Feynman’s Lectures on Physics 11-2: Translations – Why the Laws of Physics Don’t Care Where You Are

An exploration of spatial symmetry, Newtonian mechanics and the surprising consequences of assuming that space is absolute.

There is something wonderfully characteristic of Richard Feynman in the way he approaches a seemingly straightforward question: does it actually matter where we put the origin of our coordinate system?

In Lecture 11-2, Translations, from his celebrated Lectures on Physics, Feynman takes an apparently mundane feature of Newtonian mechanics and uses it to expose something rather profound about the structure of the physical universe. His argument begins with the familiar laws of motion, but quickly develops into a discussion of symmetry, the nature of space and the universality of physical laws.

This is an important idea to understand, not least because it provides an early glimpse of one of the most powerful organising principles in modern physics: symmetry.

1. Where is the centre of the universe?

Feynman begins by reminding us that Newtonian mechanics describes the motion of particles through three-dimensional space. To do this, we need a coordinate system: three mutually perpendicular axes from which we can measure position, velocity and acceleration.

But there is an immediate problem. Where should we put the origin?

Newton’s formulation of mechanics might initially suggest that there is some privileged location in the universe from which positions should be measured. Perhaps, as Feynman playfully suggests, this is the centre of the universe.

However, this raises an interesting difficulty. How would we ever identify such a position experimentally?

To investigate this, Feynman introduces two imaginary observers, Joe and Moe. Joe establishes his coordinate system at one location, while Moe chooses a different origin. Their axes remain parallel, but their coordinate systems are displaced relative to one another.

Imagine both observers watching the same particle move through space. They will assign different numerical coordinates to its position, simply because they measure from different origins.

Crucially, however, the particle itself has not changed its position in any physical sense. Only the description of that position has changed.

This is the distinction between a physical phenomenon and the coordinate system we use to describe it.

2. Why changing the origin doesn’t change the physics

Feynman’s next step is to ask whether Joe and Moe will agree on the laws governing the particle’s motion.

Although their measurements of position differ, the displacement between their origins is constant. Consequently, the particle’s velocity and acceleration are identical in both coordinate systems.

This is the key observation.

Acceleration describes how velocity changes with time. If one observer’s coordinate system is simply shifted relative to another, the particle’s velocity does not acquire an additional component, nor does its acceleration.

The same is true of force. Because Joe and Moe have parallel axes, they resolve the force into exactly the same directional components. The force acting along Joe’s horizontal axis is the same as the force acting along Moe’s corresponding horizontal axis.

They therefore arrive at identical equations of motion.

Both observers can independently apply Newton’s second law and obtain the same physical predictions.

The important consequence is that there is no experimentally distinguishable preferred origin in space.

We can move the origin of our coordinate system without changing the mathematical description of the underlying physical laws.

Feynman calls this symmetry under translational displacements.

3. The machinery thought experiment: physics is not geographically specific

Feynman then makes the argument more tangible by considering a piece of machinery.

Suppose we have a machine in one location, constructed from a collection of components that interact according to the laws of mechanics. We move an identical machine to another location.

Should we expect it to behave differently?

Provided the relevant physical conditions are equivalent, the answer is no.

Feynman’s reasoning is that the equations describing the machinery are unchanged by a translation in space. The same relationships between forces, masses and accelerations apply in the new location.

This is a deceptively important observation.

It means that the laws of physics are not local conventions that happen to work in one particular laboratory. They are universal descriptions of physical behaviour.

A pendulum does not need to be recalibrated because it has been moved several metres across a laboratory. A mechanical clock does not acquire a different fundamental mechanism because it has been transported to another building.

Of course, this assumes that the relevant physical conditions remain unchanged. Moving a pendulum to a different gravitational environment, for example, can change its period. But that is because the physical environment has changed, not because the origin of the coordinate system has moved.

This distinction matters enormously. Translational symmetry does not mean that everything behaves identically everywhere regardless of circumstances. It means that the fundamental laws themselves do not depend on an arbitrary choice of spatial origin.

4. Newton’s absolute space: an important distinction

There is a deeper philosophical question hiding within this discussion.

Newton famously proposed the existence of absolute space and absolute time. In his conception, space exists independently of the objects within it, providing an underlying framework against which physical motion can be understood.

Absolute space is not simply the coordinate system chosen by an observer. It is a deeper, supposedly objective reference structure.

This distinction becomes particularly important when we consider Newton’s mechanics more broadly.

Feynman’s discussion demonstrates that the laws of mechanics do not identify a unique spatial origin. We can translate our coordinate system without affecting the equations of motion.

However, this does not mean that Newton’s absolute space has disappeared.

Newton’s mechanics still distinguishes between inertial and non-inertial reference frames. An observer accelerating relative to an inertial frame will not generally find that Newton’s laws hold in their simplest form. Apparent forces, such as centrifugal forces, emerge when we describe motion from rotating or accelerating coordinate systems.

The distinction is therefore between two different transformations:

• Translations: Moving the origin of a coordinate system without changing its orientation or velocity. Newton’s laws remain unchanged.
• Accelerations and rotations: Changing the motion or orientation of the reference frame. Additional considerations are needed to preserve the Newtonian description.

Feynman is deliberately restricting his discussion to translations. He is not claiming that every conceivable coordinate system is equally suitable for Newtonian mechanics.

This is important because it highlights a limitation of the argument: the absence of a preferred origin does not imply the absence of a preferred state of motion.

In Newtonian physics, inertial frames remain special.

5. The consequences of absolute space and time

The implications of Newton’s assumptions extend considerably further than the choice of coordinate origin.

Absolute space provides a fixed geometrical background. Absolute time provides a universal clock, ticking at the same rate for every observer, independently of their motion or location.

Together, these assumptions produce a particular picture of the universe:

• Space is a fixed, three-dimensional arena in which events occur.
• Time progresses independently of the events taking place within it.
• Distances and durations are fundamentally objective.
• Observers may disagree about coordinates, but they share the same underlying spatial and temporal framework.
• There is a distinction between inertial motion and acceleration, with inertial frames providing the natural setting for Newton’s laws.

The consequences are profound.

For example, if two observers are moving relative to one another, Newtonian mechanics allows them to disagree about the velocity of an object. Yet they agree about the passage of time and the spatial separation between simultaneous events.

This is the world of Galilean relativity.

It works extraordinarily well for everyday mechanics, where speeds are small compared with the speed of light. But it is not the final word.

Einstein’s special theory of relativity fundamentally changes this picture. Space and time are no longer independent absolute backgrounds. Different inertial observers can disagree about the duration between events, their spatial separation and whether two spatially separated events occur simultaneously.

The speed of light, rather than absolute time, becomes the central invariant shared by inertial observers.

It is worth emphasising that Feynman’s argument in this lecture does not itself establish special relativity. Rather, it provides a useful starting point for understanding how physicists think about the invariance of physical laws under changes of description.

6. From translational symmetry to conservation laws

There is one particularly interesting consequence of Feynman’s discussion that is worth taking further.

The symmetry he describes is closely connected to one of the most important results in theoretical physics: Noether’s theorem.

Developed by mathematician Emmy Noether in 1918, the theorem establishes a profound connection between continuous symmetries of physical systems and conserved quantities.

In the case of spatial translations, the corresponding conserved quantity is linear momentum.

This is not a coincidence.

If the laws governing a physical system are unchanged when the entire system is displaced through space, then the system possesses translational symmetry. Under the appropriate conditions, this symmetry implies conservation of total linear momentum.

Consider two colliding billiard balls on an ideal frictionless table. Their individual velocities change during the collision, but their combined momentum remains constant.

The conservation of momentum is not merely an empirical observation about collisions. It can be understood as a consequence of the underlying translational symmetry of the physical laws.

This is a remarkable connection. A seemingly abstract question about where we put the origin of our coordinate system leads directly to one of the fundamental conservation principles of mechanics.

There is an important qualification: the relevant symmetry must apply to the complete physical system. If we consider only part of a system, external forces can change its momentum. Similarly, a spatially varying external potential can break translational symmetry.

Nevertheless, the conceptual connection remains one of the most powerful ideas in physics.

7. Why this seemingly simple lecture matters

What I particularly like about this section of Feynman’s lectures is the way he builds an important physical principle from an almost trivial observation.

Of course, we intuitively expect a machine to behave in the same way if we move it across a laboratory. We would probably never question the idea that the laws of mechanics should be independent of the arbitrary position of our coordinate origin.

But Feynman insists on working through the reasoning.

He distinguishes between the mathematical description of an event and the event itself. He demonstrates that changing coordinates need not change the underlying physics. And he shows how a seemingly obvious assumption can be expressed as a precise mathematical symmetry.

There is also a lovely progression in the ideas:

1. We need coordinates to describe motion.
2. The origin of those coordinates is arbitrary.
3. Changing the origin does not change the equations of motion.
4. Therefore, the laws of mechanics possess translational symmetry.
5. That symmetry has a deeper connection to the conservation of momentum.

This is precisely the kind of thinking that makes theoretical physics so rewarding. A simple question about the location of an origin leads to a principle that connects classical mechanics, modern mathematical physics and our understanding of the structure of the universe.

The important lesson is that coordinates are part of our description of reality, not necessarily part of reality itself.

And that is a rather good place to begin thinking about what physics is actually trying to describe.