Perl 6 and the Real World

Physical Modelling with Perl 6

Moritz Lenz <moritz.lenz@gmail.com>

Max Planck Institute for the Science of Light

Perl 6 and the Real World - Structure

What is a model? When is it a good model?
A simple model
Math: derivatives
Free fall, spring
Resonance

What is a Model?

physics = striving to understand (parts of) the world
the world is too complicated
models are descriptions that focus on one aspect
so Model = Simplification

Example Model

Model takes into account

gravity
inertia
initial motion
connection to anchor point

Model neglects

colors
exact shapes
size of object
friction

Is it a good model?

it's a good model if it can answer a question for us
examples "how fast is the object?", "What is the swinging period?", "Does the distance to the anchor point matter?"
accuracy of the answer important
every model needs input data. Is that available?
extensibilty

Another model: free falling

Free falling: Solved in Perl 6

use Math::Model;

my $m = Math::Model.new(
    derivatives => {
        velocity      => 'height',
        acceleration  => 'velocity',
    },
    variables   => {
        acceleration  => { $:gravity },   # m / s**2
        gravity       => { -9.81 },       # m / s**2
    },
    # ...

Free falling: Solved in Perl 6

    # ...
    initials    => {
        height        => 50,              # m
        velocity      => 0,               # m/s
    },
    captures    => ('height', 'velocity'),
);

$m.integrate(:from(0), :to(4.2), :min-resolution(0.2));
$m.render-svg('free-fall.svg', :title('Free falling'));

Model result

The model in detail

use Math::Model;

my $m = Math::Model.new(
    derivatives => {
        velocity      => 'height',
        acceleration  => 'velocity',
    },

Derivative: slope of another quantity

Common derivatives in Mechanics

Derivative          Of

velocity            position
angular velocity    angle
acceleration        velocity
power               energy
force               momentum
                    (= mass * velocity)

Common derivatives

current             charge

birth rate          population
    - mortality rate

profit              funds

Using derivatives

with Math::Model, you only need to know the derivatives, note the values derived from
you need an initial value for the derived quantity
(Ordinary Differential Equation, which Math::Model integrates for you)

Rest of the model

    variables   => {
        acceleration  => { $:gravity },   # m / s**2
        gravity       => { -9.81 },       # m / s**2
    },
    initials    => {
        height        => 50,              # m
        velocity      => 0,               # m/s
    },
    captures    => ('height', 'velocity'),
);

$m.integrate(:from(0), :to(4.2), :min-resolution(0.2));
$m.render-svg('free-fall.svg', :title('Free falling'));

Perl 6 stuff

$:height is a named parameter
Math::Model introspects code blocks for arguments
calculates dependencies => execution order
RungeKutta integration

Extending the model - Spring, damping

Spring, gravity, damping: source code

# ...
variables   => {
    acceleration  => { $:gravity + $:spring + $:damping },
    gravity       => { -9.81 },
    spring        => { - 2 * $:height },
    damping       => { - 0.2 * $:velocity },
},
# ...

Spring, gravity, damping: results

Further extensions

Let's add an external, driving force
Example: motor, coupled through a second spring

External driving force: Code

sub MAIN($freq) {
  my $m = Math::Model.new(
    # ...
    variables   => {
      acceleration  => { $:gravity + $:spring
                          + $:damping + $:ext_force },
      gravity       => { -9.81 },
      spring        => { - 2 * $:height },
      damping       => { - 0.2 * $:velocity },
      ext_force     => { sin(2 * pi * $:time * $freq) },
    },
    # ...
  );

  my %h = $m.integrate(:from(0), :to(70), :min-resolution(5));
  $m.render-svg("spring-freq-$freq.svg", 
     :title("Spring with damping, external force at $freq"));