \begin{bmatrix} \partial_{x_1} g (\bar{x}) \\ \partial_{x_2} g (\bar{x}) \\ \end{bmatrix} = \begin{bmatrix} x_2 \\ x_1 \\ \end{bmatrix} $$

Lets dig into the math!

Hamiltonian mechanics

Vector notation and spaces

My tip: code it up! E.g. python is simple to learn, get a feel for the math.

import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(-1, 1, 100)  # List of t-values

f = np.stack([t + 2, t**2])
df = np.stack([np.full_like(t, 1), 2*t])

# Plot results!
fig, axes = plt.subplots(1, 2)
axes[0].plot(f_values[0, :], f_values[1, :])
axes[1].plot(df_values[0, :], df_values[1, :])
plt.show()

Lets dig into the math!

Hamiltonian mechanics

Coordinates $\bar{q}$ and their time-derivatives $\dot{\bar{q}}$ forms a phase space $M$

For example: a pendulum (coordinate is angle)

$$ M = \mathbb{R}^{2} $$

A two body gravitational system in three dimensional space

$$ M = \mathbb{R}^{2 \cdot (3 \cdot 2)} = \mathbb{R}^{12} $$

Lets dig into the math!

Hamiltonian mechanics

Simplifying a bit: Hamiltonian is system total energy

Potential energy + kinetic energy

$$ \bar{x} = \begin{bmatrix} \bar{q} \\ \bar{p} \\ \end{bmatrix} \\ H(\bar{x}) = T + V $$

The pendulum again:

$$ H(q, p) = \frac{p^2}{2 m L^2} - g m L \cos{q} \\ p = m L^2 \dot{q} $$

Lets dig into the math!

Hamiltonian mechanics

How does it evolve with time?

$$ \frac{\mathrm{d} q_i}{\mathrm{d}t} = \frac{\partial H}{\partial p_i} $$ $$ \frac{\mathrm{d} p_i}{\mathrm{d}t} = -\frac{\partial H}{\partial q_i} $$

Lets dig into the math!

Hamiltonian mechanics

Now we go bananas (lets skip some details)

Poisson brackets

$$ \{f ,g \} = \sum_{i=1}^N \left ( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right ) $$

if $\bar{x}$ is a solution to Hamiltons equations

$$ \frac{\mathrm{d}\bar{x}}{\mathrm{d}t} = \{\bar{x},H\} \\ $$

"same" differential equation as before

$$ \frac{\mathrm{d} x_i}{\mathrm{d}t} = \{x_i,H\} = \{\cdot,H\} x_i \Leftrightarrow \\ \Leftrightarrow x_i(t) = e^{t\{\cdot,H\}}x_i(0) $$

This is in itself a differential equation

Generally no analytic solution! :(

Lets dig into the math!

Hamiltonian mechanics

But! The form hints as to some things we can try

$$ \bar{x}(\Delta t) = e^{\Delta t \{\cdot,H_V + H_T\} }\bar{x}(0) \approx \\ \approx e^{\Delta t \{\cdot,H_V\} } e^{\Delta t \{\cdot,H_T\} }\bar{x}(0) $$
$$ e^{\Delta t (\{\cdot,H_V\} + \{\cdot,H_T\}) } = \sum_{i=0}^{\infty} \frac{\Delta t^i (\{\cdot,H_V\} + \{\cdot,H_T\})^i}{i!} $$ $$ e^{\Delta t \{\cdot,H_V\}} e^{\Delta t \{\cdot,H_T\}} = \left ( \sum_{i=0}^{\infty} \frac{\Delta t^i \{\cdot,H_V\}^i}{i!} \right ) \left ( \sum_{i=0}^{\infty} \frac{\Delta t^i \{\cdot,H_T\}^i}{i!} \right ) $$

Lets dig into the math!

Hamiltonian mechanics

$$ e^{\Delta t \{\cdot,H_V\}} e^{\Delta t \{\cdot,H_T\}} - e^{\Delta t (\{\cdot,H_V\} + \{\cdot,H_T\}) } = \\ = \text{Loads of terms and general dizziness} = \\ = \mathcal{O}(\Delta t^2) $$

We have now discovered the Symplectic (semi-implicit) Euler method

We can do even better: "Leapfrog method"

$$ e^{\frac{\Delta t}{2} \{\cdot,H_{kep}\} } e^{\Delta t \{\cdot,H_{I}\} } e^{\frac{\Delta t}{2} \{\cdot,H_{kep}\} } $$

Is $\mathcal{O}(\Delta t^3)$

So what can you do with these types of methods?

Gabriel Borderes Motta - IRF

Benedikt Diemer - Assistant Professor, University of Maryland

Thank you for your attention!