Preface Conventions
Mathematical Notation.
A reference for mathematical notation used throughout the notes.
- \(\vec{u},\vec{v}\)
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Vectors in the vector space \(\mathcal{V}_n\in\mathbb{R}^n\text{.}\) These are grade-1 objects in Clifford algebra.
- \(C(A)\)
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The column space of a matrix \(A\text{.}\)
- \((\vec{q},\vec{p})\)
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A point in phase space where \(\vec{q}\) represents the generalized coordinates and \(\vec{p}\) represents the conjugate momenta.For this work, we will assume \(q=\|\vec{q}\|\) denotes the magnitude of the coordinate vector and \(p=\|\vec{p}\|\) denotes the magnitude of the momentum vector, respectively.
- \(\{A, B\}\)
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The Poisson bracket between two scalar functions \(A(\vec{q},\vec{p})\) and \(B(\vec{q},\vec{p})\text{.}\)
- \(\bivec{B}\)
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A bivector (grade-2) quantity.
Abbreviations.
- GA
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Geometric Algebra
- psd
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positive semi-definite
- PSD
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power spectral density
- PIML
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Physics-Informed Machine Learning
- PINN
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Physics-Informed Neural Networks
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