2026-09-01
Circular motion as a projection of uniform motions of higher splittings
Let's imagine a point moving uniformly along a circle. The magnitude of its velocity does not change, but the direction of its velocity continuously rotates. Therefore, even with a completely constant speed, the motion remains accelerated.
If the point is connected to the center, for example, by a thread, then in a stationary frame of reference, the connection continuously deflects it from a straight-line path. This creates a centripetal acceleration directed toward the center of the circle. An observer rotating with the point describes the same state differently: relative to him, the point is stationary, and the acceleration directed toward the center is opposed by a centrifugal inertial term.
Both accelerations have the same magnitude and opposite directions:
\[\tag{1} \mathbf A_{\mathrm{cs}} =-\frac{v^2}{R}\mathbf e_R, \qquad \mathbf A_{\mathrm{cb}} =+\frac{v^2}{R}\mathbf e_R. \] Centripetal acceleration is the acceleration of a point in a stationary frame of reference. Centrifugal acceleration arises as an inertial term in a rotating frame of reference. Therefore, they cannot be considered two simultaneously acting accelerations of the same inertial system. However, they are two opposing manifestations of a single geometry of circular motion.
Here a deeper question arises. Is it possible to represent rotation itself so that at some higher level there is neither centripetal nor centrifugal acceleration? Is it possible to construct a circle solely from uniform rectilinear motions, each of which satisfies the condition
\[\tag{2} q_n(t)=q_{n0}+u_nt, \qquad \boxed{\ddot q_n(t)=0}, \] and obtain the radial acceleration only after their joint projection?
This article will show that uniform circular motion admits an exact representation as the multiplicative projection of an infinite set of uniform linear motions. Neither initial branch has acceleration, but their common projection has centripetal acceleration, and in a rotating system, centrifugal acceleration.
This is a specific two-dimensional example of the general idea discussed in the previous part. There, an arbitrary change in the velocity magnitude was represented through uniform higher phases. Here, we will consider a different mechanism: the circular motion itself will be assembled by a product of linear branches.
1. Initial Motion of a Point along a Circle
Let the plane of motion be defined by two orthogonal directions \(\ep\) and \(\em\), and the radius of the circle be \(R\). We write the position of the point as follows
\[\tag{3} \boxed{ \mathbf r(t) =R\ep\sin(\omega t) +R\em\cos(\omega t). } \] Here \(\omega\) is the constant angular velocity. The observed coordinates of the point are
\[\tag{4} x(t)=R\sin(\omega t), \qquad y(t)=R\cos(\omega t). \] The Euclidean distance to the center remains constant:
\[\tag{5} \sqrt{x^2(t)+y^2(t)} =R. \] The speed of the point is
\[\tag{6} \dot{\mathbf r}(t) =R\omega\ep\cos(\omega t) -R\omega\em\sin(\omega t), \] therefore its module does not change:
\[\tag{7} \boxed{ v=|\dot{\mathbf r}| =R|\omega| =\operatorname{const}. } \] Repeated differentiation gives
\[\tag{8} \ddot{\mathbf r}(t) =-R\omega^2\ep\sin(\omega t) -R\omega^2\em\cos(\omega t), \] or
\[\tag{9} \boxed{ \ddot{\mathbf r}(t) =-\omega^2\mathbf r(t). } \] Therefore, in a stationary frame of reference, the acceleration is directed toward the center, and its magnitude is equal to
\[\tag{10} \boxed{ |\mathbf A_{\mathrm{cs}}| =\omega^2R =\frac{v^2}{R}. } \] In a rotating frame of reference, the opposite inertial term appears:
\[\tag{11} \boxed{ \mathbf A_{\mathrm{cb}} =+\omega^2\mathbf r, \qquad \mathbf A_{\mathrm{cs}}+\mathbf A_{\mathrm{cb}}=0. } \] 2. Why a Simple Sum of Linear Motions Is Not Enough
Let's try to obtain periodic coordinates from a finite set of uniform linear motions. Let
\[\tag{12} q_n(t)=q_{n0}+u_nt, \qquad \ddot q_n=0. \] Their linear sum is
\[\tag{13} \sum_{n=1}^{N}a_nq_n(t) =Q_0+Ut. \] This is again a linear function of time and cannot coincide with sine or cosine. Therefore, the usual sum of uniform motions does not create a circle.
Now consider the product:
\[\tag{14} P_N(t)=\prod_{n=1}^{N}q_n(t). \] Even if each factor has a zero second derivative, the product is already a nonlinear function. For a finite number of branches
\[\tag{15} \begin{aligned} \ddot P_N(t) ={}& \sum_{k=1}^{N}\ddot q_k \prod_{m\ne k}q_m \\[1mm] &+ 2\sum_{1\le k < l\le N} \dot q_k\dot q_l \prod_{m\ne k,l}q_m. \end{aligned} \] Since \(\ddot q_k=0\), only the cross products of constant velocities remain:
\[\tag{16} \boxed{ \ddot P_N(t) = 2\sum_{1\le k < l\le N} \dot q_k\dot q_l \prod_{m\ne k,l}q_m. } \] A multiplicative projection can accelerate without accelerating any of the original motion. Nonlinearity arises when branches are assembled together.
3. Infinite products for sine and cosine
Introducing a dimensionless phase
\[\tag{17} z=\omega t. \] The functions that define the coordinates of a circle have classical Euler products:
\[\tag{18} \boxed{ \sin z =z\prod_{n=1}^{\infty} \left(1-\frac{z^2}{n^2\pi^2}\right), } \] \[\tag{19} \boxed{ \cos z =\prod_{n=0}^{\infty} \left( 1-\frac{4z^2}{(2n+1)^2\pi^2} \right). } \] Every quadratic factor is the product of two linear functions:
\[\tag{20} 1-\frac{z^2}{n^2\pi^2} = \left(1+\frac{z}{n\pi}\right) \left(1-\frac{z}{n\pi}\right), \] \[\tag{21} 1-\frac{4z^2}{(2n+1)^2\pi^2} = \left(1+\frac{2z}{(2n+1)\pi}\right) \left(1-\frac{2z}{(2n+1)\pi}\right). \] Thus, sine and cosine are already represented as infinite products of pairs of linear branches.
4. Linear branches of the sine coordinate
For each level \(n=1,2,3,\ldots\), we introduce two branches:
\[\tag{22} s_{n,+}(t) =1+\frac{\omega t}{n\pi}, \qquad s_{n,-}(t) =1-\frac{\omega t}{n\pi}. \] Their velocities are constant and opposite:
\[\tag{23} \dot s_{n,+} =+\frac{\omega}{n\pi}, \qquad \dot s_{n,-} =-\frac{\omega}{n\pi}. \] Therefore, there is no acceleration in either branch:
\[\tag{24} \boxed{ \ddot s_{n,+} =\ddot s_{n,-} =0. } \] Their pair product is
\[\tag{25} s_{n,+}(t)s_{n,-}(t) =1-\frac{\omega^2t^2}{n^2\pi^2}. \] The additional factor \(s_0(t)=\omega t\), which appears before the product for the sine, also describes linear uniform motion:
\[\tag{26} s_0(t)=\omega t, \qquad \dot s_0=\omega, \qquad \ddot s_0=0. \] 5. Linear branches of the cosine coordinate
For the cosine coordinate with \(n=0,1,2,\ldots\), we define
\[\tag{27} c_{n,+}(t) =1+\frac{2\omega t}{(2n+1)\pi}, \qquad c_{n,-}(t) =1-\frac{2\omega t}{(2n+1)\pi}. \] Here also
\[\tag{28} \dot c_{n,\pm} =\pm\frac{2\omega}{(2n+1)\pi} =\operatorname{const}, \qquad \boxed{\ddot c_{n,\pm}=0}. \] The pair product is of the form
\[\tag{29} c_{n,+}(t)c_{n,-}(t) =1-\frac{4\omega^2t^2}{(2n+1)^2\pi^2}. \] Thus, each coordinate of the circle is assembled from pairs of branches diverging uniformly in opposite directions.
Note: the motions in all higher splittings have no acceleration. This only occurs after they are combined by a multiplicative projection.
6. Idempotent Notation of a Single Splitting
Let's represent each pair as a separate idempotent splitting. For the sine level \(n\), we introduce an independent hyperbolic unit.
\[\tag{30} \j_{s,n}^{,2}=1 \] and the corresponding idempotents.
\[\tag{31} p_{s,n}^{+} =\frac{1+\j_{s,n}}{2}, \qquad p_{s,n}^{-} =\frac{1-\j_{s,n}}{2}. \] They satisfy the rules
\[\tag{32} (p_{s,n}^{\pm})^2=p_{s,n}^{\pm}, \qquad p_{s,n}^{+}p_{s,n}^{-}=0, \qquad p_{s,n}^{+}+p_{s,n}^{-}=1. \] We write the state of one sine splitting as
\[\tag{33} \begin{aligned} S_n(t) &=p_{s,n}^{+}s_{n,+}(t) +p_{s,n}^{-}s_{n,-}(t)\\ &=1+\j_{s,n}\frac{\omega t}{n\pi}. \end{aligned} \] This state is linear in time:
\[\tag{34} \dot S_n(t) =\j_{s,n}\frac{\omega}{n\pi} =\operatorname{const}, \qquad \boxed{\ddot S_n(t)=0}. \] We define the conjugation \(\star_n\), which changes the sign of only the unit of a given level:
\[\tag{35} \j_{s,n}^{\star_n}=-\j_{s,n}, \qquad S_n^{\star_n}(t) =1-\j_{s,n}\frac{\omega t}{n\pi}. \] The split norm of this level is
\[\tag{36} \boxed{ \mathcal N_{s,n}(t) =S_n(t)S_n^{\star_n}(t) =1-\frac{\omega^2t^2}{n^2\pi^2}. } \] Similarly, for each cosine level, an independent unit \(\j_{c,n}\) and state are introduced.
\[\tag{37} C_n(t) =1+\j_{c,n} \frac{2\omega t}{(2n+1)\pi}. \] For it.
\[\tag{38} \boxed{ \ddot C_n(t)=0, \qquad \mathcal N_{c,n}(t) =C_n(t)C_n^{\star_n}(t) =1-\frac{4\omega^2t^2}{(2n+1)^2\pi^2}. } \] Conjugation and multiplication are performed within each individual splitations. The resulting split norm is a scalar factor, after which factors of different levels can be multiplied together.
7. The Law of Multiplicative Projection
We define the sine projection as the product of the split norms of all sine levels:
\[\tag{39} \Pi_s(t) =\omega t \prod_{n=1}^{\infty} \mathcal N_{s,n}(t). \] For the cosine coordinate, we set
\[\tag{40} \Pi_c(t) =\prod_{n=0}^{\infty} \mathcal N_{c,n}(t). \] According to Euler's products, these projections are exactly equal
\[\tag{41} \boxed{ \Pi_s(t)=\sin(\omega t), \qquad \Pi_c(t)=\cos(\omega t). } \] Therefore, the full multiplicative projection is of the form
\[\tag{42} \boxed{ \begin{aligned} \mathbf r(t)={}& R\ep\,\omega t \prod_{n=1}^{\infty} S_n(t)S_n^{\star_n}(t) \\[1mm] &+ R\em \prod_{n=0}^{\infty} C_n(t)C_n^{\star_n}(t). \end{aligned} } \] After performing the products, the original motion is restored:
\[\tag{43} \boxed{ \mathbf r(t) =R\ep\sin(\omega t) +R\em\cos(\omega t). } \] The exact diagram is obtained:
\[\tag{44} \boxed{ \begin{gathered} \ddot S_n=0, \qquad \ddot C_n=0 \qquad \text{for all }n, \\[1mm] \Downarrow\quad \text{multiplicative projection} \quad\Downarrow \\[1mm] \ddot{\mathbf r} =-\omega^2\mathbf r\ne0. \end{gathered} } \] 8. Where does radial acceleration come from?
Let's consider the simplest sine wave pair:
\[\tag{45} s_{n,+}(t)s_{n,-}(t) =1-\frac{\omega^2t^2}{n^2\pi^2}. \] Both branches have zero acceleration, but their product satisfies the equality.
\[\tag{46} \frac{d^2}{dt^2} \left[s_{n,+}(t)s_{n,-}(t)\right] =-\frac{2\omega^2}{n^2\pi^2}\ne0. \] This acceleration arises from the product of two constant velocities:
\[\tag{47} \frac{d^2}{dt^2} \left(s_{n,+}s_{n,-}\right) =2\dot s_{n,+}\dot s_{n,-}. \] After combining an infinite number of levels, the cross terms add up so that the observed projection satisfies the equation of circular motion:
\[\tag{48} \boxed{ \ddot{\mathbf r} =-\omega^2\mathbf r. } \] The minus sign indicates the direction toward the center of the circle. In a rotating frame of reference, the same modulus corresponds to the opposite inertial term:
\[\tag{49} \boxed{ \mathbf A_{\mathrm{cs}} =-\omega^2\mathbf r, \qquad \mathbf A_{\mathrm{cb}} =+\omega^2\mathbf r. } \] Centripetal and centrifugal accelerations are absent in individual higher branches. They arise only after assembling these branches into a single circular projection and selecting a reference frame.
9. Finite number of splittings
For a finite number of levels, we define partial products
\[\tag{50} \sin_N z =z\prod_{n=1}^{N} \left(1-\frac{z^2}{n^2\pi^2}\right), \] \[\tag{51} \cos_N z =\prod_{n=0}^{N} \left( 1-\frac{4z^2}{(2n+1)^2\pi^2} \right). \] The corresponding final-level motion is
\[\tag{52} \mathbf r_N(t) =R\ep\sin_N(\omega t) +R\em\cos_N(\omega t). \] Each original branch still moves uniformly, but the final products are polynomials. Therefore, for finite \(N\), the trajectory only approximates the circle and, in general, does not exactly preserve the condition \(|\mathbf r_N|=R\).
On any limited time interval \(|t|\le T\), the products and their derivatives converge to sine and cosine:
\[\tag{53} \begin{gathered} \sin_N(\omega t)\longrightarrow\sin(\omega t), \qquad \cos_N(\omega t)\longrightarrow\cos(\omega t), \\[1mm] \dot{\mathbf r}_N(t)\longrightarrow\dot{\mathbf r}(t), \qquad \ddot{\mathbf r}_N(t)\longrightarrow\ddot{\mathbf r}(t) \end{gathered} \] uniformly in \(t\) on this interval. Consequently, as the number of splittings increases, the trajectory, velocity, and centripetal acceleration are simultaneously refined.
\[\tag{54} \boxed{ N\to\infty \quad\Longrightarrow\quad \mathbf r_N(t)\to\mathbf r(t), \qquad \ddot{\mathbf r}_N(t) \to-\omega^2\mathbf r(t). } \] Exact periodic motion along the entire time axis is obtained only in the limit of an infinite hierarchy. This limitation is fundamental: the finite product of linear functions is a polynomial and cannot coincide with the sine or cosine for all time values.
10. Geometric Meaning of the Construction
Each level contains a pair of mutually opposite linear motions. Their velocities are of the form
\[\tag{55} u_{n,+}=-u_{n,-}, \qquad \dot u_{n,\pm}=0. \] Individually, these motions do not rotate or deviate from a straight line. The circle is absent in any single channel. It appears onlyo after successive multiplication of the split norms of all levels.
Therefore, the observed rotation cannot be identified with any of the higher branches. It is an integral property of their projection:
\[\tag{56} \boxed{ \text{circle} \ne \text{separate higher motion}, \qquad \text{circle} = \text{multiplicative projection of all branches}. } \] In the usual description, the curvature of the trajectory means a continuous change in the direction of the velocity and leads to radial acceleration. In the higher representation, none of the velocities change. Nonlinearity is transferred from the law of motion of individual branches to the law of their projection.
\[\tag{57} \boxed{ \begin{gathered} \text{lower level:} \quad \text{accelerated curvilinear motion}, \\[1mm] \text{higher level:} \quad \text{a set of uniform rectilinear motions}. \end{gathered} } \] 11. Mathematical Result and Physical Interpretation
The above construction is an accurate mathematical representation of circular motion. It proves the possibility of obtaining the coordinates, velocity, and radial acceleration of a point from the multiplicative projection of acceleration-free linear branches.
However, the existence of the expansion does not mean that the real centripetal interaction has already been explained. The formulas do not yet provide a ready-made method for rotating a body without force, eliminating G-forces, or creating motion without an engine. To do this, it is first necessary to prove the physical existence of the splittings themselves and the possibility of controlling their projection.
\[\tag{58} \begin{gathered} \text{what real degrees of freedom correspond to the branches;}\\ \text{why is the multiplicative projection valid;}\\ \text{what physical law determines the quantities }R\text{ and }\omega;\\ \text{is the proper acceleration of a physical body conserved?} \end{gathered} \] The construction under consideration reproduces an already given relationship
\[\tag{59} |\mathbf A|=\omega^2R=\frac{v^2}{R}, \] but in itself It still doesn't explain what interaction holds a particular body on a circle. Its immediate result is different: it provides a constructive way to lift a given circular motion to a coordinate system with zero second derivatives.
The mathematical conclusion is the possibility of a non-accelerating higher-order representation. The physical hypothesis is the assumption that such splittings are indeed the hidden geometry of the observed motion.
12. How to Use the Result in Practical Use
The result has two different levels of practical application: the mathematical algorithm already available and the possible future physical application. They must be strictly distinguished.
12.1. Constructive Algorithm
Let the radius \(R\), angular velocity \(\omega\), working time \(|t|\le T\), permissible coordinate error \(\varepsilon_r^{\max}\), and permissible acceleration error \(\varepsilon_A^{\max}\) be given. Then the circular motion can be approximately assembled from a finite number of linear channels.
First, the number of levels \(N\) is selected, after which the branches are formed.
\[\tag{60} \begin{aligned} s_{n,\pm}(t) &=1\pm\frac{\omega t}{n\pi}, &&n=1,\ldots,N, \\[1mm] c_{n,\pm}(t) &=1\pm\frac{2\omega t}{(2n+1)\pi}, &&n=0,\ldots,N. \end{aligned} \] All these signals are linear in time and can be independently calculated or reproduced by separate channels. Then they are combined by multiplication:
\[\tag{61} \begin{aligned} x_N(t) &=R\omega t \prod_{n=1}^{N} s_{n,+}(t)s_{n,-}(t), \\[1mm] y_N(t) &=R \prod_{n=0}^{N} c_{n,+}(t)c_{n,-}(t). \end{aligned} \] The resulting point
\[\tag{62} \mathbf r_N(t) =\ep x_N(t)+\em y_N(t) \] approximates circular motion. Practical accuracy is verified directly by two quantities:
\[\tag{63} \begin{aligned} \varepsilon_{r,N} &= \max_{|t|\le T} \left| \mathbf r_N(t)-\mathbf r(t) \right|, \\[1mm] \varepsilon_{A,N} &= \max_{|t|\le T} \left| \ddot{\mathbf r}_N(t)+\omega^2\mathbf r(t) \right|. \end{aligned} \] The number of levels increases until the conditions are met.
\[\tag{64} \boxed{ \varepsilon_{r,N}<\varepsilon_r^{\max}, \qquad \varepsilon_{A,N}<\varepsilon_A^{\max}. } \] Thus, the formulas define a very specific procedure: instead of a single nonlinear circular motion, a set of simple linear channels is constructed, and the required trajectory is obtained in the output multiplicative layer.
For a regular numerical calculation, calculating sine and cosine directly is, of course, easier. The value of this algorithm lies not in speeding up the calculations, but in demonstrating a different internalMotion organization: nonlinearity is concentrated in the relationships between channels, not in their own laws.
12.2. Model of a system with linear channels
The design can be implemented as a mathematical, digital, or analog model. Each channel generates a linearly varying signal, and a cascade of multipliers generates two output coordinates. There are no accelerated signals at the input of such a device:
\[\tag{65} \ddot s_{n,\pm}=0, \qquad \ddot c_{n,\pm}=0, \] while the output satisfies the approximate rotation equation:
\[\tag{66} \ddot{\mathbf r}_N(t) \approx -\omega^2\mathbf r_N(t). \] This scheme allows one to study how collective nonlinearity and acceleration arise from the interaction of simple subsystems. It can be used as a model example in control theory, signal processing, and the study of multilevel geometric projections.
12.3. Experimental Criterion for Finite Splitting
If we assume that a physical system contains a finite, rather than infinite, number of actual splittings, then its trajectory should not be a perfect circle. The deviations should have a certain form:
\[\tag{67} \delta\mathbf r_N(t) =\mathbf r_N(t)-\mathbf r(t), \qquad \delta\mathbf A_N(t) =\ddot{\mathbf r}_N(t)+\omega^2\mathbf r(t). \] Therefore, the formulas provide not only a method for constructing, but also a possible way to test the hypothesis. The measured systematic deviation from the ideal circular law can be compared with the remainders of finite products. The absence of the predicted deviation structure, on the contrary, would limit such a model. This criterion applies specifically to the assumption that the physical hierarchy is structured as a successive truncation of the Euler products constructed here; for a different splitting geometry, the correction law may be different.
12.4. What would be required for a physical application?
The most interesting possible application would be to control the observed trajectory by changing the linear velocities of the higher branches:
\[\tag{68} u_{n,\pm} \quad\longrightarrow\quad \Pi_{\times} \quad\longrightarrow\quad R,\, \omega,\, \mathbf A. \] If physical analogs of these branches exist and their multiplicative relationship can be controlled, then changing the set of constant velocities could change the radius, frequency, and acceleration of the observed projection. However, for a practical device, it is necessary to experimentally demonstrate that the system's proper acceleration actually remains zero at the higher level, and is not simply rewritten by different coordinates.
Today, the formulas provide a modeling algorithm and a verification criterion. They do not yet provide a technology for eliminating centrifugal force. Such a technology will only emerge after the discovery of the physical degrees of freedom corresponding to splittings and a way to control their multiplicative projection.
Conclusion
The practical result of this article lies not only in the assertion of the existence of some higher representation. An explicit algorithm is obtained that translates a given circular motion into a set of specific linear branches:
\[\tag{69} \boxed{ \begin{gathered} q_{n,\pm}(t)=q_{n,\pm}(0)+u_{n,\pm}t, \qquad \ddot q_{n,\pm}=0, \\[1mm] \xrightarrow{\quad\Pi_{\times}\quad} \\[1mm] \mathbf r(t) =R\ep\sin(\omega t) +R\em\cos(\omega t), \qquad \ddot{\mathbf r}=-\omega^2\mathbf r. \end{gathered} } \] Mathematically, this algorithm can already be used: select a finite number of channels, assemble an approximate trajectory, and control coordinate and acceleration errors using formulas (63)–(64). Physically, the result sets a program for further research: find real analogs of linear branches, establish the law of their multiplication, and verify the characteristic deviations of the finite splitting.
So, the direct application of the result is modeling and hypothesis testing. A possible future application is control of curvilinear projection through non-accelerated higher motions.