Symmetry, coordinate systems and the universality of Newton’s laws
One of the most powerful ideas in physics is that the fundamental laws of nature do not depend on the perspective from which we observe them. An experiment conducted in one orientation should, under the appropriate conditions, produce the same results when the apparatus is rotated. The laws of physics should not favour one direction in space over another.
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What makes this lecture particularly interesting is not simply the mathematical result, but the methodology Feynman employs to arrive at it. He begins with an apparently obvious assumption, challenges it with an everyday example, identifies the hidden physical influences that complicate the picture, and eventually constructs a mathematical argument demonstrating the underlying symmetry.
It is a small but revealing example of how theoretical physics progresses: not by accepting intuitive statements at face value, but by examining precisely what they mean.
1. The principle of rotational symmetry
Feynman begins with a proposition: it should make no difference which direction we choose for our coordinate axes.
Imagine constructing an experiment in a laboratory. We carefully assemble the apparatus, establish its orientation and observe how it behaves. Now imagine building an identical apparatus nearby, but rotating it through some angle.
Should the two experiments produce identical results?
At first glance, the answer seems obvious. If the laws of physics are universal, why should turning an apparatus change its behaviour?
However, Feynman immediately introduces a counterexample: the grandfather clock.
A pendulum clock works perfectly well when standing upright. Tilt the entire clock, however, and the pendulum may strike the side of its case. The mechanism ceases to function as intended.
This appears to contradict rotational symmetry. Yet the example is deliberately chosen to expose a subtle distinction between the symmetry of physical laws and the symmetry of a particular physical situation.
The clock is not an isolated system. Its operation depends upon gravity, and gravity establishes a preferred direction: vertically downwards towards the Earth.
The clock’s internal mechanism has not changed, but its orientation relative to the gravitational field has.
Feynman uses this example to introduce an important methodological principle: when an experiment appears to contradict a fundamental physical principle, we should investigate whether we have overlooked something external to the system.
The Earth is part of the physical explanation.
2. The importance of the external environment
The grandfather clock provides Feynman with an opportunity to distinguish between the behaviour of an apparatus and the behaviour of the apparatus together with its environment.
If we rotate the clock but leave the Earth unchanged, we alter the orientation of the clock relative to gravity. We should therefore expect its behaviour to change.
However, if we rotate the entire experimental arrangement, including the source of the gravitational field, the original relationship between the clock and its environment is restored.
This is the essence of Feynman’s argument.
He extends the thought experiment by considering clocks in different gravitational environments.
An artificial satellite provides an interesting example. In orbit, an object and its surroundings experience free fall, producing apparent weightlessness. A conventional pendulum clock cannot operate in the familiar way because there is no effective restoring force acting on its pendulum.
On Mars, meanwhile, the gravitational acceleration is different from that on Earth, so a pendulum clock would have a different period.
These examples demonstrate that the operation of a physical system cannot always be understood by examining its internal construction alone.
The surrounding environment may be essential.
There is a deeper distinction here between two ideas:
- Rotational symmetry: the fundamental laws of physics do not depend on the absolute orientation of the experimental arrangement.
- Rotational invariance of an observed situation: the physical circumstances remain unchanged when the apparatus is rotated.
The first does not automatically guarantee the second. Gravity, magnetic fields and other external influences can introduce preferred directions into a particular experiment.
This distinction is fundamental to experimental physics. It reminds us that an apparent asymmetry in an observation does not necessarily imply an asymmetry in the underlying laws.
3. Absolute orientation versus rotation in progress
Feynman then introduces an especially interesting observation concerning the Earth.
We cannot determine the Earth’s absolute angular orientation through a simple local experiment. However, we can detect that the Earth is rotating.
This distinction is subtle but important.
Imagine performing an experiment in a laboratory. We rotate the entire laboratory through some angle and then repeat the experiment. Provided the surrounding physical conditions are appropriately transformed as well, the results should be unchanged.
But suppose we perform the experiment while the laboratory is actually rotating.
Now the situation is different.
A rotating reference frame introduces additional effects that are not present in a stationary one. These include familiar phenomena such as centrifugal and Coriolis effects.
A simple example is a spinning carousel. Someone standing on the ground sees a passenger moving in a circular path. The passenger, however, experiences apparent forces associated with the rotating frame.
The distinction is between occupying a different orientation and undergoing rotation.
An object’s absolute orientation in space does not determine the laws governing its behaviour. Its angular motion, however, can have measurable physical consequences.
This is why sufficiently delicate experiments can reveal the Earth’s rotation without revealing some supposedly absolute direction in space.
Feynman is preparing the reader for an important development in the understanding of reference frames: symmetry under a fixed rotation is not the same thing as invariance under continuous rotation.
The qualification that follows in his mathematical demonstration is therefore essential. The angle between the coordinate systems must remain constant.
4. Joe and Moe: geometry as a tool for physical reasoning
Having established the conceptual argument, Feynman returns to his recurring characters, Joe and Moe.
These two observers provide a simple device for exploring how different coordinate systems describe the same physical situation.
In the previous lecture, Feynman considered translations: Joe and Moe used coordinate systems with different origins, but their axes remained parallel.
Now the origins are identical, but Moe’s coordinate axes are rotated relative to Joe’s.
Feynman restricts the initial discussion to two dimensions, allowing the geometry to be represented in a simple diagram.
A point in space has one physical position, but its coordinates depend upon the orientation of the coordinate system used to describe it.
Feynman establishes the relationship between the two descriptions by considering the projections of the point onto the rotated axes.
The horizontal and vertical coordinates in Moe’s system are constructed from the corresponding components in Joe’s system.
The important feature of this approach is that Feynman does not simply present the transformation as an abstract mathematical rule. He derives it geometrically.
This is a recurring feature of his teaching. Mathematics is introduced as a description of an identifiable physical or geometrical relationship, rather than as an independent collection of procedures.
It also reinforces a fundamental distinction in physics: the coordinates assigned to an object are not properties of the object itself. They are properties of the relationship between the object and the observer’s chosen reference frame.
The point has not moved. Only the description has changed.
5. The remarkable similarity between coordinates and forces
Feynman now applies the same geometrical reasoning to forces.
He considers a particle of mass situated at point P, acted upon by a force whose components are known to Joe.
Moe observes the same particle and the same physical force, but resolves that force along his differently oriented axes.
The components must therefore be transformed in exactly the same geometrical way as the coordinates of the point.
This produces one of the most interesting observations in the lecture: the transformation rules for coordinates and force components have identical mathematical forms.
This is not an arbitrary coincidence.
Both position and force are vector quantities. A vector has magnitude and direction, and its components depend upon the coordinate axes used to represent it.
Consider a simple example.
Imagine a force acting on a particle horizontally towards the right. Joe, whose horizontal axis points towards the right, describes the entire force as a horizontal component.
Moe, whose axes are rotated by 45 degrees, describes the same force using both a horizontal and a vertical component.
Neither observer is wrong. They are simply resolving the same physical vector in different directions.
The force itself has not changed.
This is a particularly important conceptual step because it establishes that the mathematics of coordinate transformations is not restricted to describing position. It applies equally to other vector quantities.
Velocity, acceleration, momentum and force all have components that transform according to the same geometrical principles.
Feynman is gradually building towards a much broader understanding of physical laws: equations involving vectors can retain their form even when the coordinate systems used to express them change.
6. Testing Newton’s laws
The central mathematical argument now begins.
Feynman asks whether Newton’s laws remain valid for Moe, whose coordinate system is rotated relative to Joe’s.
This is a more demanding question than simply asking whether the two observers can describe the same point or force.
The real test is whether Newton’s second law retains its form.
Joe observes the particle and applies Newton’s second law. The mass multiplied by the particle’s acceleration in each direction is equal to the corresponding force component.
Can Moe do the same?
Feynman’s method is to calculate the two sides of the law independently and compare them.
First, he considers the acceleration.
The relationship between Joe’s and Moe’s coordinates has already been established. Since the rotation angle is constant, Feynman can differentiate these coordinate relationships with respect to time.
The resulting expressions show how the components of acceleration transform between the two coordinate systems.
He then considers the force.
Because force components transform in the same way as coordinate components, he can substitute Joe’s expressions for force using Newton’s second law.
The resulting expressions are identical to those obtained from the acceleration calculation.
This is the crucial result.
The force components measured by Moe correspond exactly to the accelerations he calculates using his own coordinate system.
Consequently, if Newton’s laws are correct for Joe, they must also be correct for Moe.
The importance of this result is not merely that Newton’s laws work in two particular coordinate systems. It is that the argument applies to any fixed rotation of the axes.
The laws of mechanics do not depend upon the arbitrary orientation chosen by the observer.
7. Why the constant-angle assumption matters
One of the most important details in Feynman’s derivation is his assumption that the angle between Joe’s and Moe’s coordinate systems remains constant.
It would be easy to overlook this qualification, but it is essential.
If Moe’s coordinate system is simply rotated relative to Joe’s, the transformation is fixed. Differentiating the coordinates does not introduce additional terms associated with the changing orientation of the axes.
However, if Moe’s coordinate system is itself rotating, the situation becomes considerably more complicated.
The axes change direction as time progresses. Consequently, the components of a vector can change even when the vector itself remains constant in the original reference frame.
This introduces additional terms into the mathematical description of velocity and acceleration.
It is the origin of the apparent forces encountered in rotating reference frames.
For example, consider a ball rolling across the floor of a rotating carousel.
An observer standing outside sees the ball follow a path influenced by its actual motion and the rotation of the platform. An observer on the carousel must account for the additional effects associated with the rotating coordinate system.
These are not violations of Newton’s laws. Rather, they reflect the additional mathematical terms required when applying Newtonian mechanics in a non-inertial reference frame.
Feynman’s distinction between changing orientation and changing angular motion therefore anticipates a much wider subject in classical mechanics.
It is also a useful reminder that the assumptions behind a mathematical derivation are just as important as the result.
8. The historical context: from Galileo to Newton
Feynman’s discussion belongs to a long historical development in the understanding of space, motion and reference frames.
The question of whether physical laws depend upon an observer’s position or orientation is closely connected to the work of Galileo Galilei.
In the early seventeenth century, Galileo developed arguments challenging the idea that the motion of the Earth should be detectable through ordinary mechanical experiments.
His famous thought experiment involving a ship is particularly relevant.
Imagine being below deck on a smoothly moving ship. Flies move around, fish swim in a tank, and objects fall to the floor. Provided the ship moves uniformly, the ordinary mechanical experiments produce the same results as they would if the ship were stationary.
The observer cannot determine the ship’s uniform motion simply by performing experiments within the enclosed cabin.
This is the principle of Galilean relativity.
Newton subsequently formalised classical mechanics in his Philosophiæ Naturalis Principia Mathematica, published in 1687.
However, Newton adopted a more complicated position concerning space and time.
He distinguished between absolute space and relative space, and between absolute and relative motion. In his conception, absolute space provided a fundamental framework against which true motion could be understood.
This created a tension that would become increasingly important in later physics.
If the laws of mechanics do not depend upon the position or orientation of an experiment, what exactly is physically meaningful about the coordinate system in which we describe it?
Feynman’s treatment helps separate the mathematical description from the physical content.
Coordinate axes are human constructions. They provide a convenient way of describing physical events, but they should not be mistaken for privileged directions built into the laws of mechanics.
The development of this idea would eventually become central to Einstein’s theories of relativity.
It is important, however, not to conflate Newtonian rotational symmetry with the full principles of Einsteinian relativity. Feynman is working within classical mechanics, where the transformations between coordinate systems are governed by Euclidean geometry and Newtonian time.
Nevertheless, the underlying question is closely related: which features of our descriptions reflect physical reality, and which are merely consequences of our choice of reference frame?
9. From rotational symmetry to the universality of physics
Feynman concludes by drawing out the consequences of the argument.
If Newton’s laws remain valid under both translations and fixed rotations of the coordinate axes, then no particular position or orientation can claim special physical significance within classical mechanics.
This has a practical consequence for experimental science.
A completely self-contained piece of equipment, containing all the mechanisms necessary to generate the forces involved in its operation, should function in the same way regardless of its orientation.
Imagine a mechanical device containing its own springs, motors, electrical supplies and control systems. If it relies on no external directional influence, rotating the device should not alter its fundamental operation.
The qualification about external influences remains important. A compass, for example, depends upon the Earth’s magnetic field. A solar panel depends upon the direction of incoming sunlight. A pendulum depends upon gravity.
These devices are not counterexamples to rotational symmetry. Their behaviour depends upon physical fields outside the apparatus.
The distinction is between the intrinsic laws governing the equipment and the environmental conditions in which it operates.
This is a powerful idea for experimental design. If a device behaves differently when rotated, we should investigate whether the difference arises from an external influence, an overlooked interaction or a genuine directional dependence in the physical system.
10. The wider significance: symmetry as a guiding principle
Although this lecture is concerned with a relatively straightforward mathematical transformation, it introduces an idea that has become one of the organising principles of modern theoretical physics.
Symmetry is not simply an aesthetic preference for elegant equations. It provides information about the physical world.
The requirement that the laws of physics remain unchanged under a transformation places constraints on the mathematical form those laws can take.
Rotational symmetry, for example, is fundamental to our understanding of angular momentum.
In classical mechanics, the conservation of angular momentum is intimately connected to the rotational symmetry of physical systems.
This connection was developed into a general mathematical framework by the German mathematician Emmy Noether, whose celebrated theorem was published in 1918.
Noether demonstrated that continuous symmetries of the action governing a physical system correspond to conservation laws.
In broad terms:
- Translational symmetry in space is associated with conservation of linear momentum.
- Rotational symmetry is associated with conservation of angular momentum.
- Translational symmetry in time is associated with conservation of energy.
This is one of the most profound connections in theoretical physics.
A property that initially appears to be about the arbitrary choice of coordinate axes turns out to be connected to quantities that can be measured and conserved in physical processes.
Feynman does not develop Noether’s theorem in this lecture, but his discussion establishes the conceptual foundations for appreciating why symmetry matters so much.
He is teaching the reader to recognise that the mathematical structure of physical laws is constrained by the transformations under which those laws remain unchanged.
Conclusion: changing our perspective without changing the physics
Lecture 11-3 is a carefully constructed demonstration of an important principle: the laws of classical mechanics do not depend upon the fixed orientation of the coordinate system used to describe them.
Feynman’s methodology is particularly effective because he moves from intuition to complication, and then from geometry to mathematical proof.
The grandfather clock introduces the problem of external influences. The discussion of the Earth distinguishes absolute orientation from actual rotation. Joe and Moe provide a geometrical framework for transforming coordinates and forces. Finally, the comparison of the two sides of Newton’s second law establishes the mathematical result.
The most important lesson is perhaps that physics must distinguish between the world itself and the descriptions we construct of it.
A particle does not know which way our coordinate axes point. A force does not change simply because we choose to resolve it in different directions. The physical relationships remain the same, even when our mathematical descriptions change.
Feynman is therefore doing more than demonstrating a property of Newton’s laws. He is introducing a way of thinking about physics in which symmetry becomes a test of our understanding.
The challenge is not simply to write down an equation that works in one particular arrangement. It is to understand why the same law should work in every equivalent arrangement, and to identify the circumstances under which that equivalence breaks down.
That is the deeper significance of rotational symmetry: the universe does not need to know how we have chosen to look at it.
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