Verlet implementation in Common Lisp (#608)

Author: Trashtalk217

contents/verlet_integration/code/clisp/verlet.lisp

@@ -0,0 +1,45 @@
+;;;; Verlet integration implementation in Common Lisp
+
+(defun verlet (pos acc dt)
+  "Integrates Newton's equation for motion while pos > 0 using Verlet integration."
+  (loop
+    with prev-pos = pos
+    for time = 0 then (incf time dt)
+    while (> pos 0)
+    ;; The starting speed is assumed to be zero.
+    do (psetf
+         pos (+ (* pos 2) (- prev-pos) (* acc dt dt))
+         prev-pos pos)
+    finally (return time)))
+
+(defun stormer-verlet (pos acc dt)
+  "Integrates Newton's equation for motion while pos > 0 using the Stormer-Verlet method."
+  (loop
+    with prev-pos = pos
+    for time = 0 then (incf time dt)
+    for vel = 0 then (incf vel (* acc dt))
+    while (> pos 0)
+    ;; Variables are changed simultaneously by 'psetf', so there's no need for a temporary variable.
+    do (psetf
+         pos (+ (* pos 2) (- prev-pos) (* acc dt dt))
+         prev-pos pos)
+    finally (return (list time vel))))
+
+(defun velocity-verlet (pos acc dt)
+  "Integrates Newton's equation for motion while pos > 0 using the velocity in calculations."
+  (loop
+    for time = 0 then (incf time dt)
+    for vel = 0 then (incf vel (* acc dt))
+    for p = pos then (incf p (+ (* vel dt) (* 0.5 acc dt dt)))
+    while (> p 0)
+    finally (return (list time vel))))
+
+(format T "Time for Verlet integration: ~d~%" (verlet 5 -10 0.01))
+
+(defvar stormer-verlet-result (stormer-verlet 5 -10 0.01))
+(format T "Time for Stormer Verlet integration is: ~d~%" (first stormer-verlet-result))
+(format T "Velocity for Stormer Verlet integration is: ~d~%" (second stormer-verlet-result))
+
+(defvar velocity-verlet-result (velocity-verlet 5 -10 0.01))
+(format T "Time for velocity Verlet integration is: ~d~%" (first velocity-verlet-result))
+(format T "Velocity for velocity Verlet integration is: ~d~%" (second velocity-verlet-result))

contents/verlet_integration/verlet_integration.md

@@ -69,6 +69,8 @@ Unfortunately, this has not yet been implemented in LabVIEW, so here's Julia cod
 [import:3-15, lang:"kotlin"](code/kotlin/verlet.kt)
 {% sample lang="nim" %}
 [import:1-14, lang:"nim"](code/nim/verlet.nim)
+{% sample lang="lisp" %}
+[import:3-13, lang:"lisp"](code/clisp/verlet.lisp)
 {% endmethod %}
 
 Now, obviously this poses a problem; what if we want to calculate a term that requires velocity, like the kinetic energy, $$\frac{1}{2}mv^2$$? In this case, we certainly cannot get rid of the velocity! Well, we can find the velocity to $$\mathcal{O}(\Delta t^2)$$ accuracy by using the Stormer-Verlet method, which is the same as before, but we calculate velocity like so
@@ -125,6 +127,8 @@ Unfortunately, this has not yet been implemented in LabVIEW, so here's Julia cod
 [import:17-30, lang:"kotlin"](code/kotlin/verlet.kt)
 {% sample lang="nim" %}
 [import:16-32, lang:"nim"](code/nim/verlet.nim)
+{% sample lang="lisp" %}
+[import:15-26, lang:"lisp"](code/clisp/verlet.lisp)
 {% endmethod %}
 
 
@@ -195,6 +199,8 @@ Unfortunately, this has not yet been implemented in LabVIEW, so here's Julia cod
 [import:32-42, lang:"kotlin"](code/kotlin/verlet.kt)
 {% sample lang="nim" %}
 [import:34-45, lang:"nim"](code/nim/verlet.nim)
+{% sample lang="lisp" %}
+[import:28-35, lang:"lisp"](code/clisp/verlet.lisp)
 {% endmethod %}
 
 Even though this method is more widely used than the simple Verlet method mentioned above, it unforunately has an error term of $$\mathcal{O}(\Delta t^2)$$, which is two orders of magnitude worse. That said, if you want to have a simulaton with many objects that depend on one another --- like a gravity simulation --- the Velocity Verlet algorithm is a handy choice; however, you may have to play further tricks to allow everything to scale appropriately. These types of simulatons are sometimes called *n-body* simulations and one such trick is the Barnes-Hut algorithm, which cuts the complexity of n-body simulations from $$\sim \mathcal{O}(n^2)$$ to $$\sim \mathcal{O}(n\log(n))$$.
@@ -256,6 +262,8 @@ Submitted by P. Mekhail
 [import, lang:"kotlin"](code/kotlin/verlet.kt)
 {% sample lang="nim" %}
 [import, lang="nim"](code/nim/verlet.nim)
+{% sample lang="lisp" %}
+[import, lang:"lisp"](code/clisp/verlet.lisp)
 {% endmethod %}