Note 1.1.1.
The conventions established in this subsection are based on standard conventions. However, they are not exhaustive and are not necessarily consistent with popular conventions used outside the scope of There and Back Again.
Section 1.1 🪡 Convention Stitching
There currently is a bit of a epistemic crisis regarding redundant and/or inconsistent mathematical notation. This section doesn’t aim to derive a universal mathematical language, but rather aims to repair cognitive confusion the author has personally experienced during her years studying mathematics across various scientific disciplines.
The purpose of this subsection is to create a consistent reference list of conventions for writing various mathematical expressions that have inconsistent conventional representations in various contexts.
Context 1.1.1 Symplectic algebra and calculus of variations
| Object | What it means | Intuition | Example |
|---|---|---|---|
| \(q, p\) |
scalars/ coordinates
|
Where is the system?
|
\(q=2, \, p=3\)
|
| \(\mathbf{v}\) |
displacement vector
|
Which direction does the system move?
How far did the system move?
|
\(\mathbf{v}=2\mathbf{e}_q+3\mathbf{e}_p\)
|
| \(dq, dp\) |
1-forms
|
How do I measure coordinate change when the system moves?
|
\(dq(\mathbf{v})=2\)
|
| \(d\mathbf{z}\) |
tangent/ displacement vector
|
What is the infinitesimal direction and magnitude in which the system moves?
|
\(d\mathbf{z}=dq\mathbf{e}_q + dp\mathbf{e}_p\)
|
| \(dq\wedge dp\) |
2-form
|
How do I measure a rotating system’s ’orientation’?
|
\((dq\wedge dp)(\mathbf{v_1}, \mathbf{v_2})\)
|
| \(X_H\) |
vector field
|
Which direction does the system move at each point?
|
\(X_H=\dot{q}\mathbf{e}_q+\dot{p}\mathbf{e}_p\)
|
| \(d\mathcal{J}\) |
differential of a functional
|
How does the functional respond to the direction in which the system changes?
|
\(d\mathcal{J}=\delta\mathcal{J}\)
|
| \(\eta(x)\) |
An arbitrary function that has at least one derivative and vanishes at the endpoints of a perturbation.
|
Which way does the system move through function space?
|
\(y\rightarrow y+\epsilon\eta\)
|
| Concept | Mathematical expression | Meaning |
|---|---|---|
|
A vector has coordinate components
|
\(\mathbf{v}=v_q\mathbf{e}_q+v_p\mathbf{e}_p\) |
A direction can be decomposed into \(q\) and \(p\) coordinates
|
|
A 1-form (or covector) measures a coordinate component of a a vector
|
\(dq(\mathbf{v})=v_q\) |
\(dq\) extracts the \(q\) component of \(\mathbf{v}\)
|
|
Coordinate basis duality
|
\(dq(\mathbf{e}_q)=1, \quad dq(\mathbf{e}_p)=0\) |
\(dq\) measures \(q\)-direction and ignores \(p\)-direction
|
|
Infinitesimal displacement vector
|
\(d\mathbf{z}=dq(\mathbf{z})\mathbf{e}_q + dp(\mathbf{z})\mathbf{e}_p\) |
The coordinate changes combine into a displacement vector
|
|
Two vectors form an area
|
\(\mathbf{v}\wedge\mathbf{w}\) |
The wedge product of two vectors forms an anti-commutative oriented area called a bivector
|
|
Two 1-forms form an area-measuring object
|
\(dq\wedge dp\) | |
|
Hamiltonian generates motion
|
\(\dot{A} = \{A, H\} \) |
\(H\) generates time evolution
|
|
Hamiltonian vector field
|
\(X_H = \dot{q}\mathbf{e}_q + \dot{p}\mathbf{e}_p \) |
The direction of Hamiltonian motion
|
|
|
\(\) |
|
|
|
\(\) |
|
|
|
\(\) |
|
| \(y_\epsilon = y+\epsilon\eta\) |
\(\eta\) is a direction in function space
|
|
| \(\delta\mathcal{J}=d\mathcal{J}[n]\) |
Measures how the functional \(\mathcal{J}\) changes in the \(\eta\) direction
|
|
|
Extremum
|
\(d\mathcal{J}[\eta]=0 \, \forall \, \eta\) |
No first-order change in any admissible direction
|
