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Newton’s law of motion is derived from Kepler’s laws of planetary motion. This is achieved by applying a simple system identification method using numerical data from the planet’s orbits in conjunction with the inverse square law for the attractive force between celestial bodies and the concepts of the derivative and differential equation. The identification procedure yields the differential equation of motion of a body under the action of an applied force as stated by Newton. Moreover, the employed procedure, besides validating the inverse square law, permits the evaluation of the gravitational mass (standard gravitational parameter), paving thus the way for establishing Newton’s law of universal gravitation. As the employed mathematical tools and the theory were available before 1686, we are allowed to state that the equation of motion for a body with constant mass could have been established from Kepler’s law of planetary motion, before Newton had published his law of motion.

A force is said to be central under two conditions. First, the direction of the force must always be toward or away from a fixed point (see Fig. 9.1). This point is known as the center of the force. Second, the magnitude of the force should only be proportional to the distance r between the particle and the center of the force.

We show that the photon trajectory calculated on the basis of the Newton laws of mechanics coincides with that calculated on the basis of laws of the geometrical optics on assumption that the momentum of the photon in matter increases by n times as compared with that in free space. The force applied to the photon is calculated from the third Newton law as the force that is opposite to the net force produced by the electrical field of the photon in the matter. It is shown that there is no Lorentz force and the Maxwell force produced by an electrical filed in a dielectric is responsible for a change of the photon momentum.

The motion of a mass undergoing free horizontal motion on a planet’s surface is analyzed in an inertial frame by using a potential well presentation. The following results from the conventional analysis in rotating coordinates are also found in inertial coordinates. Under typical circumstances the mass oscillates about the latitude where its angular momentum per unit mass matches that of the planet’s surface. For masses with nonzero angular momentum, conservation of angular momentum bounds the mass away from the pole. Motion with zero angular momentum is confined to a fixed meridian plane on which the mass may oscillate across the pole. Motion across the equator is possible only for masses with sufficient energy. Masses with angular momentum per unit mass greater than that of the planet at the equator oscillate about the equator with a limiting period determined by the excess angular momentum.

The two important conservation laws are conservation of energy and conservation of momentum. From the concept of linear momentum and forces, impacts can be considered. ‘Newton’s Cradle’ and the ‘Galilean Cannon’ are each illustrated with a simple JavaScript simulation. But we must also deal with angular momentum and couples. It is shown that a displacement and a rotation can be combined into a rotation about another axis, or into a ‘screw’. In the same way, a couple and a force can either be combined into a force with a different line of action, or more generally into a ‘wrench’. Finally linear and angular momentum are combined with an impulse in finding the ‘sweet spot’ of a cricket bat.

The relevance of the Central - force motion in the macroscopic and microscopic frames warrants a detailed study of the theoretical mechanics associated with it. So far, researchers have only considered central - force motion, as motion only in the translational and rotational elliptical plane with polar coordinates. However, the theoretical knowledge advanced by these researchers in line with this type of motion is scientifically restricted as several possibilities are equally applicable. In order to make the mechanics of a Central - force motion sufficiently meaningful, we have in this work extended the theory which has only been that of translational and rotational in the elliptical plane, by including fictitious radii and spin oscillations of the body about the axis of rotation. In this work, we used the methods of Newtonian mechanics to establish the new central-force field obeyed by the motion of a body, when the effect of spin oscillation is added. The new central-force field comprises of the radial accelerations, translational orbital angular velocity and the oscillating spin angular velocities. The energy conveyed in the spin oscillating phase increases as the orbital oscillating angles above or below the horizon of the elliptical plane is increased.

International audience; Two spherical balls are connected by a taught string passing through a small hole in a perfectly planar table: the first ball, subject to a central force, moves without friction on a two-dimensional plane, while the second ball moves only along the vertical axis directly below the hole. The pedagogical aspects of this novel two-body problem are given particular attention: Newton's laws, central force motion, conservation laws, angular momentum, constraints, etc. The dynamics of the system is considered under various initial conditions wherein the ball on the table moves qualitatively in rotating ellipses or hypotrochoids. The conditions for closed or periodic orbits are examined. The more complex case of the inclined plane is then considered, revealing a rich variety of periodic, aperiodic and chaotic solutions as a function of the ballmass ratio and the plane inclination angle. The associated Poincar'e phase-space maps are discussed.

In a previous paper we have shown that Newton’s third law cannot strictly hold in a distributed system of which the different parts are at a finite distance from each other. This is due to the finite speed of signal propagation which cannot exceed the speed of light in vacuum, which in turn means that when summing the total force in the system the force does not add up to zero. This was demonstrated in a specific example of two current loops with time-dependent currents, the above analysis led to the suggestion of a relativistic engine. Since the system is affected by a total force for a finite period of time this means that the system acquires momentum, the question then arises if we need to abandon the law of momentum conservation. In this paper we make a detailed calculation and show that any momentum gained by the material part of the system is equal in magnitude and opposite in direction to the momentum gained by the electromagnetic field. Hence the total momentum of the system is conserved.

Newton's laws of motion are three physical laws that together, laid the foundation for classical three dimensional mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. Kepler's laws of planetary motion are also three scientific laws describing the motion of planets around the Sun. Although the differential calculus was invented by Newton, Kepler established his famous laws 70 years earlier by using the same idea, namely to find a path in a non uniform field of force by small steps. Even though neither the force was known nor its relation to motion, he could determine the differential equations of motion from observation. This is one of the most important achievements in the history of physics. In this paper, we will see that these laws are a consequence of Newton’s second law even in multidimensional Euclidian spaces.Keywords: Central vector field; Differential equation.

For a nonrelativistic particle acted on by a spatially uniform but possibly time-dependent force, or for a harmonic oscillator also subject to such a uniform force, the wave function differs from that without the force only by a spatial displacement and a phase factor. The shifts in position and phase are simply derived from a solution of the classical equations of motion.

In this paper the laws of motion of classical particles have been investigated in a non-commutative phase space. The corresponding non-commutative relations contain not only spatial non-commutativity but also momentum non-commutativity. First, new Poisson brackets have been defined in non-commutative phase space. They contain corrections due to the non-commutativity of coordinates and momenta. On the basis of this new Poisson brackets, a new modified second law of Newton has been obtained. For two cases, the free particle and the harmonic oscillator, the equations of motion are derived on basis of the modified second law of Newton and the linear transformation (Phys. Rev. D, 2005, 72: 025010). The consistency between both methods is demonstrated. It is shown that a free particle in commutative space is not a free particle with zero-acceleration in the non-commutative phase space, but it remains a free particle with zero-acceleration in non-commutative space if only the coordinates are non-commutative.

If the force causing accelerated motion of a particle passes through a fixed point, then this yields central force motion

The concept of areal velocity is intrinsically and historically connected with that of angular momentum. For central force fields the area swept out in the orbital plane by a particle’s radius vector is proportional to the time. For non-planar problems, it is shown that a particle’s position vector sweeps out a general conical surface and that interesting kinematical relations hold. The areal velocity vector is always perpendicular to the conical surface and is proportional to the angular momentum of the particle. In some problems, the magnitude of the areal velocity is constant while its direction changes. In others, only a component of the areal velocity vector is constant. A moving orthonormal basis associated with the areal velocity is defined and a matrix equation for the angular velocities of the basis vectors is found to have an elegant form, involving “sweeping” and “tilting” components. Illustrative examples are provided and some historical background is included.

We address the curious problem of quadratic Zeeman effect at the classical mechanical level. The problem has been very well understood for decades, but an analytical solution of the equations of motion is still to be found. This state of a. airs persists because the simultaneous presence of the Coulombic and quadratic terms lowers the dynamical symmetry. Energy and orbital angular momentum are still constants of motion. We find the exact solutions by introducing the concept of an image ellipse. The quadratic effect leads to a dilation of space-time, and a one-to-one correspondence is observed for pairs of physical quantities like energy and angular momentum, and the maximum and minimum distances from the Coulomb center for the Zeeman orbit and the corresponding pairs for the image ellipse. Thus, instead of finding additional conserved quantities, we find constants of motion for an additional dynamics, namely, the image problem. The trajectory is open, in agreement with Bertrand's theorem, but necessarily bound. A stable unbound trajectory does not exist for real values of energy and angular momentum. The radial distance, the angle covered in the plane of the orbit, and the time are uniquely determined by introducing further the concept of an image circle. While the radial distance is defined in a closed form as a transcendental function of the image-circular angle, the corresponding orbit angle and time variables are found in the form of two convergent series expansions. The latter two variables are especially contracted, thereby leading to a precession of the open cycles around the Coulomb center. It is expected that the space time dilation effect observed here would somehow influence the solution of the quantum mechanical problem at the non-relativistic level.

This chapter presents orbital angular momentum as the generator or rotations in space. The operators which carry out rotations in space form a group, which is a symmetry group of the Hamiltonian when the interaction is spherically symmetric. In such a case, there are certain useful statements which we can make about the role of invariant and irreducible subspaces of the Hilbert space under the group operations and how to calculate matrix elements of tensor operators within such spaces, that is, the Wigner-Eckart theorem.

It is known that for the magnetic force due to a closed circuit, the Weber force law can be identical to the Lorentz force law. In this investigation it is shown that for both the electric and the magnetic force of the quasi-static case, the Riemann force law can be identical to the Lorentz force law, while the former is based on a potential energy depending on a relative speed and is in accord with Newton's law of action and reaction.

The motion of the planets, asteroids and comets in the Solar System are controlled by gravity and are well explained in classical mechanics by Newton's and Kepler's laws. Using these laws, one can easily calculate some of the unknown variables such as the period or the angular momentum of the planets. However, neither Newton's laws nor Kepler's laws explain the relationship between the angular momentum of the orbits in the Solar System. The purpose of this paper is to demonstrate that the specific angular momentum (angular momentum per unit mass) of any planet in the Solar System can be expressed in terms of the specific angular momentum of the planet Mercury and, consequently, to prove that the semi-major axis of any planet, asteroid or comet can be expressed as a multiple of the semi-major axis of Mercury.

A survey of topics of recent interest in Hamiltonian and Lagrangian dynamical systems, including accessible discussions of regularization of the central force problem; inequivalent Lagrangians and Hamiltonians; constants of central force motion; a general discussion of higher-order Lagrangians and Hamiltonians with examples from Bohmian quantum mechanics, the Korteweg-de Vries equation and the logistic equation; gauge theories of Newtonian mechanics; classical spin, Grassmann numbers, and pseudomechanics., Comment: Einstein Centennial Review Article, 48 pages

The equation of motion for a mass that moves under the influence of a central, inverse-square force is formulated and solved as a problem in complex variables. To find the solution, the constancy of angular momentum is first established using complex variables. Next, the complex position coordinate and complex velocity of the particle are assumed to depend on time in an implicit manner only, through the polar angle ϕ, and angular momentum conservation is then imposed in the complex equation of motion itself, from the start. The general solution follows after performing but a single elementary integral, , and the trajectory is shown to be a conic section. Energy conservation is also established using complex variables. The proposed method of solution is rigorous, yet concise and relatively simple to solve, so it could prove useful in a pedagogical context.

In a brief handwritten but undated memorandum entitled "A True state of the Case and Controversy between Sr Isaak Newton & Dr Robert. Hooke as the Priority of that Noble Hypothesis of Motion of ye Planets about ye Sun as their Centers,""25 Hooke recounted his hypothesis for the physics of orbital motion and his theory of universal gravitation. Hooke's memorandum, which remained unpublished during his lifetime, is historically quite accurate, contradicting numerous criticisms of his contemporaries and historians of science that Hooke always claimed for himself more credit than he actually deserved. In fact, to support his "priority" Hooke quoted verbatim from several extant documents: the transcript of his lecture on Planetary Movements as a Mechanical Problem given at the Royal Society on May 23, 1666,26 his short (28 pages) monograph, An Attempt to prove the motion of the Earth by Observations published in 1674,27 and his lengthy correspondence in the Fall of 1679 with Isaac Newton.28 However, Hooke did not mention his remarkable geometrical implementation of orbital motion for central force motion, (see Fig. 1), based on the application of his physical principles, which was found only recently in a manuscript dated September 1685.29 Unfortunately

Central force motion for power laws of the form F(r)=−krn for n>−3 (n not necessarily an integer) and k>0 is discussed in terms of the energy, angular momentum, and orbital radius of an arbitrary circular orbit. A proof is given that orbits for these power laws must be everywhere concave toward the origin. The only two cases where the orbits are closed, n=1 and n=−2, are discussed.

The computation of structures moving in central force fields generally requires long‐time integration including geometrically nonlinear behavior (large rotations) as such, e.g. satellite structures move for a long time. To achieve a numerically stable computation the energy momentum method which fulfills linear and angular momentum as well as energy conservation within the time step is chosen for the time integration. The focus in the contribution is on Hamiltonian systems. A formulation for the gravitational force in a central force field as external force on a rigid or flexible satellite is given. The presented formulation enables the computation of the exact spatial distribution of the gravitational forces acting on a structure using the FE‐discretization which is necessary to analyze, e.g. the orientation of a satellite in a gravitational field. The fulfillment of the conservation laws within the time step is proved. The necessity for considering the spatial distribution of the gravitational forces is discussed based on numerical examples.

The trajectory of the isotropic three-dimensional oscillator in relativistic classical mechanics is derived. Its form depends on the values of the angular momentum l. It is a rosette for 0 < l < l0 and a circle for l = l0, where l0 is the maximum value of the angular momentum for a given energy. According to expectation, in the non-relativistic limit, the rosette reduces to an ellipse.

A statement is formulated which leads to the general formula for the force of interaction between two bodies. The distance between their centers of inertia varies with time. According to the third Kepler's law, the well-known Newton's law of gravitation follows from the formula. Some examples are presented and discussed

The purpose of this paper is to analyze the way in which Newton uses his polygon model and passes to the limit in Proposition I, Book I of his Principia . It will be evident from his method that the limit of the polygon is indeed the orbital arc of the body and that his approximation of the actual continuous force situation by a series of impulses passes correctly in the limit into the continuous centripetal force situation. The analysis of the polygon model is done in two ways: (1) using the modern concepts of force, linear momentum, linear impulse, and velocity, and (2) using Newton's concepts of motive force and quantity of motion. It should be clearly understood that the term “force” without the adjective “motive,” is used in the modern sense, which is that force is a vector which is the time rate of change of the linear momentum. Newton did not use the word “force” in this modern sense. The symbol F denotes modern force. For Newton “force” was “motive force,” which is measured by the change in the quantity of motion of a body. Newton's “quantity of motion” is proportional to the magnitude of the modern vector momentum. Motive force is a scalar and the symbol F m is used for motive force.

In the 1687 Principia, Newton gave a solution to the direct problem (given the orbit and center of force, find the central force) for a conic-section with a focal center of force (answer: a reciprocal square force) and for a spiral orbit with a polar center of force (answer: a reciprocal cube force). He did not, however, give solutions for the two corresponding inverse problems (given the force and center of force, find the orbit). He gave a cryptic solution to the inverse problem of a reciprocal cube force, but offered no solution for the reciprocal square force. Some take this omission as an indication that Newton could not solve the reciprocal square, for, they ask, why else would he not select this important problem? Others claim that ``it is child's play'' for him, as evidenced by his 1671 catalogue of quadratures (tables of integrals). The answer to that question is obscured for all who attempt to work through Newton's published solution of the reciprocal cube force because it is done in the synthetic geometric style of the 1687 Principia rather than in the analytic algebraic style that Newton employed until 1671. In response to a request from David Gregory in 1694, however, Newton produced an analytic version of the body of the proof, but one which still had a geometric conclusion. Newton's charge is to find both ``the orbit'' and ``the time in orbit.'' In the determination of the dependence of the time on orbital position, t(r), Newton evaluated an integral of the form ∫dx/x n to calculate a finite algebraic equation for the area swept out as a function of the radius, but he did not write out the analytic expression for time t = t(r), even though he knew that the time t is proportional to that area. In the determination of the orbit, θ (r), Newton obtained an integral of the form ∫dx/√(1−x2) for the area that is proportional to the angle θ, an integral he had shown in his 1669 On Analysis by Infinite Equations to be equal to the arcsin(x). Since the solution must therefore contain a transcendental function, he knew that a finite algebraic solution for θ=θ(r) did not exist for ``the orbit'' as it had for ``the time in orbit.'' In contrast to these two solutions for the inverse cube force, however, it is not possible in the inverse square solution to generate a finite algebraic expression for either ``the orbit'' or ``the time in orbit.'' In fact, in Lemma 28, Newton offers a demonstration that the area of an ellipse cannot be given by a finite equation. I claim that the limitation of Lemma 28 forces Newton to reject the inverse square force as an example and to choose instead the reciprocal cube force as his example in Proposition 41.

In this essay I analyze the solutions given by Isaac Newton and Johann Bernoulli to a well-posed mathematical problem known in the eighteenth century as the ‘inverse problem of central forces’. This work prompted polemical exchanges between two rival groups: the first, whose leader was Newton, was based in Oxford, Cambridge, and London; the second, whose leading representatives were Leibniz and Bernoulli, was scattered througout Europe, though centered in Basel and Paris.

Graduate level physics curricula in many countries around the world, as well as senior-level undergraduate ones in some major institutions, include Classical Mechanics courses, mostly based on Goldstein's textbook masterpiece. During the discussion of central force motion, however, the Kepler problem is virtually the only serious application presented. In this paper, we present another problem that is also soluble, namely the interaction of Schwinger's dual-charged (dyon) particles. While the electromagnetic interaction of magnetic monopoles and electric charges was studied in detail some 40 years ago, we consider that a pedagogical discussion of it from an essentially classical mechanics point of view is a useful contribution for students. Following a path that generalizes Kepler's problem and Rutherford scattering, we show that they exhibit remarkable properties such as stable non-planar orbits, as well as rainbow and glory scattering, which are not present in the ordinary scattering of two singly charged particles. Moreover, it can be extended further to the relativistic case and to a semi-classical quantization, which can also be included in the class discussion., Comment: 23 pages, 4 figures, accepted for publication on European Journal of Physics, 2015

It is shown presently that the Newtonian axiomatics by inference includes the concept of angular momentum and its conservation, and that this leads to the conservation of energy theorem as a base for development of a set of equations of motion for dynamic systems, which are almost identical with those obtained by the Lagrangian Methodology. This is done utilizing the formulations of the kinetic energy as a quantitative measure, and as a function of the state of motion of the systems. The simplicity of the Generalized Energy Methodology makes it possible to identify a number of imperfections of the equations of motion obtained by means of either methodology.

DYNAMICS The Galileo-Newton Formulation of Dynamics Galilean relativity Newton's dynamical laws Gravitational and inertial mass Particle Dynamics in One Dimension Motion of a particle under a force Potential energy diagrams Tension Friction Resistive motion Escape velocity Oscillations Hooke's law Simple harmonic motion Period of small oscillations Damped simple harmonic motion Damped simple harmonic motion with a forcing term The LCR circuit Particle Dynamics in Two and Three Dimensions Projectiles Energy and force Charged particles in an electromagnetic field Central Forces and Orbits Central forces and angular momentum Circular motion Orbital motion The inverse square law The orbital equation Perturbed orbits Kepler's laws of planetary motion The perihelion precession of Mercury Rutherford scattering Multi-Particle Systems Conservation of linear momentum Conservation of angular momentum The centre of mass frame The two-body problem Collisions Inelastic collisions Variable mass problems Rigid Bodies Rotation of a rigid body about a fixed axis Planar motion of a rigid body Rotating Reference Frames Rates of change in a rotating frame Newton's second law in a rotating frame The centrifugal force The Coriolis force RELATIVITY Special Relativity Inception Einstein's postulates of special relativity Lorentz transformations Minkowski diagrams (space-time diagrams) Relativistic kinematics Space-Time The light cone Proper time The four-component vector formalism Relativistic Mechanics 4-momentum Relativistic energy Massless particles Aberration Particle collisions Appendix: Conic Sections Solutions Index Exercises appear at the end of each chapter.

We describe a new undergraduate mechanics laboratory experiment that illustrates principles of linear and angular momentum conservation. Students observe that angular momentum is conserved in systems where the initial conditions involve only translational motion, and yet the final conditions include rotational motion. To do this, students make a digital video of a two-object collision on an air table. One object has an asymmetric mass distribution and gains rotational kinetic energy as a result of the collision. Using their digital video, students create a stroboscopic image of the two objects’ trajectories and show that both linear and angular momentum are conserved to within a few percent, the limit of experimental uncertainty. Students also show that angular momentum conservation is independent of the choice of origin, that the center of mass of the asymmetric object follows a linear trajectory, and that the center of mass of the whole system follows a linear trajectory.

Spinning skaters and satellite orbits are usually discussed in terms of angular momentum and other powerful formalisms. This paper presents an alternative, more primitive analysis based directly on Newton’s second law.

Cross-term conservation relationships for electromagnetic energy, linear momentum, and angular momentum are derived and discussed here. When two or more sources of electromagnetic fields are present, these relationships connect the cross terms that appear in the traditional expressions for the electromagnetic (1) energy, (2) linear momentum, and (3) angular momentum, over to, respectively, (1) the sum of the rates of work, (2) the sum of the forces, and (3) the sum of the torques, that are due to the fields of each charge or current source acting upon the other charge and current sources. These relationships, although not new, appear to be rarely recognized and used in the physics literature. As shown here, they can be extremely helpful for solving and gaining a deeper physical understanding into a rather diverse range of interesting problems in electrodynamics, including (1) aspects of Poynting's theorem when applied to charged point particles, (2) the detailed physical basis of electrostatic analysis, (3) understanding the connection between different techniques used in the past for solving Casimir force problems, and (4) reconciling the invalidity of Newton's third law in electrodynamics.

This paper deals with the mechanics and control for multi-particle systems from a geometric point of view. The centre-of-mass system is viewed as a principal fibre bundle with structure group SO(3), the base space of which is called the internal or shape space. A natural connection and a natural Riemannian metric are both defined on the centre-of-mass system. The equations of motion for the multi-particle system are derived in the Lagrangian formalism adapted to the bundle structure, and then reduced with the conserved total angular momentum. In contrast with this, the control problem is studied with non-holonomic constraints, i.e. with the vanishing total angular momentum, and equations of motion are determined for an optimally controlled multi-particle system. The resultant equations derived in each of the mechanical and control systems are to be compared.

The force field described by a potential function of the form U = ?n k=1 ak/rk (r = distance between particles, ak = real parameters) models various concrete situations belonging to astronomy, physics, mechanics, astrodynamics, etc. The two-body problem is being tackled in such a field. The motion equations and the first integrals of energy and angular momentum are established. The McGehee-type coordinates are used to blow up the collision singularity and to paste the resulting manifold on the phase space. The flow on the collision manifold is depicted. Then, using the rotational symmetry of the problem and the angular momentum integral, the local flow near collision is described and interpreted in terms of physical motion.

The laws of mechanics can be set forth in different equivalent formulations. Frequently, one formulation can be deduced from another. For example, Lagrange's equations can be deduced both from Newton's law and from Hamilton's principle. One task never undertaken to date is that of deducing the principles of linear and angular momentum from a formulation of analytical mechanics. This task is undertaken in the present note in connection with simple nonholonomic systems.

This chapter delves into some systems of differential equations arising in mechanics. The first is the famous Newton's second law, force = mass × acceleration. This leads to the more general type of systems known as conservative systems, which are special examples of Hamiltonian systems as described in Chapter 9 . The main example here is given by central force fields and, specifically, the Newtonian central force system. Specific examples treated include Kepler's first law and the two-body problem from celestial mechanics. A new technique called blowing up the singularity provides a tool for understanding these systems near places where the system is undefined–that is, collisions between the various bodies. Explorations include classical limits of quantum mechanical systems and the motion of a glider.

Based on the equations of planar motion with drag used in point dynamics, a simple derivation is presented that shows how a constant Keplerian (elliptical) orbit can be obtained with a non-central force field and variable angular momentum.

We consider the planar dynamics of a body under the simultaneous influence of a force given by the spatial gradient of a potential function and a force that remains perpendicular to the body's translational velocity. Either force can vary with time. We describe a physical example of such a system and analyze the structure of the underlying equations of motion, demonstrating that they exemplify a special class of almost Hamiltonian systems. We show that when the potential function varies periodically with time, the momentum of the body can evolve chaotically.

This chapter applies a combination of vector mechanics and Lagrangian mechanics to the general system of two point masses interacting by a central force depending only on their separation. The special case of an inverse square central force reproduces Newton's achievement in modern notation. Newton's Principia presented proof that elliptical orbits are a natural consequence of an inverse square gravitational attraction between the sun and the planets, and launched the topic of discussion in this chapter. The central force problem is thus central to the early history of analytical mechanics. An inverse square central force is also found between charged particles in electrodynamics. The orbital motion of an electron around a proton underlies both the Bohr model of the hydrogen atom and the old quantum theory that was superseded by Schroedinger's quantum mechanics.

Through the use of the optical‐mechanical analogy, Newton’s law of motion may be cast into the same form as the equation for the ray in the geometrical optics of gradient‐index media. The resulting equation is called the ray form of Newton’s law of motion. The same equation may be derived by taking the geometrical optics limit of quantum mechanics. The ray form of Newton’s law of motion is derived in three different ways and is applied in the solution of several problems.

Advantages of a neo‐Cartesian approach to classical mechanics are noted. If conservation of linear momentum is the fundamental principle, Newton’s three laws become theorems. A minor paradox in static Newtonian mechanics is identified, and solved by reinterpreting force as a current of momentum. Contact force plays the role of a mere midwife in the exchange of momentum; however, force cannot be eliminated from physics because it provides the numerical value for momentum current. In this sense, in a neo‐Cartesian formulation of mechanics the concept of force becomes strengthened rather than weakened.