Force-free Newtonian mechanics
Contents
- The Question of Whether Force Is Fundamental
- Momentum as the Primitive Dynamical Quantity
- Recovering Newton's Laws
- Work Without Force
- Energy as a Consequence of Momentum Transfer
- Potential Energy and Conservative Interactions
- Collisions, Circular Motion, and the Centre of Mass
- Angular Momentum and Rotational Mechanics
- What Becomes of Force?
- How Much of Newtonian Mechanics Survives?
- What the Reformulation Does and Does Not Establish
- From a Change of Vocabulary to a Different Framework
Part 1 - The Question of Whether Force Is Fundamental
Newtonian mechanics is ordinarily introduced through the concept of force. A body moves because forces act upon it, its acceleration is proportional to the applied force, and interactions between bodies are expressed through equal and opposite forces. This vocabulary is so deeply embedded in elementary mechanics that it is easy to mistake it for a statement about what nature fundamentally contains rather than what happens to be a particularly successful way of organizing the theory.
There is, however, an alternative starting point available inside Newtonian physics itself. Momentum is already a central dynamical quantity and a conserved quantity, while Newton's second law can be written in the general form
This equation admits an interpretation that is conceptually different from the usual force-centered formulation. Instead of regarding force as a primitive entity that subsequently causes a change in momentum, one can regard momentum transfer as the physical event and force as the rate at which that transfer occurs. The mathematics is unchanged. What changes is the conceptual order in which the quantities are introduced.
That distinction matters because a change in the momentum of one body need not be the same thing as a change in the total momentum of an interacting system. During an interaction, momentum can be transferred from one body to another even though the total momentum of the isolated system remains unchanged. Thus the central concept here is not simply "net momentum change." It is momentum transfer during interaction.
The question, therefore, is not whether Newtonian mechanics can be written without the symbol \(F\). Obviously it can. The more interesting question is whether force can be removed as a primitive concept altogether, while enough of Newtonian mechanics can still be reconstructed from principles concerning momentum transfer, motion, and energy. If this can be done, force becomes a derived rate description rather than a primitive ingredient of the formulation.
This is a claim about the structure of a formulation, not yet a claim about metaphysical reality. Showing that momentum transfer can be taken as primitive does not by itself prove that momentum is ontologically more fundamental than force. It shows something more precise: force is not required as a primitive variable in order to reconstruct a substantial portion of Newtonian dynamics.
This possibility becomes especially interesting when considering work. The conventional expression for mechanical work is
But if force is the rate of momentum transfer, then the same expression can be rewritten entirely in terms of momentum:
This suggests a different conceptual hierarchy. An interaction transfers momentum. The recipient moves with some velocity while that transfer occurs. The energetic consequence of the interaction is then determined by the component of the momentum transfer along that motion. Force is not necessary to express this relationship.
Part 2 - Momentum as the Primitive Dynamical Quantity
A force-centered formulation begins with a statement such as "a force acts on a body." A momentum-centered formulation can instead begin with the more concrete statement that an interaction transfers momentum between physical systems. For an interaction between two systems \(A\) and \(B\), the elementary relation is
The equation says that momentum transferred to one participant is accompanied by an equal and opposite transfer from the other participant. The interaction therefore redistributes momentum rather than creating a net amount of momentum within the isolated pair.
This distinction is essential. The individual momenta may change substantially:
Yet the total momentum can remain unchanged:
Thus "momentum change" and "momentum transfer" should not be treated as synonyms. A body can undergo a momentum change because it receives momentum from another body, while the total momentum of the complete isolated system does not change at all. The primitive event proposed here is the transfer occurring between the participants.
For an isolated collection of systems, the total momentum therefore satisfies
This provides an economical description of interaction. Rather than introducing an independent entity called force and then asking what effect it produces, one begins by describing the dynamical exchange itself: momentum is transferred between systems.
For a body of constant mass,
Consequently, a change in momentum is a change in the body's velocity. No separate concept of force is required to state this. The interaction changes the momentum, and the momentum determines the state of motion.
We may then introduce a useful derived quantity,
where \(\mathbf{q}\) represents the rate of momentum transfer to the body. If we choose to give this quantity the conventional name "force," then
Nothing new has been added physically by this definition. We have simply assigned a compact name to the rate at which momentum is transferred. The conventional force formulation and the momentum-centered formulation are therefore mathematically equivalent when related in this way. They differ in explanatory priority: one takes force as the starting dynamical quantity, while the other takes momentum transfer as the starting dynamical event.
Part 3 - Recovering Newton's Laws
The first test of any force-free reformulation is whether it can reproduce Newton's three laws without quietly reintroducing force under another name. The result is surprisingly straightforward.
Newton's First Law
Suppose no momentum is transferred to a body. Then
Therefore the body's momentum remains constant. For constant mass this means that its velocity remains constant:
Its position consequently evolves as
Newton's first law is thus the special case in which no interaction transfers momentum to the body. The familiar statement that an object remains in uniform motion unless acted upon can be replaced by the more direct statement that, in the absence of momentum transfer, momentum remains unchanged.
Newton's Second Law
For constant mass,
If the rate of momentum transfer is denoted by \(\mathbf{q}\), then
Calling \(\mathbf{q}\) force immediately recovers the familiar Newtonian equation
The important point is that the force concept has not been required to obtain the dynamics. It has appeared only after the dynamics have already been expressed in terms of momentum transfer.
Newton's Third Law
The third law becomes especially transparent when expressed as momentum exchange. If an interaction transfers momentum from \(B\) to \(A\), then the corresponding change of momentum of the other participant is equal and opposite:
Dividing by time gives
If these rates are called forces, this becomes Newton's familiar action and reaction relation. But in the momentum-centered formulation, the more primitive statement is that the interaction redistributes momentum between its participants.
This also reveals an important conceptual point about the third law. In an appropriate isolated interaction, equal and opposite momentum transfers express conservation of total momentum. The familiar paired-force language is therefore one compact representation of a deeper exchange relation.
Part 4 - Work Without Force
The most revealing test is work, because work is usually defined explicitly in terms of force and displacement. If force can genuinely be removed from the framework, work must be expressible without it.
The displacement of a moving body is
Meanwhile, the momentum transferred during the same interval is
The mechanical energy transferred by the interaction can therefore be written as
This equation is mathematically equivalent to the conventional work expression, but it gives the quantity a different conceptual interpretation. Work is the energetic aspect of momentum transfer occurring while the recipient has a particular velocity. Momentum transfer is the broader dynamical event; work describes the portion of that event that transfers mechanical energy.
The dot product is significant. Only momentum transfer with a component in the direction of motion contributes directly to mechanical work. Momentum can be transferred perpendicular to the velocity without changing the kinetic energy.
This makes circular motion particularly instructive. During uniform circular motion, the momentum vector continually changes direction, so \(d\mathbf{p}\neq0\), yet the momentum transfer is perpendicular to the velocity. Consequently,
The interaction is therefore dynamically real without producing mechanical work. The momentum-centered description makes the distinction immediate: momentum is being transferred and redirected, but no component of that transfer contributes to an increase in kinetic energy.
This example also demonstrates why the phrase "momentum change" is insufficient as the central concept. The body has a changing momentum, but the isolated system can have no net momentum change. What matters dynamically is the transfer taking place between the interacting systems.
Part 5 - Energy as a Consequence of Momentum Transfer
For a particle of constant mass,
Substituting into the momentum-based expression for work gives
Since
we obtain
Integration therefore gives
The conventional kinetic energy theorem has been recovered without treating force as fundamental. Kinetic energy itself can consequently be regarded as a derived quantity associated with the momentum state of a moving body:
This also clarifies why momentum transfer and energy transfer cannot simply be identified. The same quantity of momentum transferred at different velocities can correspond to different quantities of energy transfer. In differential form,
Momentum transfer is therefore the dynamical event, while work measures its energetic consequence. They are related, but they are not the same physical quantity.
This distinction is important to the overall proposal. The aim is not to replace work with momentum or to claim that energy is reducible to momentum. The claim is that the relationship between momentum transfer and mechanical energy can be expressed directly, without force occupying the conceptual middle position.
Part 6 - Potential Energy and Conservative Interactions
The same framework can accommodate potential energy without introducing force. Suppose an interaction transfers momentum in a way that depends on the configuration of a system. If the mechanical work associated with that interaction depends only on the initial and final configurations, the interaction is conservative.
We can then define a potential energy \(V\) by
Since the work itself is already expressed as
potential energy represents stored capacity for future energy transfer through interaction. The familiar conservation law follows:
Consider a spring. Instead of saying that a spring exerts a force \(\mathbf{F}=-k\mathbf{x}\), the momentum-centered description can say that the spring transfers momentum toward its equilibrium configuration at a rate proportional to the displacement:
Since \(d\mathbf{x}=\mathbf{v}\,dt\), the associated energy transfer is
Integration gives
and therefore the stored potential energy is
Nothing essential has been lost. The spring still produces exactly the same motion and energy transformations. What has disappeared is the assumption that the force description must be the starting point.
Part 7 - Collisions, Circular Motion, and the Centre of Mass
Collisions are perhaps the simplest example of why momentum is a natural primitive. Consider two bodies that interact over a short period. The interaction transfers momentum according to
Consequently,
The entire collision can be described without knowing or even defining the instantaneous interaction force. This is particularly valuable because the details of the interaction may be extremely complicated while the total momentum transfer remains simple.
The distinction between transfer and net change is especially clear here. During the collision, each body undergoes a momentum change, while the isolated two-body system undergoes no net momentum change. The interaction is therefore best described as an internal redistribution of momentum.
The same perspective extends naturally to many-particle systems. Define the total momentum as
Momentum transferred internally between members of the system cancels in pairs. Consequently, only momentum transferred across the boundary of the system can change its total momentum. For an isolated system,
For total mass \(M\) and centre of mass velocity \(\mathbf{V}_{\rm CM}\),
Thus
for an isolated system. Centre of mass motion therefore follows naturally from the conservation and transfer of momentum.
Circular motion provides the complementary case. A body can continuously receive momentum while its kinetic energy remains unchanged. If the transferred momentum is perpendicular to its velocity,
The interaction therefore redirects the body's motion rather than increasing its speed. What conventional mechanics calls centripetal force becomes the rate of momentum redirection. The distinction between changing the direction of momentum and changing its magnitude becomes explicit.
Part 8 - Angular Momentum and Rotational Mechanics
A force-free formulation should also reproduce rotational mechanics. Angular momentum provides a natural extension because it is itself constructed from position and momentum:
Taking the differential gives
Because \(d\mathbf{r}=\mathbf{v}\,dt\) and momentum is parallel to velocity for a particle of constant mass,
Therefore,
The rate of angular momentum transfer is consequently
Conventionally, the right hand side is called torque:
But the momentum-centered formulation gives a more primitive interpretation: torque is the rate of angular momentum transfer. The same conceptual move that demotes force to the rate of linear momentum transfer demotes torque to the rate of angular momentum transfer.
For rigid bodies, translational and rotational dynamics can therefore be expressed in terms of the transfer of linear and angular momentum. The conventional force and torque equations remain available as compact shorthand, but neither needs to be regarded as primitive within this formulation.
Part 9 - What Becomes of Force?
At this point force has not disappeared from physics. Something more subtle has happened. Its status within the formulation has changed.
Force can be defined as
Under this definition, force is the rate of momentum transfer. It remains an extremely useful quantity because it compresses a temporal process into a single vector. Saying that an interaction transfers momentum at a rate of \(10\) newtons is much more concise than repeatedly describing the corresponding change in momentum per unit time.
This explains why force has remained so successful in practical mechanics. The argument is not that the force concept is mathematically defective or physically useless. It is that its usefulness does not establish its fundamentality.
There is an important distinction between mathematical convenience and explanatory priority. If two formulations generate exactly the same trajectories and conservation laws, neither formulation has thereby been proved to describe a different physical reality. The present proposal is instead that one formulation may place the dynamical concepts in a more direct explanatory order.
Force can therefore be viewed as a highly efficient representation of how quickly momentum is being transferred. If the goal is compact calculation, force is difficult to beat. If the goal is identifying the underlying dynamical event, momentum transfer may be the more primitive starting point.
Part 10 - How Much of Newtonian Mechanics Survives?
The result of the reconstruction is significant. Beginning with momentum and momentum transfer, together with ordinary kinematics, we have recovered the essential structure of a large portion of Newtonian mechanics.
The framework gives uniform inertial motion whenever momentum is not transferred. It gives acceleration when momentum changes. It gives Newton's third law as momentum conservation within an interaction. It gives impulse as the quantity of momentum transferred. It gives collisions and centre of mass motion through total momentum conservation. It gives kinetic energy through the relationship between momentum and velocity. It gives work through \(\mathbf{v}\cdot d\mathbf{p}\). It gives potential energy for conservative interactions. It gives circular motion through momentum redirection. It gives angular momentum and rotational dynamics through the transfer of momentum at a lever arm.
What remains to be supplied is not primarily the concept of force. What remains are the interaction laws: rules specifying how momentum is transferred under particular physical circumstances. Gravity, elasticity, electromagnetic interaction, friction, and other phenomena require additional laws describing their particular momentum-transfer behaviour.
This is an important distinction. Newtonian mechanics by itself never explains why every conceivable interaction has the particular form it does. It provides a general dynamical structure within which interaction laws can be inserted. The momentum-centered formulation changes the form in which that structure is expressed, but it does not eliminate the need for empirical laws governing particular interactions.
The reconstruction therefore establishes something substantial without establishing everything. A large part of Newtonian mechanics can be formulated without taking force as primitive. It does not follow merely from this that momentum is metaphysically fundamental, or that every possible mechanical theory can be reduced to momentum transfer alone.
Part 11 - What the Reformulation Does and Does Not Establish
The strongest version of the proposal should be stated carefully. It does not claim that force does not exist. Nor does it claim that force is merely an arbitrary word. Force is a perfectly well-defined and experimentally useful quantity. The claim is instead that nothing in the dynamical structure of Newtonian mechanics requires force to be taken as the primitive dynamical concept.
Likewise, the proposal does not claim that momentum change and momentum transfer are identical. A single body can undergo a momentum change because it receives momentum from another system, while the complete isolated system undergoes no net momentum change. The distinction is essential:
There can therefore be substantial momentum transfer even when the total momentum of the isolated system remains constant. The proposed primitive is the interaction-level transfer, not merely the initial-to-final difference of a chosen system's total momentum.
The proposal also does not identify momentum with work or energy. Momentum, momentum transfer, work, and energy remain distinct physical quantities. What the reformulation establishes is a direct relation among them:
This relation shows that work can be described as the energetic consequence of momentum transfer occurring through motion. It does not turn momentum into energy, nor does it make every momentum transfer a transfer of mechanical work.
There is also a broader methodological point. A successful reformulation does not automatically prove that its chosen primitive is ontologically superior. It shows that the theory can be organized coherently around that primitive. Whether this organization is more explanatory, more fundamental, or merely more illuminating is a further philosophical and physical question.
The modest claim is therefore already sufficient to be interesting: force need not be assumed as a primitive in order to reconstruct a substantial Newtonian framework. Momentum transfer can serve as the primitive dynamical concept, while force can be recovered as its rate description.
Part 12 - From a Change of Vocabulary to a Different Framework
It would be easy to dismiss this proposal as a change of terminology. After all, if we define
then replacing every occurrence of \(d\mathbf{p}/dt\) with \(F\) restores ordinary Newtonian mechanics. In that narrow mathematical sense, nothing new has been created.
But mathematical equivalence does not make the conceptual organization irrelevant. Two mathematically equivalent formulations can differ in which quantities they treat as primitive, which relations they regard as fundamental, and which phenomena they present as consequences of more basic ones.
The force-centered picture tends to make interaction appear as the action of a quantity called force, with momentum change appearing as its consequence. The momentum-centered picture reverses that explanatory order. Interaction is described as momentum transfer. Momentum change is the resulting change in the state of an individual system. Force is the compact rate description of that transfer.
The same transfer also has an energetic aspect:
Mathematically, that relationship is captured by
This equation may be the most revealing feature of the entire reformulation. It places momentum and energy within a single description of interaction. Momentum transfer describes the dynamical exchange itself, while its component along the velocity determines the associated mechanical energy transfer.
The resulting hierarchy is compact:
This does not make conventional Newtonian mechanics wrong. On the contrary, it shows how much of Newtonian mechanics can be recovered without taking force as primitive. The familiar equations remain intact, but their conceptual order has been rearranged.
The ultimate test is therefore not whether this framework can reproduce \(F=ma\). It obviously can, because \(F\) can be recovered as the rate of momentum transfer. The deeper test is whether the remaining domains of Newtonian mechanics can be reconstructed without force quietly returning through the back door. Constrained systems, friction, variable mass, non-inertial reference frames, multi-body interactions, and continuum mechanics provide progressively stronger tests.
These tests also clarify what would count as genuine progress. If every apparently force-based construction can be reformulated directly in terms of momentum transfer, then force has been shown to be unnecessary as a primitive variable through a broad range of mechanics. If some domain requires an additional structure that cannot naturally be expressed in this language, that limitation would identify a genuine boundary of the reformulation rather than merely a failure of terminology.
If the framework survives those tests, the conclusion would be considerably more interesting than the claim that "force can be renamed." It would show that a large part of Newtonian mechanics can be organized around a different explanatory priority: interactions redistribute momentum, force describes the rate of that redistribution, and work describes the energetic consequence of momentum transfer occurring through motion.
The resulting perspective is therefore not that force is meaningless, nor that momentum and work are the same quantity. It is that the direction of explanation can be reversed. Instead of beginning with force and deriving momentum change, one can begin with momentum transfer and derive force. Instead of beginning with force and displacement to define work, one can begin with momentum transfer and velocity:
The central proposal can therefore be stated in its strongest defensible form: momentum transfer can be taken as the primitive dynamical concept in an alternative formulation of Newtonian mechanics. Force is then the rate description of that transfer, while work is the energetic consequence of the transfer occurring through motion.
The conceptual hierarchy is consequently:
and, for the energetic aspect of the same process,
Force has therefore not been destroyed. It has been placed in a different conceptual position: not as the thing that must be assumed in order for dynamics to exist, but as a remarkably useful shorthand for describing how rapidly momentum is exchanged.
