Feynman’s Lectures on Physics: Chapter 11-6 – Newton’s Laws in Vector Notation

From three equations to one universal law

One of the great achievements of Newtonian mechanics was to establish a mathematical framework capable of describing the motion of objects, from falling stones to orbiting planets. Yet the way we express these laws mathematically has an important bearing on how we understand their universality. In Chapter 11-6 of The Feynman Lectures on Physics, Richard Feynman brings together the ideas developed in the preceding sections on vectors, vector addition and rotations to show how Newton’s laws can be expressed in a form that is independent of our choice of coordinate system.

The central idea is deceptively simple: Newton’s laws are not fundamentally statements about motion along the x, y and z axes. They are statements about physical quantities in space. By expressing them using vectors, we can write a single law that applies equally well regardless of how we orient our coordinate system.

Acceleration as a vector

We are accustomed to thinking of acceleration as the rate at which velocity changes. However, velocity is itself a vector, possessing both magnitude and direction. It follows that acceleration must also be a vector, describing how the velocity vector changes with time.

This is an important distinction. An object can accelerate not only because it is travelling faster or slower, but also because its direction of travel is changing. Acceleration therefore encompasses two physically different kinds of change: a change in speed and a change in direction.

In Cartesian coordinates, acceleration can be described by considering the motion along each of the three perpendicular axes. Each component is determined by how the corresponding velocity component changes with time. Equivalently, each is the second rate of change of position along its respective axis.

This establishes acceleration as a genuine vector quantity. Its components transform in exactly the way required of a vector when we rotate our coordinate system. We can therefore describe acceleration without having to specify any particular orientation of the axes.

Newton’s second law in vector form

Once acceleration has been established as a vector, Newton’s second law can be expressed in a remarkably compact form. The force acting on an object is equal to its mass multiplied by its acceleration. If force and acceleration are both represented as vectors, the law becomes a single vector equation.

This is more than a convenient shorthand. It expresses something fundamental about the structure of mechanics.

Previously, we might have written three separate equations: one describing the forces and acceleration along the x-axis, another along the y-axis, and a third along the z-axis. Each equation would describe the same physical principle, but applied to a different direction in space.

Vector notation allows us to combine these three statements into one. Of course, the three equations have not disappeared. They are contained within the vector equation itself, since equality between two vectors requires equality between all their corresponding components.

The advantage is that we no longer need to choose a particular set of axes before expressing the law. The equation has the same form whatever coordinate system we use.

The invariance of Newton’s laws under rotation

This brings Feynman to an important question concerning the relationship between mathematics and physical reality. If we rotate our coordinate system, do Newton’s laws change?

The answer is no, provided that the quantities involved are correctly represented as vectors.

Consider Newton’s second law. We have established that acceleration is a vector, and we assume that force is also a vector. The mass of an object is a scalar quantity, meaning that its value is independent of the orientation of our coordinate system. Multiplying a vector by a scalar produces another vector. Consequently, the relationship between force and acceleration retains exactly the same form after any rotation of the axes.

This is an example of rotational invariance: the mathematical expression of a physical law remains unchanged when we rotate our coordinate system.

It is worth appreciating what this means physically. There is no privileged direction in ordinary space. We can choose to describe a falling object using vertical and horizontal axes, or we can rotate those axes through any angle we please. The numerical components of its position, velocity, acceleration and force will change, but the underlying physical relationship will not.

This is an important characteristic of the laws of nature. The coordinate system is something we impose upon space to make calculations easier; it is not an intrinsic feature of the physical world.

Acceleration along a curved path

Feynman now moves from the general mathematical formulation to a particularly useful application: understanding the acceleration of an object moving along a complicated curved path.

Imagine a particle travelling along an irregular trajectory. At one instant, its velocity is represented by a vector pointing in a particular direction. A short time later, its velocity has changed. It may have increased or decreased in magnitude, changed direction, or done both simultaneously.

To calculate the acceleration, we need to determine the change in velocity during the time interval. This requires subtracting the two velocity vectors.

Here Feynman makes an important geometrical point. Vector subtraction is not quite as straightforward as it might initially appear. We cannot simply draw a line between the ends of two vectors in an arbitrary diagram and assume that this represents their difference.

For vector subtraction to be meaningful, the vectors must first be represented with a common starting point, or equivalently arranged using the appropriate head-to-tail construction. The difference vector is then drawn from the tip of the original vector to the tip of the new one.

This geometrical construction allows us to see that the change in velocity has two distinct contributions.

One contribution is parallel to the direction of motion, representing a change in the magnitude of the velocity. The other is perpendicular to the direction of motion, representing a change in its direction.

These two contributions correspond to two familiar forms of acceleration.

Tangential acceleration: changing speed

The first component is tangential acceleration. It acts along the tangent to the particle’s path and describes how quickly its speed changes.

If an object is travelling along a straight road and the driver presses the accelerator, its speed increases. The acceleration is directed along the road, in the direction of motion. If the driver applies the brakes, the acceleration acts in the opposite direction.

In both cases, the direction of travel can remain unchanged while the magnitude of the velocity changes.

The important point is that tangential acceleration concerns the length of the velocity vector, rather than its orientation.

For an object travelling along a curved path, the same principle applies. At every instant, we can identify the tangent to the trajectory and consider the component of acceleration acting along that tangent. This tells us whether the particle is speeding up or slowing down.

Normal acceleration: changing direction

The second component is more subtle. An object can have acceleration even when its speed remains perfectly constant.

Consider a particle travelling around a circular track at constant speed. Its velocity vector is continuously changing direction, even though its magnitude never changes. Since velocity is a vector, this change in direction necessarily represents an acceleration.

This acceleration acts perpendicular to the instantaneous direction of motion, towards the centre of curvature of the path. It is known as normal acceleration, and in the special case of circular motion it is also called centripetal acceleration.

Feynman develops this result geometrically by considering two velocity vectors separated by a very short interval of time. If the particle’s speed remains approximately constant, the tips of these vectors lie on a circle in velocity space, because their magnitudes are equal while their directions differ.

The small change in direction of the velocity vector can therefore be related to the small distance travelled along the particle’s actual trajectory.

If the path is approximated locally by a circle, its radius of curvature provides the geometrical connection between the distance travelled and the change in direction. The particle travels a short distance along the path during the time interval, while its velocity vector rotates through a corresponding small angle.

Combining these geometrical relationships leads to the familiar result: the normal acceleration is proportional to the square of the speed and inversely proportional to the radius of curvature.

This is the same result obtained when studying uniform circular motion. The significance here is that Feynman has arrived at it using the general vector definition of acceleration, rather than treating circular motion as an entirely separate problem.

The significance of curvature

An especially interesting consequence is that the acceleration associated with changing direction depends upon the curvature of the trajectory.

A particle moving rapidly around a tight bend experiences a much greater normal acceleration than one travelling at the same speed around a gentle curve. Similarly, doubling the speed produces four times the normal acceleration if the radius of curvature remains unchanged.

This explains why a vehicle travelling around a bend must exert a substantial sideways force on its occupants. It also explains why a satellite in orbit is continually accelerating towards the body it orbits, despite potentially maintaining an almost constant orbital speed.

The same geometrical principle applies to any sufficiently smooth curved trajectory. At each instant, we can approximate the path by a circle whose radius is the local radius of curvature. This allows us to calculate the acceleration associated with the changing direction of motion, even when the overall trajectory is far more complicated than a circle.

Two components, one acceleration

The two components of acceleration can be considered independently. Tangential acceleration describes changes in speed, while normal acceleration describes changes in direction.

Together, they completely describe the instantaneous acceleration of a particle moving along a smooth curved path.

This is a powerful way of thinking about motion. Rather than treating acceleration as a single quantity that is difficult to visualise, we can decompose it into two geometrically meaningful contributions. One tells us how quickly the particle is moving faster or slower; the other tells us how sharply its direction is changing.

For a particle travelling in a straight line, there is no curvature and therefore no normal acceleration. For uniform circular motion, the speed is constant and there is no tangential acceleration. In the general case, both components may be present.

Conclusion: the power of vector thinking

Feynman’s discussion in this section illustrates two important achievements of vector notation.

First, it allows Newton’s laws to be expressed independently of the coordinate system we happen to choose. The laws of mechanics are not tied to a particular set of axes, and vector notation makes this independence explicit.

Second, it provides a natural way of understanding complicated motion. By recognising that acceleration is the rate of change of a velocity vector, we can separate changes in speed from changes in direction and understand both within a single mathematical framework.

There is a broader lesson here about the relationship between mathematics and physics. A good mathematical representation does more than make calculations shorter. It reveals the underlying structure of a physical problem. The vector formulation of Newton’s laws is a particularly elegant example: three apparently separate equations become one universal statement, while the geometry of vectors allows us to understand motion along even the most complicated paths.

What initially appears to be a change in mathematical notation is therefore a significant conceptual advance. We have moved away from describing motion in terms of its projections onto arbitrary axes and towards describing the physical relationships themselves.