double t = 0.0;
    double dt = 1.0/60.0;

    while ( !quit )
    {
         integrate( state, t, dt );
         render( state );
         t += dt;
    }

In many ways this code is ideal. If you are lucky enough to have your physics delta time match the display rate and you can ensure that your update loop takes less than one frame then you have the perfect solution for updating your physics simulation.
But in the real world you may not know the display refresh rate ahead of time, VSYNC could be turned off, or perhaps you could be running on a slow computer which cannot update and render your frame fast enough to present it at 60fps.
In these cases your simulation will run faster or slower than you intended.
Variable delta time
Fixing this *seems* simple. Just measure how long the previous frame takes, then feed that value back in as the delta time for the next frame. This makes sense because of course, because if the computer is too slow to update at 60HZ and has to drop down to 30fps, you’ll automatically pass in 1/30 as delta time. Same thing for a display refresh rate of 75HZ instead of 60HZ or even the case where VSYNC is turned off on a fast computer:

    double t = 0.0;

    double currentTime = hires_time_in_seconds();

    while ( !quit )
    {
         double newTime = hires_time_in_seconds();
         double frameTime = newTime - currentTime;
         currentTime = newTime;

         integrate( state, t, frameTime );
         t += frameTime;

         render( state );
    }

But there is a huge problem with this approach which I will now explain. The problem is that the behavior of your physics simulation depends on the delta time you pass in. The effect could be subtle as your game having a slightly different “feel” depending on framerate or it could be as extreme as your spring simulation exploding to infinity, fast moving objects tunneling through walls and the player falling through the floor!
One thing is for certain though and that is that it’s utterly unrealistic to just expect your simulation to correctly handle *any* delta time passed into it. To understand why, consider what would happen if you passed in 1/10th of a second as delta time?How about one second?10 seconds?100?Eventually you’ll find a breaking point.
Semi-fixed timestep
It’s much more realistic to say that your simulation is well behaved only if delta time is less than or equal to some maximum value. You can use mathematics to find this exact delta time value given your simulation (there is a whole field on it called numerical analysis, try researching “interval arithmetic” as well if you are keen), or you can arrive at it using a process of experimentation, or by simply tuning your simulation at some ideal framerate that you determine ahead of time. This is usually significantly easier in practice than attempting to make your simulation bulletproof at a wide range of delta time values.
With this knowledge at hand, here is a simple trick to ensure that you never pass in a delta time greater than the maximum value, while still running at the correct speed on different machines:

    double t = 0.0;
    double dt = 1/60.0;

    double currentTime = hires_time_in_seconds();

    while ( !quit )
    {
         double newTime = hires_time_in_seconds();
         double frameTime = newTime - currentTime;
         currentTime = newTime;
          
         while ( frameTime > 0.0 )
         {
             const float deltaTime = min( frameTime, dt );
             integrate( state, t, deltaTime );
             frameTime -= deltaTime;
             t += deltaTime;
         }

         render( state );
    }

The benefit of this approach is of course that we now have an upper bound on delta time. It’s never larger than this value because if it is we subdivide the timestep. The disadvantage is that we’re now taking multiple steps per-display update including one additional step to consume any the remainder of frame time not divisible by dt. This is no problem if you are render bound, but if your simulation is the most expensive part of your frame you could run into problems including the so called “spiral of death”.
What exactly is this spiral of death?It’s what happens when your physics simulation cannot keep up with the steps it’s asked to take. For example, if your simulation is told: “OK, please simulate X seconds worth of physics” and if it takes Y seconds of real time to do so where Y > X, then it doesn’t take Einstein to realize that over time your simulation falls behind. It’s called the spiral of death because ironically being behind causes your update to simulate more steps, which causes you to fall further behind, which makes you simulate more steps…
So how do we avoid this?In order to ensure a stable update I recommend leaving some headroom. You really need to ensure that it takes *significantly less* than X seconds of real time to update X seconds worth of physics simulation. If you can do this then your physics engine can “catch up” from any temporary spike by simulating more frames. Alternatively you can clamp at a maximum # of steps per-frame and the simulation will appear to slow down under heavy load. Arguably this is better than spiraling to death, assuming of course that the heavy load is just a temporary spike.
Free the physics
Now lets take it one step further. What if you want exact reproducibility from one run to the next given the same inputs?This comes in handy when trying to network your physics simulation using deterministic lockstep, but it’s also generally a nice thing to know that your simulation behaves exactly the same from one run to the next without any potential for different behavior depending on the render framerate.
But you ask why is it necessary to have fully fixed delta time to do this?Surely the semi-fixed delta time with the small remainder step is “good enough”?And yes, you are right. It is good enough in most cases but it is not *exactly the same*. It takes only a basic understanding of floating point numbers to realize that (v*dt) + (v*dt) is not necessarily equal to v*2*dt due to to the limited precision of floating point arithmetic, so it follows that in order to get exactly the same result (and I mean exact down to the floating point bits) it is necessary to use a fixed delta time value.
So what we want is the best of both worlds: a fixed delta time value for the simulation plus the ability to render at different framerates. These two things seem completely at odds, and they are – unless we can find a way to decouple the simulation and rendering framerates.
Here’s how to do it. Advance the physics simulation ahead in fixed dt time steps while also making sure that it keeps up with the timer values coming from the renderer so that the simulation advances at the correct rate. For example, if the display framerate is 50fps and the simulation runs at 100fps then we need to take two physics steps every display update.
What if the display framerate is 200fps?Well in this case it would seem that we need to take half a physics step each display update, but we can’t do that, we must advance with constant dt; So instead we must take one physics step every two display updates. Even trickier, what if the display framerate is 60fps, but we want our simulation to run at 100fps?There is no easy multiple here. Finally, what if VSYNC is disabled and the display frame rate fluctuates from frame to frame?
If you head just exploded don’t worry, all that is needed to solve this is to change your point of view. Instead of thinking that you have a certain amount of frame time you must simulate before rendering, flip your viewpoint upside down and think of it like this: The renderer produces time and the simulation consumes it in discrete dt sized chunks.
Again:
The renderer produces time and the simulation consumes it in discrete dt sized chunks.

    double t = 0.0;
    const double dt = 0.01;

    double currentTime = hires_time_in_seconds();
    double accumulator = 0.0;

    while ( !quit )
    {
         double newTime = hires_time_in_seconds();
         double frameTime = newTime - currentTime;
         currentTime = newTime;

         accumulator += frameTime;

         while ( accumulator >= dt )
         {
              integrate( state, t, dt );
              accumulator -= dt;
              t += dt;
         }

         render(state);
    }

Notice that unlike the semi-fixed timestep we only ever integrate with steps sized dt so it follows that in the common case we have some unsimulated time left over at the end of each frame. This is important! This left over time is passed on to the next frame via the accumulator variable and is not thrown away.
The final touch
But what do to with this remaining time?It seems incorrect somehow doesn’t it?
To understand what is going on consider a situation where the display framerate is 60fps and the physics is running at 50fps. There is no nice multiple so the accumulator causes the simulation to alternate between mostly taking one and occasionally two physics steps per-frame when the remainders “accumulate” above dt.
Now consider that in general all render frames will have some small remainder of frame time left in the accumulator that cannot be simulated because it is less than dt. What this means is that we’re displaying the state of the physics simulation at a time value slightly different from the render time. This causes a subtle but visually unpleasant stuttering of the physics simulation on the screen known as temporal aliasing.
The solution is to interpolate between the previous and current physics state based on how much time is left in the accumulator:

    double t = 0.0;
    const double dt = 0.01;

    double currentTime = hires_time_in_seconds();
    double accumulator = 0.0;

    State previous;
    State current;

    while ( !quit )
    {
         double newTime = time();
         double frameTime = newTime - currentTime;
         if ( frameTime > 0.25 )
              frameTime = 0.25;	  // note: max frame time to avoid spiral of death
         currentTime = newTime;

         accumulator += frameTime;

         while ( accumulator >= dt )
         {
              previousState = currentState;
              integrate( currentState, t, dt );
              t += dt;
              accumulator -= dt;
         }

         const double alpha = accumulator/dt;

         State state = currentState*alpha + previousState * ( 1.0 - alpha );

         render( state );
    }

This looks pretty complicated but here is a simple way to think about it. Any remainder in the accumulator is effectively a measure of just how much more time is required before another whole physics step can be taken. For example, a remainder of dt/2 means that we are currently halfway between the current physics step and the next. A remainder of dt*0.1 means that the update is 1/10th of the way between the current and the next state.
We can use this remainder value to get a blending factor between the previous and current physics state simply by dividing by dt. This gives an alpha value in the range [0,1] which is used to perform a linear interpolation between the two physics states to get the current state to render. This interpolation is easy to do for single values and for vector state values. You can even use it with full 3D rigid body dynamics if you store your orientation as a quaternion and use a spherical linear interpolation (slerp) to blend between the previous and current orientations.
Click here to download the source code for this article.
Further reading
Integration Basics
Integration is used to determine the motion of an object over time. In this article I show how to correctly integrate the equations of motion using an RK4 integrator instead of starting off on the wrong foot with a stupid Euler integrator.
Physics in 3D
Leap ahead from integrating single values to integrating the entire physics state for a cube in three dimensions. Introduces rotational physics concepts including orientation in 3D, angular velocity and momentum, inertia and torque.
Spring Physics
Explains the physics of springs and how to apply them to control physics simulations. Learn how to implement joints, constraints, motors and basic collision response.
Networked Physics
How do network games synchronize physics over the network?This article explains the core techniques used in today’s first person shooters and shows how you can apply these techniques to network your own physics simulations.