For example, a simulating a double pendulum system by simulating the progression of the angles of each bob:
You only need:
Your generalized coordinates (in this case,
θ2), and equations to convert them to cartesian coordinates of your objects:
x1 = sin θ1 y1 = -cos θ1 x2 = sin θ1 + sin θ2 / 2 -- second pendulum is half-length y2 = -cos θ1 - cos θ2 / 2
The masses/inertias of each of those cartesian coordinates (
A potential energy function for your objects:
U = (m1 y1 + m2 y2) * g
And that's it! Hamiltonian mechanics steps your generalized coordinates (
θ2) through time, without needing to do any simulation involving
y2! And you don't need to worry about tension or any other
stuff like that. All you need is a description of your coordinate system
itself, and the potential energy!
doublePendulum :: System 4 2 doublePendulum = mkSystem' (vec4 m1 m1 m2 m2) -- masses (\(V2 θ1 θ2) -> V4 (sin θ1) (-cos θ1) (sin θ1 + sin θ2/2) (-cos θ1 - cos θ2/2) ) -- coordinates (\(V4 _ y1 _ y2) -> (m1 * y1 + m2 * y2) * g) -- potential
Alexander William Rowan Hamilton, we can express our
system parameterized by arbitrary coordinates and get back equations of motions
as first-order differential equations. This library solves those first-order
differential equations for you using automatic differentiation and some matrix
Let's turn our double pendulum (with the second pendulum half as long) into an
actual running program. Let's say that
g = 5,
m1 = 1, and
m2 = 2.
First, the system:
import Numeric.LinearAlgebra.Static import qualified Data.Vector.Sized as V doublePendulum :: System 4 2 doublePendulum = mkSystem' masses coordinates potential where masses :: R 4 masses = vec4 1 1 2 2 coordinates :: Floating a => V.Vector 2 a -> V.Vector 4 a coordinates (V2 θ1 θ2) = V4 (sin θ1) (-cos θ1) (sin θ1 + sin θ2/2) (-cos θ1 - cos θ2/2) potential :: Num a => V.Vector 4 a -> a potential (V4 _ y1 _ y2) = (y1 + 2 * y2) * 5 -- some helper patterns to pattern match on sized vectors pattern V2 :: a -> a -> V.Vector 2 a pattern V2 x y <- (V.toList->[x,y]) where V2 x y = fromJust (V.fromList [x,y]) pattern V4 :: a -> a -> a -> a -> V.Vector 4 a pattern V4 x y z a <- (V.toList->[x,y,z,a]) where V4 x y z a = fromJust (V.fromList [x,y,z,a])
Neat! Easy, right?
Okay, now let's run it. Let's pick a starting configuration (state of the
config0 :: Config 2 config0 = Cfg (vec2 1 0 ) -- initial positions (vec2 0 0.5) -- initial velocities
Configurations are nice, but Hamiltonian dynamics is all about motion through phase space, so let's convert this configuration-space representation of the state into a phase-space representation of the state:
phase0 :: Phase 2 phase0 = toPhase doublePendulum config0
And now we can ask for the state of our system at any amount of points in time!
ghci> evolveHam doublePendulum phase0 [0,0.1 .. 1] -- result: state of the system at times 0, 0.1, 0.2, 0.3 ... etc.
Or, if you want to run the system step-by-step:
evolution :: [Phase 2] evolution = iterate (stepHam 0.1 doublePendulum) phase0
And you can get the position of the coordinates as:
positions :: [R 2] positions = phsPositions <$> evolution
And the position in the underlying cartesian space as:
positions' :: [R 4] positions' = underlyingPos doublePendulum <$> positions
Example App runner
$ git clone https://github.com/mstksg/hamilton $ cd hamilton $ stack install
$ hamilton-examples [EXAMPLE] (options) $ hamilton-examples --help $ hamilton-examples [EXAMPLE] --help
The example runner is a command line application that plots the progression of several example system through time.
||Double pendulum, described above||
||Masses of each bob|
||Initial angle and velocity of bob|
||Object bounding around walled room||
||Initial launch angle of object|
||Two gravitationally attracted bodies, described below||
||Masses of bodies and initial angular veocity|
||Spring hanging from a block on a rail, holding up a weight||
||Masses of block, weight, spring constant, initial compression|
||Bead sliding at constant velocity along bezier curve||
||Control points for arbitrary bezier curve|
[EXAMPLE] --help) for more information.
Two-body system under gravity
The generalized coordinates are just:
r, the distance between the two bodies
θ, the current angle of rotation
x1 = m2/(m1+m2) * r * sin θ -- assuming (0,0) is the center of mass y1 = m2/(m1+m2) * r * cos θ x2 = -m1/(m1+m2) * r * sin θ y2 = -m1/(m1+m2) * r * cos θ
The masses/inertias are again
The potential energy function is the classic gravitational potential:
U = - m1 * m2 / r
And...that's all you need!
Here is the actual code for the two-body system, assuming
twoBody :: System 4 2 twoBody = mkSystem masses coordinates potential where masses :: R 4 masses = vec4 100 100 1 1 coordinates :: Floating a => V.Vector 2 a -> V.Vector 4 a coordinates (V2 r θ) = V4 (r1 * cos θ) (r1 * sin θ) (r2 * cos θ) (r2 * sin θ) where r1 = r * 1 / 101 r2 = - r * 100 / 101 potential :: Num a => V.Vector 4 a -> a potential (V2 r _) = - 100 / r
Time-dependent systems: Shouldn't be an problem in theory/math; just add a time parameter before all of the functions. This opens a lot of doors, like deriving inertial forces for free (like the famous Coriolis force and centrifugal force).
The only thing is that it makes the API pretty inconvenient, because it'd require all of the functions to also take a time parameter. Of course, the easy way out/ugly solution would be to just offer two versions of the same function (one for time-independent systems and one for time-dependent systems. But this is un-ideal.
Velocity-dependent potentials: Would give us the ability to model systems with velocity-dependent Lagrangians like a charged particle in an electromagnetic field, and also dissipative systems, like systems with friction (dependent on
signum v) and linear & quadratic wind resistance.
This issue is much harder, theoretically. It involves inverting arbitrary functions
forall a. RealFloat a => V.Vector n a -> V.Vector m a. It might be possible with the help of some bidirectionalization techniques, but I can't get the bff package to compile, and I'm not sure how to get bff-mono to work with numeric functions.
If anyone is familiar with bidirectionalization techniques and is willing to help out, please send me a message or open an issue! :)