Verlet Integration
Verlet integration is essentially a solution to the kinematic equation for the motion of any object,
where is the position, is the velocity, is the acceleration, is the often forgotten jerk term, and is time. This equation is a central equation to almost every Newtonian physics solver and brings up a class of algorithms known as force integrators. One of the first force integrators to work with is Verlet Integration.
So, let's say we want to solve for the next timestep in . To a close approximation (actually performing a Taylor Series Expansion about ), that might look like this:
This means that if we need to find the next , we need the current , , , etc. However, because few people calculate the jerk term, our error is typically . That said, we can calculate with less knowledge and higher accuracy if we play a trick! Let's say we want to calculate of the previous timestep. Again, to a close approximation, that might look like this:
Now, we have two equations to solve for two different timesteps in x, one of which we already have. If we add the two equations together and solve for , we find that
So, this means we can find our next simply by knowing our current , the before that, and the acceleration! No velocity necessary! In addition, this drops the error to , which is great! Here is what it looks like in code:
function verlet(pos::Float64, acc::Float64, dt::Float64)
prev_pos = pos
time = 0.0
while (pos > 0)
time += dt
temp_pos = pos
pos = pos * 2 - prev_pos + acc * dt * dt
prev_pos = temp_pos
end
return time
end
double verlet(double pos, double acc, double dt) {
double prev_pos = pos;
double time = 0;
while (pos > 0) {
time += dt;
double next_pos = pos * 2 - prev_pos + acc * dt * dt;
prev_pos = pos;
pos = next_pos;
}
return time;
}
void verlet(double *time, double pos, double acc, double dt) {
double prev_pos, temp_pos;
prev_pos = pos;
*time = 0.0;
while (pos > 0) {
*time += dt;
temp_pos = pos;
pos = pos * 2 - prev_pos + acc * dt * dt;
prev_pos = temp_pos;
}
}
static double verlet(double pos, double acc, double dt) {
// Note that we are using a temp variable for the previous position
double prev_pos, temp_pos, time;
prev_pos = pos;
time = 0;
while (pos > 0) {
time += dt;
temp_pos = pos;
pos = pos*2 - prev_pos + acc * dt * dt;
prev_pos = temp_pos;
}
return time;
}
def verlet(pos, acc, dt):
prev_pos = pos
time = 0
while pos > 0:
time += dt
next_pos = pos * 2 - prev_pos + acc * dt * dt
prev_pos, pos = pos, next_pos
return time
type Method = Model -> Time -> Particle -> Particle -> Particle
verlet :: Method
verlet acc dt (xOld, _, _, _) (x, _, a, t) = (x', v', a', t + dt)
where
x' = 2 * x - xOld + a * dt ^ 2
v' = 0
a' = acc (x', v', a, t + dt)
function verlet(pos, acc, dt) {
let prevPos = pos;
let time = 0;
let tempPos;
while (pos > 0) {
time += dt;
tempPos = pos;
pos = pos * 2 - prevPos + acc * dt * dt;
prevPos = tempPos;
}
return time;
}
fn verlet(mut pos: f64, acc: f64, dt: f64) -> f64 {
let mut prev_pos = pos;
let mut time = 0.0;
while pos > 0.0 {
time += dt;
let temp_pos = pos;
pos = pos * 2.0 - prev_pos + acc * dt * dt;
prev_pos = temp_pos;
}
time
}
func verlet(pos: Double, acc: Double, dt: Double) -> Double {
var pos = pos
var temp_pos, time: Double
var prev_pos = pos
time = 0.0
while (pos > 0) {
time += dt
temp_pos = pos
pos = pos*2 - prev_pos + acc * dt * dt
prev_pos = temp_pos
}
return time
}
SUBROUTINE verlet(pos, acc, dt, time)
IMPLICIT NONE
REAL(8), INTENT(INOUT) :: pos, acc, dt, time
REAL(8) :: prev_pos, next_pos
prev_pos = pos
time = 0d0
DO
IF (pos > 0d0) THEN
time = time + dt
next_pos = pos * 2d0 - prev_pos + acc * dt ** 2
prev_pos = pos
pos = next_pos
ELSE
EXIT
END IF
END DO
END SUBROUTINE verlet
def verlet(pos, acc, dt)
prev_pos = pos
time = 0
while pos > 0 do
time += dt
temp_pos = pos
pos = pos*2 - prev_pos + acc * dt * dt
prev_pos = temp_pos
end
return time
end
func verlet(pos, acc, dt float64) (time float64) {
prevPos := pos
time = 0
for pos > 0 {
time += dt
nextPos := pos*2 - prevPos + acc*dt*dt
prevPos, pos = pos, nextPos
}
return
}
# xmm0 - pos
# xmm1 - acc
# xmm2 - dt
# RET xmm0 - time
verlet:
pxor xmm7, xmm7 # Holds 0 for comparisons
pxor xmm3, xmm3 # Holds time value
comisd xmm0, xmm7 # Check if pos is greater then 0.0
jbe verlet_return
movsd xmm6, xmm1 # xmm6 = acc * dt * dt
mulsd xmm6, xmm2
mulsd xmm6, xmm2
movsd xmm5, xmm0 # Holds previous position
verlet_loop:
addsd xmm3, xmm2 # Adding dt to time
movsd xmm4, xmm0 # Hold old value of posistion
addsd xmm0, xmm0 # Calculating new position
subsd xmm0, xmm5
addsd xmm0, xmm6
movsd xmm5, xmm4
comisd xmm0, xmm7 # Check if position is greater then 0.0
ja verlet_loop
verlet_return:
movsd xmm0, xmm3 # Saving time value
ret
fun verlet(_pos: Double, acc: Double, dt: Double): Double {
var pos = _pos // Since function parameter are val and can't be modified
var prevPos = pos
var time = 0.0
while (pos > 0) {
time += dt
val nextPos = pos * 2 - prevPos + acc * dt * dt
prevPos = pos
pos = nextPos
}
return time
}
func verlet(pos_in, acc, dt: float): float =
var
pos: float = pos_in
prevPos: float = pos
time: float = 0.0
tempPos: float
while pos > 0.0:
time += dt
tempPos = pos
pos = pos * 2 - prevPos + acc * dt * dt
prevPos = tempPos
time
(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)))
Now, obviously this poses a problem; what if we want to calculate a term that requires velocity, like the kinetic energy, ? In this case, we certainly cannot get rid of the velocity! Well, we can find the velocity to accuracy by using the Stormer-Verlet method. We have the equations for and above, so let's start there. If we subtract the latter from the former, we get the following:
When we solve for , we get
Note that the 2 in the denominator makes sense because we are going over 2 timesteps. It's essentially solving . In addition, we can calculate the velocity of the next timestep like so
However, the error for this is , which is quite poor, but it gets the job done in a pinch. Here's what it looks like in code:
function stormer_verlet(pos::Float64, acc::Float64, dt::Float64)
prev_pos = pos
time = 0.0
vel = 0.0
while (pos > 0.0)
time += dt
temp_pos = pos
pos = pos * 2 - prev_pos + acc * dt * dt
prev_pos = temp_pos
# Because acceleration is constant, velocity is straightforward
vel += acc * dt
end
return time, vel
end
timestep stormer_verlet(double pos, double acc, double dt) {
double prev_pos = pos;
double time = 0;
double vel = 0;
while (pos > 0) {
time += dt;
double next_pos = pos * 2 - prev_pos + acc * dt * dt;
prev_pos = pos;
pos = next_pos;
// The acceleration is constant, so the velocity is
// straightforward
vel += acc * dt;
}
return timestep { time, vel };
}
void stormer_verlet(double *time, double *vel,
double pos, double acc, double dt) {
double prev_pos, temp_pos;
prev_pos = pos;
*vel = 0.0;
*time = 0.0;
while (pos > 0) {
*time += dt;
temp_pos = pos;
pos = pos * 2 - prev_pos + acc * dt * dt;
prev_pos = temp_pos;
*vel += acc * dt;
}
}
static VerletValues stormer_verlet(double pos, double acc, double dt) {
// Note that we are using a temp variable for the previous position
double prev_pos, temp_pos, time, vel;
prev_pos = pos;
vel = 0;
time = 0;
while (pos > 0) {
time += dt;
temp_pos = pos;
pos = pos*2 - prev_pos + acc * dt * dt;
prev_pos = temp_pos;
// The acceleration is constant, so the velocity is straightforward
vel += acc*dt;
}
return new VerletValues(time, vel);
}
def stormer_verlet(pos, acc, dt):
prev_pos = pos
time = 0
vel = 0
while pos > 0:
time += dt
next_pos = pos * 2 - prev_pos + acc * dt * dt
prev_pos, pos = pos, next_pos
vel += acc * dt
return time, vel
stormerVerlet :: Method
stormerVerlet acc dt (xOld, _, _, _) (x, _, a, t) = (x', v', a', t + dt)
where
x' = 2 * x - xOld + a * dt ^ 2
v' = (x' - x) / dt
a' = acc (x', v', a, t + dt)
function stormerVerlet(pos, acc, dt) {
let prevPos = pos;
let time = 0;
let vel = 0;
let tempPos;
while (pos > 0) {
time += dt;
tempPos = pos;
pos = pos * 2 - prevPos + acc * dt * dt;
prevPos = tempPos;
vel += acc * dt;
}
return { time, vel };
}
fn stormer_verlet(mut pos: f64, acc: f64, dt: f64) -> (f64, f64) {
let mut prev_pos = pos;
let mut time = 0.0;
let mut vel = 0.0;
while pos > 0.0 {
time += dt;
let temp_pos = pos;
pos = pos * 2.0 - prev_pos + acc * dt * dt;
prev_pos = temp_pos;
// Because acceleration is constant, velocity is
// straightforward
vel += acc * dt;
}
(time, vel)
}
func stormerVerlet(pos: Double, acc: Double, dt: Double) -> (time: Double, vel: Double) {
var pos = pos
var temp_pos, time, vel: Double
var prev_pos = pos
vel = 0
time = 0
while (pos > 0) {
time += dt
temp_pos = pos
pos = pos*2 - prev_pos + acc * dt * dt
prev_pos = temp_pos
vel += acc*dt
}
return (time:time, vel:vel)
}
SUBROUTINE stormer_verlet(pos, acc, dt, time, vel)
IMPLICIT NONE
REAL(8), INTENT(INOUT) :: pos, acc, dt, time, vel
REAL(8) :: prev_pos, next_pos
prev_pos = pos
time = 0d0
vel = 0d0
DO
IF (pos > 0d0) THEN
time = time + dt
next_pos = pos * 2 - prev_pos + acc * dt ** 2
prev_pos = pos
pos = next_pos
vel = vel + acc * dt
ELSE
EXIT
END IF
END DO
END SUBROUTINE stormer_verlet
def stormer_verlet(pos, acc, dt)
prev_pos = pos
vel = 0
time = 0
while pos > 0 do
time += dt
temp_pos = pos
pos = pos*2 - prev_pos + acc * dt * dt
prev_pos = temp_pos
vel += acc*dt
end
return time, vel
end
func stormerVerlet(pos, acc, dt float64) (time, vel float64) {
prevPos := pos
time, vel = 0, 0
for pos > 0 {
time += dt
vel += acc * dt
nextPos := pos*2 - prevPos + acc*dt*dt
prevPos, pos = pos, nextPos
}
return
}
# xmm0 - pos
# xmm1 - acc
# xmm2 - dt
# RET xmm0 - time
# RET xmm1 - velocity
stormer_verlet:
pxor xmm7, xmm7 # Holds 0 for comparisons
pxor xmm3, xmm3 # Holds time value
comisd xmm0, xmm7 # Check if pos is greater then 0.0
jbe stormer_verlet_return
movsd xmm6, xmm1 # xmm6 = acc * dt * dt
mulsd xmm6, xmm2
mulsd xmm6, xmm2
movsd xmm5, xmm0 # Holds previous position
stormer_verlet_loop:
addsd xmm3, xmm2 # Adding dt to time
movsd xmm4, xmm0 # Hold old value of posistion
addsd xmm0, xmm0 # Calculating new position
subsd xmm0, xmm5
addsd xmm0, xmm6
movsd xmm5, xmm4
comisd xmm0, xmm7 # Check if position is greater then 0.0
ja stormer_verlet_loop
stormer_verlet_return:
movsd xmm0, xmm3 # Saving time and velocity
mulsd xmm3, xmm1
movsd xmm1, xmm3
ret
fun stormerVerlet(_pos: Double, acc: Double, dt: Double): VerletValues {
var pos = _pos
var prevPos = pos
var time = 0.0
var vel = 0.0
while (pos > 0) {
time += dt
val nextPos = pos * 2 - prevPos + acc * dt * dt
prevPos = pos
pos = nextPos
vel += acc * dt
}
return VerletValues(time, vel)
}
func stormerVerlet(pos_in, acc, dt: float): (float, float) =
var
pos: float = pos_in
prevPos: float = pos
time: float = 0.0
vel: float = 0.0
tempPos: float
while pos > 0.0:
time += dt
tempPos = pos
pos = pos * 2 - prevPos + acc * dt * dt
prevPos = tempPos
vel += acc * dt
(time, vel)
(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))))
Now, let's say we actually need the velocity to calculate out next timestep. Well, in this case, we simply cannot use the above approximation and instead need to use the Velocity Verlet algorithm.
Velocity Verlet
In some ways, this algorithm is even simpler than above. We can calculate everything like
which is literally the kinematic equation above, solving for , , and every timestep. You can also split up the equations like so
Here is the velocity Verlet method in code:
function velocity_verlet(pos::Float64, acc::Float64, dt::Float64)
prev_pos = pos
time = 0.0
vel = 0.0
while (pos > 0.0)
time += dt
pos += vel * dt + 0.5 * acc * dt * dt;
vel += acc * dt;
end
return time, vel
end
timestep velocity_verlet(double pos, double acc, double dt) {
double time = 0;
double vel = 0;
while (pos > 0) {
time += dt;
pos += vel * dt + 0.5 * acc * dt * dt;
vel += acc * dt;
}
return timestep { time, vel };
}
void velocity_verlet(double *time, double *vel,
double pos, double acc, double dt) {
*vel = 0.0;
*time = 0.0;
while (pos > 0) {
*time += dt;
pos += (*vel) * dt + 0.5 * acc * dt * dt;
*vel += acc * dt;
}
}
static VerletValues velocity_verlet(double pos, double acc, double dt) {
// Note that we are using a temp variable for the previous position
double time, vel;
vel = 0;
time = 0;
while (pos > 0) {
time += dt;
pos += vel*dt + 0.5*acc * dt * dt;
vel += acc*dt;
}
return new VerletValues(time, vel);
}
def velocity_verlet(pos, acc, dt):
time = 0
vel = 0
while pos > 0:
time += dt
pos += vel * dt + 0.5 * acc * dt * dt
vel += acc * dt
return time, vel
velocityVerlet :: Method
velocityVerlet acc dt (xOld, _, aOld, _) (x, v, a, t) = (x', v', a', t + dt)
where
x' = 2 * x - xOld + a * dt ^ 2
v' = v + 0.5 * (aOld + a) * dt
a' = acc (x', v', a, t + dt)
function velocityVerlet(pos, acc, dt) {
let time = 0;
let vel = 0;
while (pos > 0) {
time += dt;
pos += vel * dt + 0.5 * acc * dt * dt;
vel += acc * dt;
}
return { time, vel };
}
fn velocity_verlet(mut pos: f64, acc: f64, dt: f64) -> (f64, f64) {
let mut time = 0.0;
let mut vel = 0.0;
while pos > 0.0 {
time += dt;
pos += vel * dt + 0.5 * acc * dt * dt;
vel += acc * dt;
}
(time, vel)
}
func velocityVerlet(pos: Double, acc: Double, dt: Double) -> (time: Double, vel: Double) {
var pos = pos
var time, vel : Double
vel = 0
time = 0
while (pos > 0) {
time += dt
pos += vel*dt + 0.5*acc * dt * dt
vel += acc*dt
}
return (time:time, vel:vel)
}
SUBROUTINE velocity_verlet(pos, acc, dt, time, vel)
IMPLICIT NONE
REAL(8), INTENT(INOUT) :: pos, acc, dt, time, vel
time = 0d0
vel = 0d0
DO
IF (pos > 0d0) THEN
time = time + dt
pos = pos + vel * dt + 0.5d0 * acc * dt ** 2
vel = vel + acc * dt
ELSE
EXIT
END IF
END DO
END SUBROUTINE velocity_verlet
def velocity_verlet(pos, acc, dt)
vel = 0
time = 0
while pos > 0 do
time += dt
pos += vel*dt + 0.5*acc * dt * dt
vel += acc*dt
end
return time, vel
end
func velocityVerlet(pos, acc, dt float64) (time, vel float64) {
time, vel = 0, 0
for pos > 0 {
time += dt
pos += vel*dt + .5*acc*dt*dt
vel += acc * dt
}
return
}
# xmm0 - pos
# xmm1 - acc
# xmm2 - dt
# RET xmm0 - time
# RET xmm1 - velocity
velocity_verlet:
pxor xmm7, xmm7 # Holds 0 for comparisons
pxor xmm3, xmm3 # Holds the velocity value
pxor xmm4, xmm4 # Holds the time value
comisd xmm0, xmm7 # Check if pos is greater then 0.0
jbe velocity_verlet_return
movsd xmm5, half # xmm5 = 0.5 * dt * dt * acc
mulsd xmm5, xmm2
mulsd xmm5, xmm2
mulsd xmm5, xmm1
velocity_verlet_loop:
movsd xmm6, xmm3 # Move velocity into register
mulsd xmm6, xmm2 # Calculate new position
addsd xmm6, xmm5
addsd xmm0, xmm6
addsd xmm4, xmm2 # Incrementing time
movsd xmm3, xmm4 # Updating velocity
mulsd xmm3, xmm1
comisd xmm0, xmm7
ja velocity_verlet_loop
velocity_verlet_return:
movsd xmm0, xmm4 # Saving time and velocity
movsd xmm1, xmm3
ret
fun velocityVerlet(_pos: Double, acc: Double, dt: Double): VerletValues {
var pos = _pos
var time = 0.0
var vel = 0.0
while (pos > 0) {
time += dt
pos += vel * dt + 0.5 * acc * dt * dt
vel += acc * dt
}
return VerletValues(time, vel)
}
func velocityVerlet(pos_in, acc, dt: float): (float, float) =
var
pos: float = pos_in
time: float = 0.0
vel: float = 0.0
while pos > 0.0:
time += dt
pos += vel * dt + 0.5 * acc * dt * dt
vel += acc * dt
(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))))
Even though this method is more widely used than the simple Verlet method mentioned above, it unfortunately has an error term of , which is two orders of magnitude worse. That said, if you want to have a simulation 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 simulations are sometimes called n-body simulations and one such trick is the Barnes-Hut algorithm, which cuts the complexity of n-body simulations from to .
Video Explanation
Here is a video describing Verlet integration:
Example Code
Both of these methods work simply by iterating timestep-by-timestep and can be written straightforwardly in any language. For reference, here are snippets of code that use both the classic and velocity Verlet methods to find the time it takes for a ball to hit the ground after being dropped from a given height.
function verlet(pos::Float64, acc::Float64, dt::Float64)
prev_pos = pos
time = 0.0
while (pos > 0)
time += dt
temp_pos = pos
pos = pos * 2 - prev_pos + acc * dt * dt
prev_pos = temp_pos
end
return time
end
function stormer_verlet(pos::Float64, acc::Float64, dt::Float64)
prev_pos = pos
time = 0.0
vel = 0.0
while (pos > 0.0)
time += dt
temp_pos = pos
pos = pos * 2 - prev_pos + acc * dt * dt
prev_pos = temp_pos
# Because acceleration is constant, velocity is straightforward
vel += acc * dt
end
return time, vel
end
function velocity_verlet(pos::Float64, acc::Float64, dt::Float64)
prev_pos = pos
time = 0.0
vel = 0.0
while (pos > 0.0)
time += dt
pos += vel * dt + 0.5 * acc * dt * dt;
vel += acc * dt;
end
return time, vel
end
function main()
time = verlet(5.0, -10.0, 0.01);
println("[#]\nTime for Verlet integration is:")
println("$(time)")
time, vel = stormer_verlet(5.0, -10.0, 0.01);
println("[#]\nTime for Stormer Verlet integration is:")
println("$(time)")
println("[#]\nVelocity for Stormer Verlet integration is:")
println("$(vel)")
time, vel = velocity_verlet(5.0, -10.0, 0.01);
println("[#]\nTime for velocity Verlet integration is:")
println("$(time)")
println("[#]\nVelocity for velocity Verlet integration is:")
println("$(vel)")
end
main()
#include <iomanip>
#include <iostream>
struct timestep {
double time;
double vel;
};
double verlet(double pos, double acc, double dt) {
double prev_pos = pos;
double time = 0;
while (pos > 0) {
time += dt;
double next_pos = pos * 2 - prev_pos + acc * dt * dt;
prev_pos = pos;
pos = next_pos;
}
return time;
}
timestep stormer_verlet(double pos, double acc, double dt) {
double prev_pos = pos;
double time = 0;
double vel = 0;
while (pos > 0) {
time += dt;
double next_pos = pos * 2 - prev_pos + acc * dt * dt;
prev_pos = pos;
pos = next_pos;
// The acceleration is constant, so the velocity is
// straightforward
vel += acc * dt;
}
return timestep { time, vel };
}
timestep velocity_verlet(double pos, double acc, double dt) {
double time = 0;
double vel = 0;
while (pos > 0) {
time += dt;
pos += vel * dt + 0.5 * acc * dt * dt;
vel += acc * dt;
}
return timestep { time, vel };
}
int main() {
std::cout << std::fixed << std::setprecision(8);
// Note that depending on the simulation, you might want to have the
// Verlet loop outside.
// For example, if your acceleration chages as a function of time,
// you might need to also change the acceleration to be read into
// each of these functions.
double time = verlet(5.0, -10, 0.01);
std::cout << "[#]\nTime for Verlet integration is:\n" \
<< time << std::endl;
timestep timestep_sv = stormer_verlet(5.0, -10, 0.01);
std::cout << "[#]\nTime for Stormer Verlet integration is:\n" \
<< timestep_sv.time << std::endl;
std::cout << "[#]\nVelocity for Stormer Verlet integration is:\n" \
<< timestep_sv.vel << std::endl;
timestep timestep_vv = velocity_verlet(5.0, -10, 0.01);
std::cout << "[#]\nTime for velocity Verlet integration is:\n" \
<< timestep_vv.time << std::endl;
std::cout << "[#]\nVelocity for velocity Verlet integration is:\n" \
<< timestep_vv.vel << std::endl;
return 0;
}
#include <stdio.h>
void verlet(double *time, double pos, double acc, double dt) {
double prev_pos, temp_pos;
prev_pos = pos;
*time = 0.0;
while (pos > 0) {
*time += dt;
temp_pos = pos;
pos = pos * 2 - prev_pos + acc * dt * dt;
prev_pos = temp_pos;
}
}
void stormer_verlet(double *time, double *vel,
double pos, double acc, double dt) {
double prev_pos, temp_pos;
prev_pos = pos;
*vel = 0.0;
*time = 0.0;
while (pos > 0) {
*time += dt;
temp_pos = pos;
pos = pos * 2 - prev_pos + acc * dt * dt;
prev_pos = temp_pos;
*vel += acc * dt;
}
}
void velocity_verlet(double *time, double *vel,
double pos, double acc, double dt) {
*vel = 0.0;
*time = 0.0;
while (pos > 0) {
*time += dt;
pos += (*vel) * dt + 0.5 * acc * dt * dt;
*vel += acc * dt;
}
}
int main() {
double time, vel;
verlet(&time, 5.0, -10, 0.01);
printf("[#]\nTime for Verlet integration is:\n");
printf("%lf\n", time);
stormer_verlet(&time, &vel, 5.0, -10, 0.01);
printf("[#]\nTime for Stormer Verlet integration is:\n");
printf("%lf\n", time);
printf("[#]\nVelocity for Stormer Verlet integration is:\n");
printf("%lf\n", vel);
velocity_verlet(&time, &vel, 5.0, -10, 0.01);
printf("[#]\nTime for velocity Verlet integration is:\n");
printf("%lf\n", time);
printf("[#]\nVelocity for Stormer Verlet integration is:\n");
printf("%lf\n", vel);
return 0;
}
public class Verlet {
private static class VerletValues {
public double time;
public double vel;
public VerletValues(double time, double vel) {
this.time = time;
this.vel = vel;
}
}
static double verlet(double pos, double acc, double dt) {
// Note that we are using a temp variable for the previous position
double prev_pos, temp_pos, time;
prev_pos = pos;
time = 0;
while (pos > 0) {
time += dt;
temp_pos = pos;
pos = pos*2 - prev_pos + acc * dt * dt;
prev_pos = temp_pos;
}
return time;
}
static VerletValues stormer_verlet(double pos, double acc, double dt) {
// Note that we are using a temp variable for the previous position
double prev_pos, temp_pos, time, vel;
prev_pos = pos;
vel = 0;
time = 0;
while (pos > 0) {
time += dt;
temp_pos = pos;
pos = pos*2 - prev_pos + acc * dt * dt;
prev_pos = temp_pos;
// The acceleration is constant, so the velocity is straightforward
vel += acc*dt;
}
return new VerletValues(time, vel);
}
static VerletValues velocity_verlet(double pos, double acc, double dt) {
// Note that we are using a temp variable for the previous position
double time, vel;
vel = 0;
time = 0;
while (pos > 0) {
time += dt;
pos += vel*dt + 0.5*acc * dt * dt;
vel += acc*dt;
}
return new VerletValues(time, vel);
}
public static void main(String[] args) {
double verletTime = verlet(5.0, -10, 0.01);
System.out.println("[#]\nTime for Verlet integration is:");
System.out.println(verletTime);
VerletValues stormerVerlet = stormer_verlet(5.0, -10, 0.01);
System.out.println("[#]\nTime for Stormer Verlet integration is:");
System.out.println(stormerVerlet.time);
System.out.println("[#]\nVelocity for Stormer Verlet integration is:");
System.out.println(stormerVerlet.vel);
VerletValues velocityVerlet = velocity_verlet(5.0, -10, 0.01);
System.out.println("[#]\nTime for velocity Verlet integration is:");
System.out.println(velocityVerlet.time);
System.out.println("[#]\nVelocity for velocity Verlet integration is:");
System.out.println(velocityVerlet.vel);
}
}
def verlet(pos, acc, dt):
prev_pos = pos
time = 0
while pos > 0:
time += dt
next_pos = pos * 2 - prev_pos + acc * dt * dt
prev_pos, pos = pos, next_pos
return time
def stormer_verlet(pos, acc, dt):
prev_pos = pos
time = 0
vel = 0
while pos > 0:
time += dt
next_pos = pos * 2 - prev_pos + acc * dt * dt
prev_pos, pos = pos, next_pos
vel += acc * dt
return time, vel
def velocity_verlet(pos, acc, dt):
time = 0
vel = 0
while pos > 0:
time += dt
pos += vel * dt + 0.5 * acc * dt * dt
vel += acc * dt
return time, vel
def main():
time = verlet(5, -10, 0.01)
print("[#]\nTime for Verlet integration is:")
print("{:.10f}".format(time))
time, vel = stormer_verlet(5, -10, 0.01)
print("[#]\nTime for Stormer Verlet integration is:")
print("{:.10f}".format(time))
print("[#]\nVelocity for Stormer Verlet integration is:")
print("{:.10f}".format(vel))
time, vel = velocity_verlet(5, -10, 0.01)
print("[#]\nTime for velocity Verlet integration is:")
print("{:.10f}".format(time))
print("[#]\nVelocity for velocity Verlet integration is:")
print("{:.10f}".format(vel))
if __name__ == '__main__':
main()
-- submitted by Jie
type Time = Double
type Position = Double
type Speed = Double
type Acceleration = Double
type Particle = (Position, Speed, Acceleration, Time)
type Model = Particle -> Acceleration
type Method = Model -> Time -> Particle -> Particle -> Particle
verlet :: Method
verlet acc dt (xOld, _, _, _) (x, _, a, t) = (x', v', a', t + dt)
where
x' = 2 * x - xOld + a * dt ^ 2
v' = 0
a' = acc (x', v', a, t + dt)
stormerVerlet :: Method
stormerVerlet acc dt (xOld, _, _, _) (x, _, a, t) = (x', v', a', t + dt)
where
x' = 2 * x - xOld + a * dt ^ 2
v' = (x' - x) / dt
a' = acc (x', v', a, t + dt)
velocityVerlet :: Method
velocityVerlet acc dt (xOld, _, aOld, _) (x, v, a, t) = (x', v', a', t + dt)
where
x' = 2 * x - xOld + a * dt ^ 2
v' = v + 0.5 * (aOld + a) * dt
a' = acc (x', v', a, t + dt)
trajectory :: Method -> Model -> Time -> Particle -> [Particle]
trajectory method acc dt p0@(x, v, a, t0) = traj
where
traj = p0 : p1 : zipWith (method acc dt) traj (tail traj)
p1 = (x', v', acc (x', v', a, t0 + dt), t0 + dt)
x' = x + v * dt + 0.5 * a * dt ^ 2
v' = v + a * dt
main :: IO ()
main = do
let p0 = (5, 0, -10, 0)
dt = 0.001
freefall _ = -10
aboveGround (x, _, _, _) = x > 0
timeVelocity m =
let (_, v, _, t) = last $ takeWhile aboveGround $ trajectory m freefall dt p0
in (show t, show v)
putStrLn "[#]\nTime for Verlet integration is:"
putStrLn $ fst $ timeVelocity verlet
putStrLn "[#]\nTime for Stormer Verlet integration is:"
putStrLn $ fst $ timeVelocity stormerVerlet
putStrLn "[#]\nVelocity for Stormer Verlet integration is:"
putStrLn $ snd $ timeVelocity stormerVerlet
putStrLn "[#]\nTime for velocity Verlet integration is:"
putStrLn $ fst $ timeVelocity velocityVerlet
putStrLn "[#]\nVelocity for velocity Verlet integration is:"
putStrLn $ snd $ timeVelocity velocityVerlet
function verlet(pos, acc, dt) {
let prevPos = pos;
let time = 0;
let tempPos;
while (pos > 0) {
time += dt;
tempPos = pos;
pos = pos * 2 - prevPos + acc * dt * dt;
prevPos = tempPos;
}
return time;
}
function stormerVerlet(pos, acc, dt) {
let prevPos = pos;
let time = 0;
let vel = 0;
let tempPos;
while (pos > 0) {
time += dt;
tempPos = pos;
pos = pos * 2 - prevPos + acc * dt * dt;
prevPos = tempPos;
vel += acc * dt;
}
return { time, vel };
}
function velocityVerlet(pos, acc, dt) {
let time = 0;
let vel = 0;
while (pos > 0) {
time += dt;
pos += vel * dt + 0.5 * acc * dt * dt;
vel += acc * dt;
}
return { time, vel };
}
const time = verlet(5, -10, 0.01);
console.log(`[#]\nTime for Verlet integration is:`);
console.log(`${time}`);
const stormer = stormerVerlet(5, -10, 0.01);
console.log(`[#]\nTime for Stormer Verlet integration is:`);
console.log(`${stormer.time}`);
console.log(`[#]\nVelocity for Stormer Verlet integration is:`);
console.log(`${stormer.vel}`);
const velocity = velocityVerlet(5, -10, 0.01);
console.log(`[#]\nTime for velocity Verlet integration is:`);
console.log(`${velocity.time}`);
console.log(`[#]\nVelocity for velocity Verlet integration is:`);
console.log(`${velocity.vel}`);
fn verlet(mut pos: f64, acc: f64, dt: f64) -> f64 {
let mut prev_pos = pos;
let mut time = 0.0;
while pos > 0.0 {
time += dt;
let temp_pos = pos;
pos = pos * 2.0 - prev_pos + acc * dt * dt;
prev_pos = temp_pos;
}
time
}
fn stormer_verlet(mut pos: f64, acc: f64, dt: f64) -> (f64, f64) {
let mut prev_pos = pos;
let mut time = 0.0;
let mut vel = 0.0;
while pos > 0.0 {
time += dt;
let temp_pos = pos;
pos = pos * 2.0 - prev_pos + acc * dt * dt;
prev_pos = temp_pos;
// Because acceleration is constant, velocity is
// straightforward
vel += acc * dt;
}
(time, vel)
}
fn velocity_verlet(mut pos: f64, acc: f64, dt: f64) -> (f64, f64) {
let mut time = 0.0;
let mut vel = 0.0;
while pos > 0.0 {
time += dt;
pos += vel * dt + 0.5 * acc * dt * dt;
vel += acc * dt;
}
(time, vel)
}
fn main() {
let time_v = verlet(5.0, -10.0, 0.01);
let (time_sv, vel_sv) = stormer_verlet(5.0, -10.0, 0.01);
let (time_vv, vel_vv) = velocity_verlet(5.0, -10.0, 0.01);
println!("[#]\nTime for Verlet integration is:");
println!("{}", time_v);
println!("[#]\nTime for Stormer Verlet integration is:");
println!("{}", time_sv);
println!("[#]\nVelocity for Stormer Verlet integration is:");
println!("{}", vel_sv);
println!("[#]\nTime for velocity Verlet integration is:");
println!("{}", time_vv);
println!("[#]\nVelocity for velocity Verlet integration is:");
println!("{}", vel_vv);
}
func verlet(pos: Double, acc: Double, dt: Double) -> Double {
var pos = pos
var temp_pos, time: Double
var prev_pos = pos
time = 0.0
while (pos > 0) {
time += dt
temp_pos = pos
pos = pos*2 - prev_pos + acc * dt * dt
prev_pos = temp_pos
}
return time
}
func stormerVerlet(pos: Double, acc: Double, dt: Double) -> (time: Double, vel: Double) {
var pos = pos
var temp_pos, time, vel: Double
var prev_pos = pos
vel = 0
time = 0
while (pos > 0) {
time += dt
temp_pos = pos
pos = pos*2 - prev_pos + acc * dt * dt
prev_pos = temp_pos
vel += acc*dt
}
return (time:time, vel:vel)
}
func velocityVerlet(pos: Double, acc: Double, dt: Double) -> (time: Double, vel: Double) {
var pos = pos
var time, vel : Double
vel = 0
time = 0
while (pos > 0) {
time += dt
pos += vel*dt + 0.5*acc * dt * dt
vel += acc*dt
}
return (time:time, vel:vel)
}
func main() {
let verletTime = verlet(pos: 5.0, acc: -10.0, dt: 0.01)
print("[#]\nTime for Verlet integration is:")
print("\(verletTime)")
let stormer = stormerVerlet(pos: 5.0, acc: -10.0, dt: 0.01);
print("[#]\nTime for Stormer Verlet integration is:")
print("\(stormer.time)")
print("[#]\nVelocity for Stormer Verlet integration is:")
print("\(stormer.vel)")
let velVerlet = velocityVerlet(pos: 5.0, acc: -10, dt: 0.01)
print("[#]\nTime for velocity Verlet integration is:")
print("\(velVerlet.time)")
print("[#]\nVelocity for velocity Verlet integration is:")
print("\(velVerlet.vel)")
}
main()
SUBROUTINE verlet(pos, acc, dt, time)
IMPLICIT NONE
REAL(8), INTENT(INOUT) :: pos, acc, dt, time
REAL(8) :: prev_pos, next_pos
prev_pos = pos
time = 0d0
DO
IF (pos > 0d0) THEN
time = time + dt
next_pos = pos * 2d0 - prev_pos + acc * dt ** 2
prev_pos = pos
pos = next_pos
ELSE
EXIT
END IF
END DO
END SUBROUTINE verlet
SUBROUTINE stormer_verlet(pos, acc, dt, time, vel)
IMPLICIT NONE
REAL(8), INTENT(INOUT) :: pos, acc, dt, time, vel
REAL(8) :: prev_pos, next_pos
prev_pos = pos
time = 0d0
vel = 0d0
DO
IF (pos > 0d0) THEN
time = time + dt
next_pos = pos * 2 - prev_pos + acc * dt ** 2
prev_pos = pos
pos = next_pos
vel = vel + acc * dt
ELSE
EXIT
END IF
END DO
END SUBROUTINE stormer_verlet
SUBROUTINE velocity_verlet(pos, acc, dt, time, vel)
IMPLICIT NONE
REAL(8), INTENT(INOUT) :: pos, acc, dt, time, vel
time = 0d0
vel = 0d0
DO
IF (pos > 0d0) THEN
time = time + dt
pos = pos + vel * dt + 0.5d0 * acc * dt ** 2
vel = vel + acc * dt
ELSE
EXIT
END IF
END DO
END SUBROUTINE velocity_verlet
PROGRAM verlet_integration
IMPLICIT NONE
REAL(8) :: pos,acc, dt, time, vel
INTERFACE
SUBROUTINE verlet(pos, acc, dt, time)
REAL(8), INTENT(INOUT) :: pos, acc, dt, time
REAL(8) :: prev_pos, next_pos
END SUBROUTINE
END INTERFACE
INTERFACE
SUBROUTINE stormer_verlet(pos, acc, dt, time, vel)
REAL(8), INTENT(INOUT) :: pos, acc, dt, time, vel
REAL(8) :: prev_pos, next_pos
END SUBROUTINE
END INTERFACE
INTERFACE
SUBROUTINE velocity_verlet(pos, acc, dt, time, vel)
REAL(8), INTENT(INOUT) :: pos, acc, dt, time, vel
REAL(8) :: prev_pos, next_pos
END SUBROUTINE
END INTERFACE
pos = 5d0
acc = -10d0
dt = 0.01d0
! Verlet
CALL verlet(pos, acc, dt, time)
WRITE(*,*) '[#]'
WRITE(*,*) 'Time for Verlet integration:'
WRITE(*,*) time
! stormer Verlet
pos = 5d0
CALL stormer_verlet(pos, acc, dt, time, vel)
WRITE(*,*) '[#]'
WRITE(*,*) 'Time for Stormer Verlet integration:'
WRITE(*,*) time
WRITE(*,*) '[#]'
WRITE(*,*) 'Velocity for Stormer Verlet integration:'
WRITE(*,*) vel
! Velocity Verlet
pos = 5d0
CALL velocity_verlet(pos, acc, dt, time, vel)
WRITE(*,*) '[#]'
WRITE(*,*) 'Time for velocity Verlet integration:'
WRITE(*,*) time
WRITE(*,*) '[#]'
WRITE(*,*) 'Velocity for velocity Verlet integration:'
WRITE(*,*) vel
END PROGRAM verlet_integration
def verlet(pos, acc, dt)
prev_pos = pos
time = 0
while pos > 0 do
time += dt
temp_pos = pos
pos = pos*2 - prev_pos + acc * dt * dt
prev_pos = temp_pos
end
return time
end
def stormer_verlet(pos, acc, dt)
prev_pos = pos
vel = 0
time = 0
while pos > 0 do
time += dt
temp_pos = pos
pos = pos*2 - prev_pos + acc * dt * dt
prev_pos = temp_pos
vel += acc*dt
end
return time, vel
end
def velocity_verlet(pos, acc, dt)
vel = 0
time = 0
while pos > 0 do
time += dt
pos += vel*dt + 0.5*acc * dt * dt
vel += acc*dt
end
return time, vel
end
puts "[#]\nTime for Verlet integration is:"
p verlet(5.0, -10, 0.01)
time, vel = stormer_verlet(5.0, -10, 0.01)
puts "[#]\nTime for Stormer Verlet integration is:"
p time
puts "[#]\nVelocity for Stormer Verlet integration is:"
p vel
time, vel = velocity_verlet(5.0, -10, 0.01)
puts "[#]\nTime for velocity Verlet integration is:"
p time
puts "[#]\nVelocity for velocity Verlet integration is:"
p vel
package main
import "fmt"
func verlet(pos, acc, dt float64) (time float64) {
prevPos := pos
time = 0
for pos > 0 {
time += dt
nextPos := pos*2 - prevPos + acc*dt*dt
prevPos, pos = pos, nextPos
}
return
}
func stormerVerlet(pos, acc, dt float64) (time, vel float64) {
prevPos := pos
time, vel = 0, 0
for pos > 0 {
time += dt
vel += acc * dt
nextPos := pos*2 - prevPos + acc*dt*dt
prevPos, pos = pos, nextPos
}
return
}
func velocityVerlet(pos, acc, dt float64) (time, vel float64) {
time, vel = 0, 0
for pos > 0 {
time += dt
pos += vel*dt + .5*acc*dt*dt
vel += acc * dt
}
return
}
func main() {
time := verlet(5., -10., .01)
fmt.Println("[#]\nTime for Verlet integration is:")
fmt.Println(time)
time, vel := stormerVerlet(5., -10., .01)
fmt.Println("[#]\nTime for Stormer Verlet integration is:")
fmt.Println(time)
fmt.Println("[#]\nVelocity for Stormer Verlet integration is:")
fmt.Println(vel)
time, vel = velocityVerlet(5., -10., .01)
fmt.Println("[#]\nTime for velocity Verlet integration is:")
fmt.Println(time)
fmt.Println("[#]\nVelocity for velocity Verlet integration is:")
fmt.Println(vel)
}
.intel_syntax noprefix
.section .rodata
zero: .double 0.0
two: .double 2.0
half: .double 0.5
verlet_fmt: .string "[#]\nTime for Verlet integration is:\n%lf\n"
stormer_fmt: .string "[#]\nTime for Stormer Verlet Integration is:\n%lf\n[#]\nVelocity for Stormer Verlet Integration is:\n%lf\n"
velocity_fmt: .string "[#]\nTime for Velocity Verlet Integration is:\n%lf\n[#]\nVelocity for Velocity Verlet Integration is:\n%lf\n"
pos: .double 5.0
acc: .double -10.0
dt: .double 0.01
.section .text
.global main
.extern printf
# xmm0 - pos
# xmm1 - acc
# xmm2 - dt
# RET xmm0 - time
verlet:
pxor xmm7, xmm7 # Holds 0 for comparisons
pxor xmm3, xmm3 # Holds time value
comisd xmm0, xmm7 # Check if pos is greater then 0.0
jbe verlet_return
movsd xmm6, xmm1 # xmm6 = acc * dt * dt
mulsd xmm6, xmm2
mulsd xmm6, xmm2
movsd xmm5, xmm0 # Holds previous position
verlet_loop:
addsd xmm3, xmm2 # Adding dt to time
movsd xmm4, xmm0 # Hold old value of posistion
addsd xmm0, xmm0 # Calculating new position
subsd xmm0, xmm5
addsd xmm0, xmm6
movsd xmm5, xmm4
comisd xmm0, xmm7 # Check if position is greater then 0.0
ja verlet_loop
verlet_return:
movsd xmm0, xmm3 # Saving time value
ret
# xmm0 - pos
# xmm1 - acc
# xmm2 - dt
# RET xmm0 - time
# RET xmm1 - velocity
stormer_verlet:
pxor xmm7, xmm7 # Holds 0 for comparisons
pxor xmm3, xmm3 # Holds time value
comisd xmm0, xmm7 # Check if pos is greater then 0.0
jbe stormer_verlet_return
movsd xmm6, xmm1 # xmm6 = acc * dt * dt
mulsd xmm6, xmm2
mulsd xmm6, xmm2
movsd xmm5, xmm0 # Holds previous position
stormer_verlet_loop:
addsd xmm3, xmm2 # Adding dt to time
movsd xmm4, xmm0 # Hold old value of posistion
addsd xmm0, xmm0 # Calculating new position
subsd xmm0, xmm5
addsd xmm0, xmm6
movsd xmm5, xmm4
comisd xmm0, xmm7 # Check if position is greater then 0.0
ja stormer_verlet_loop
stormer_verlet_return:
movsd xmm0, xmm3 # Saving time and velocity
mulsd xmm3, xmm1
movsd xmm1, xmm3
ret
# xmm0 - pos
# xmm1 - acc
# xmm2 - dt
# RET xmm0 - time
# RET xmm1 - velocity
velocity_verlet:
pxor xmm7, xmm7 # Holds 0 for comparisons
pxor xmm3, xmm3 # Holds the velocity value
pxor xmm4, xmm4 # Holds the time value
comisd xmm0, xmm7 # Check if pos is greater then 0.0
jbe velocity_verlet_return
movsd xmm5, half # xmm5 = 0.5 * dt * dt * acc
mulsd xmm5, xmm2
mulsd xmm5, xmm2
mulsd xmm5, xmm1
velocity_verlet_loop:
movsd xmm6, xmm3 # Move velocity into register
mulsd xmm6, xmm2 # Calculate new position
addsd xmm6, xmm5
addsd xmm0, xmm6
addsd xmm4, xmm2 # Incrementing time
movsd xmm3, xmm4 # Updating velocity
mulsd xmm3, xmm1
comisd xmm0, xmm7
ja velocity_verlet_loop
velocity_verlet_return:
movsd xmm0, xmm4 # Saving time and velocity
movsd xmm1, xmm3
ret
main:
push rbp
movsd xmm0, pos # Calling verlet
movsd xmm1, acc
movsd xmm2, dt
call verlet
mov rdi, OFFSET verlet_fmt # Print output
mov rax, 1
call printf
movsd xmm0, pos # Calling stormer_verlet
movsd xmm1, acc
movsd xmm2, dt
call stormer_verlet
mov rdi, OFFSET stormer_fmt # Print output
mov rax, 1
call printf
movsd xmm0, pos # Calling velocity_verlet
movsd xmm1, acc
movsd xmm2, dt
call velocity_verlet
mov rdi, OFFSET velocity_fmt # Print output
mov rax, 1
call printf
pop rbp
xor rax, rax # Set exit code to 0
ret
data class VerletValues(val time: Double, val vel: Double)
fun verlet(_pos: Double, acc: Double, dt: Double): Double {
var pos = _pos // Since function parameter are val and can't be modified
var prevPos = pos
var time = 0.0
while (pos > 0) {
time += dt
val nextPos = pos * 2 - prevPos + acc * dt * dt
prevPos = pos
pos = nextPos
}
return time
}
fun stormerVerlet(_pos: Double, acc: Double, dt: Double): VerletValues {
var pos = _pos
var prevPos = pos
var time = 0.0
var vel = 0.0
while (pos > 0) {
time += dt
val nextPos = pos * 2 - prevPos + acc * dt * dt
prevPos = pos
pos = nextPos
vel += acc * dt
}
return VerletValues(time, vel)
}
fun velocityVerlet(_pos: Double, acc: Double, dt: Double): VerletValues {
var pos = _pos
var time = 0.0
var vel = 0.0
while (pos > 0) {
time += dt
pos += vel * dt + 0.5 * acc * dt * dt
vel += acc * dt
}
return VerletValues(time, vel)
}
fun main(args: Array<String>) {
val verletTime = verlet(5.0, -10.0, 0.01)
println("[#]\nTime for Verlet integration is:")
println("$verletTime")
val stormerVerlet = stormerVerlet(5.0, -10.0, 0.01)
println("[#]\nTime for Stormer Verlet integration is:")
println("${stormerVerlet.time}")
println("[#]\nVelocity for Stormer Verlet integration is:")
println("${stormerVerlet.vel}")
val velocityVerlet = velocityVerlet(5.0, -10.0, 0.01)
println("[#]\nTime for Velocity Verlet integration is:")
println("${velocityVerlet.time}")
println("[#]\nVelocity for Velocity Verlet integration is:")
println("${velocityVerlet.vel}")
}
func verlet(pos_in, acc, dt: float): float =
var
pos: float = pos_in
prevPos: float = pos
time: float = 0.0
tempPos: float
while pos > 0.0:
time += dt
tempPos = pos
pos = pos * 2 - prevPos + acc * dt * dt
prevPos = tempPos
time
func stormerVerlet(pos_in, acc, dt: float): (float, float) =
var
pos: float = pos_in
prevPos: float = pos
time: float = 0.0
vel: float = 0.0
tempPos: float
while pos > 0.0:
time += dt
tempPos = pos
pos = pos * 2 - prevPos + acc * dt * dt
prevPos = tempPos
vel += acc * dt
(time, vel)
func velocityVerlet(pos_in, acc, dt: float): (float, float) =
var
pos: float = pos_in
time: float = 0.0
vel: float = 0.0
while pos > 0.0:
time += dt
pos += vel * dt + 0.5 * acc * dt * dt
vel += acc * dt
(time, vel)
when isMainModule:
let timeV = verlet(5.0, -10.0, 0.01)
echo "[#]\nTime for Verlet integration is:"
echo timeV
let (timeSV, velSV) = stormerVerlet(5.0, -10.0, 0.01)
echo "[#]\nTime for Stormer Verlet integration is:"
echo timeSV
echo "[#]\nVelocity for Stormer Verlet integration is:"
echo velSV
let (timeVV, velVV) = velocityVerlet(5.0, -10.0, 0.01)
echo "[#]\nTime for velocity Verlet integration is:"
echo timeVV
echo "[#]\nVelocity for velocity Verlet integration is:"
echo velVV
;;;; 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:~%")
(format T "~d~%" (verlet 5 -10 0.01))
(defvar stormer-verlet-result (stormer-verlet 5 -10 0.01))
(format T "[#]~%Time for Stormer Verlet integration is:~%")
(format T "~d~%" (first stormer-verlet-result))
(format T "[#]~%Velocity for Stormer Verlet integration is:~%")
(format T "~d~%" (second stormer-verlet-result))
(defvar velocity-verlet-result (velocity-verlet 5 -10 0.01))
(format T "[#]~%Time for velocity Verlet integration is:~%")
(format T "~d~%" (first velocity-verlet-result))
(format T "[#]~%Velocity for velocity Verlet integration is:~%")
(format T "~d~%" (second velocity-verlet-result))
License
Code Examples
The code examples are licensed under the MIT license (found in LICENSE.md).
Text
The text of this chapter was written by James Schloss and is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.
Pull Requests
After initial licensing (#560), the following pull requests have modified the text or graphics of this chapter:
- none