Expression Shape

The Expression shape is the mathematical heart of every Modellus model. It is where you define equations, parameters, differential equations, recurrences, and conditional logic that drive the simulation.

Overview

An Expression shape contains one or more mathematical statements written in LaTeX-like syntax using a MathLive math field. Each statement occupies one line. All Expression shapes in a model share a single calculator system — term names must be globally unique across shapes.

Use multiple small Expression shapes grouped by theme (parameters, equations, derived quantities) to keep your model organized and readable.

Toolbar

The toolbar appears when you select an Expression shape. It provides:

Color picker — Change the shape accent color
Shortcuts picker — Quick insertion of common mathematical templates
Delete — Remove the shape from the board

Statement Types

Each line in an Expression shape is parsed as one of these statement types:

Type	Syntax	Example	Description
Parameter	`name = value`	`g = 9.8`	A constant value. No dependency on other terms or the independent variable.
Function	`name = expression`	`E = ½·m·v²`	A computed value depending on other terms or the independent variable.
Differential	`dx/dt = expression`	`dx/dt = v`	First-order ODE. Requires an initial value in shape properties.
Recurrence	`x_{n} = expression`	`x_{n} = x_{n-1} + v·Δt`	Iterative step definition. Subscript determines step offset.
Conditional	`name = \begin{cases}...\end{cases}`	`v = { 0 & t=0 \\ a·t & otherwise }`	Piecewise definition. At least two branches required.
Function with argument	`f(t) = expression`	`h(t) = v₀·t - ½·g·t²`	Explicit argument notation (cosmetic; behavior same as Function).
Display	`expression`	`F = ma`	Visual label only. Not computed by the calculator.

Arithmetic Operators

Operator	LaTeX	Keystroke	Description
Addition	`+`	`+`	Sum of two expressions
Subtraction	`-`	`-`	Difference of two expressions
Multiplication	`\cdot`	`*`	Explicit multiplication (· symbol)
Implicit multiplication	`2x`	`2x`	Coefficient followed by name or parenthesis
Inline division	`/`	`/`	a/b inline notation
Fraction	`\frac{a}{b}`	`/` (in numerator position)	Display-quality vertical fraction
Power	`^{n}`	`^`	Exponentiation
Square root	`\sqrt{x}`	`\sqrt`	Square root
Negation	`-x`	`-`	Unary negation

Comparison Operators

Used in conditional (\begin{cases}) branches:

Operator	LaTeX	Meaning
=	`=`	Equal to
>	`>`	Greater than
<	`<`	Less than
≥	`\ge` or `\geq`	Greater than or equal
≤	`\le` or `\leq`	Less than or equal
≠	`\neq`	Not equal to

Conditions can be chained (e.g., 0 < x < 1) or combined with \lor (OR) and \land (AND). The \text{otherwise} keyword acts as an else branch.

Built-in Functions

Category	Function	LaTeX	Arguments
Trigonometric	Sine	`\sin\left(x\right)`	1
	Cosine	`\cos\left(x\right)`	1
	Tangent	`\tan\left(x\right)`	1
	Cotangent	`\cot\left(x\right)`	1
	Secant	`\sec\left(x\right)`	1
	Cosecant	`\csc\left(x\right)`	1
Inverse Trig	Arc sine	`\arcsin\left(x\right)`	1
	Arc cosine	`\arccos\left(x\right)`	1
	Arc tangent	`\arctan\left(x\right)`	1
Hyperbolic	Hyperbolic sine	`\sinh\left(x\right)`	1
	Hyperbolic cosine	`\cosh\left(x\right)`	1
	Hyperbolic tangent	`\tanh\left(x\right)`	1
Logarithmic	Natural log	`\ln\left(x\right)`	1
Logarithmic	Logarithm base 10	`\log\left(x\right)`	1
Two-argument	Maximum	`\max\left(a, b\right)`	2
	Minimum	`\min\left(a, b\right)`	2
	Modulo	`\mod\left(a, b\right)`	2
Utility	Sign	`sign\left(x\right)`	1 — returns -1, 0, or 1
	Random	`rnd\left(x\right)`	1 — random float in [0, x]
	Integer random	`irnd\left(x\right)`	1 — random integer in [0, x]
	Integer part	`int\left(x\right)`	1 — truncate to integer
	Round	`round\left(x\right)`	1 — round to nearest integer
Other	Determinant	`\det\left(x\right)`	1

All function calls use \left( and \right) delimiters. Bare parentheses are not valid in function calls. The math field inserts these automatically.

Constants

Name	LaTeX	Value
Pi	`\pi`	3.14159…
Euler's number	`e` or `\E`	2.71828…

g is not a built-in constant. Define it as a parameter: g = 9.8.

Special Syntax

Finite Difference (Δ)

Syntax	Meaning
`\Delta x`	x(current) − x(previous). For the independent variable: the step size.
`\Delta\left(expr\right)`	Finite difference applied to a sub-expression.

Subscripts (Step References)

Syntax	Meaning
`x_{n-1}`	Value of x from the previous iteration
`x_{n}`	Current iteration (LHS of recurrence)
`x_0`	Initial value of x (bare digit, no braces)
`x_{0}`	Initial value (braces form)

Units Annotation

Append display-only units after any statement:

v = 10 \quad\textcolor{gray}{\mathrm{m/s}}

Units are cosmetic — the engine does not validate or convert them.

Dotted Names (Child References)

Reference terms from child shapes inside a Referential: body.x, particle.velocity.

Keyboard Shortcuts

The MathLive math field supports these keyboard shortcuts for fast input:

Action	macOS	Windows
Fraction	`⌘` + `/`	`Ctrl` + `/`
Superscript (power)	`^`	`^`
Subscript	`_`	`_`
Square root	`⌘` + `Shift` + `R`	`Ctrl` + `Shift` + `R`
Greek letter	Type `\` + name (e.g., `\alpha`)	Same
Multiplication dot	`*`	`*`
Greater or equal	Type `\ge`	Same
Less or equal	Type `\le`	Same
Not equal	Type `\neq`	Same
Delta	Type `\Delta`	Same
Pi	Type `\pi`	Same
New line (new statement)	`Enter`	`Enter`
Navigate out of fraction	`Tab`	`Tab`
Navigate into next placeholder	`Tab`	`Tab`

Shortcuts Picker

Expression shape selected with toolbar visible

Click the button in the toolbar to open the shortcuts picker. It provides pre-built templates for common expressions including:

Differential equations (dx/dt = ...)
Recurrences (x_n = x_{n-1} + ...)
Piecewise functions
Common physics formulas
Trigonometric identities

Properties Panel

When an Expression shape is selected, the properties panel shows:

Property	Description
Independent variable	The variable of iteration (default: `t`)
Step	The increment per iteration (Δt)
Start / End	Range of the independent variable
Initial values	Starting values for differential equations and recurrences
Cases	Multiple sets of initial values for parameter studies

Examples

Simple Kinematics

g = 9.8
v₀ = 20
θ = 45
vx = v₀·\cos\left(θ\right)
vy = v₀·\sin\left(θ\right) - g·t
\frac{dx}{dt} = vx
\frac{dy}{dt} = vy

Population Growth (Recurrence)

r = 0.05
P_{n} = P_{n-1}·\left(1 + r\right)

Piecewise Velocity

v = \begin{cases} 0 & t = 0 \\ a·t & 0 < t < 5 \\ a·5 & \text{otherwise} \end{cases}