# Newton’s laws and their duals

The following is based on Classical Mechanics by Kibble and Berkshire, 5th ed., Imperial College Press, 2004, with the dual version indented and changes italicized.

p.2 The most fundamental assumptions of physics are probably those concerned with the concepts of space and time. We assume that space and time are continuous, that it is meaningful to say that an event occurred at a specific spacepoint and a specific timepoints, and that there are universal standards of length and time (in the sense that observers in different places and at different times can make meaningful comparisons of their measurements).

The most fundamental assumptions of physics are probably those concerned with the concepts of length and duration. We assume that length and duration are continuous, that it is meaningful to say that an event occurred at a specific point in length space and a specific timepoint of duration space, and that there are universal standards of length and duration (in the sense that observers in different places and at different times can make meaningful comparisons of their measurements).

In ‘classical’ physics, we assume further that there is a universal time scale (in the sense that two observers who have synchronized their clocks will always agree about the time of any event), that the geometry of space is Euclidean, and that there is no limit in principle to the accuracy with which we can measure all positions and velocities.

In dual ‘classical’ physics, we assume further that there is a universal length scale (in the sense that two observers who have synstancized their clocks will always agree about the stance of any event), that the geometry of time is Euclidean, and that there is no limit in principle to the accuracy with which we can measure all chronations and lenticities.

p.3-4 To specify positions and time, each observer may choose a zero of the time scale, a spatial origin, and a set of three Cartesian co-ordinate axes. We shall refer to these collectively as a frame of reference. The position and time of any event may they be specified with respect to this frame by the three Cartesian co-ordinates x, y, z and the time t. … The frames used by unaccelerated observers are called inertial frames.

p.3-4 To specify chronations and base, each observer may choose a zero of the stance scale, an origin in 3D duration, and a set of three Cartesian co-ordinate axes. We shall refer to these collectively as a time frame of reference. The chronation and stance of any event may they be specified with respect to this time frame by the three Cartesian co-ordinates ξ, η, ζ and the stance r. … The time frames used by unretardated observers are called facile time frames.

Formally, an inertial frame may be defined to be one with respect to which any isolated body, far removed from all other matter, would move with uniform velocity.

Formally, a facile time frame may be defined to be one with respect to which any isolated body, far removed from all other motion, would move with uniform lenticity.

p.4-5 It is often convenient to use a notation which does not refer explicitly to a particular set of co-ordinate axes. Instead of using Cartesian co-ordinates x, y, z we may specify the position of a point P with respect to a given origin O by the length and direction of the line OP. A quantity which is specified by a magnitude and a direction is called a vector; in this case the position vector r of P with respect to O.

p.4-5 It is often convenient to use a notation which does not refer explicitly to a particular set of co-ordinate axes. Instead of using Cartesian co-ordinates ξ, η, ζ we may specify the chronation of an event Q with respect to a given origin O by the duration and direction of the line OQ. A quantity which is specified by a magnitude and a direction is called a vector; in this case the chronation vector t of Q with respect to O.

We shall therefore consider first only small bodies which can be effectively located at a point, so that the position of each, at time t, can be specified by a position vector r(t).

We shall therefore consider first only small bodies which can be effectively located at an instant, so that the chronation of each, at stance r, can be specified by a chronation vector t(r).

p.6 Each body is characterized by a scalar constant, its mass, mi. Its momentum pi is defined to be mass × velocity:

pi = mi vi.

The equation of motion, which specified how the body will move is Newton’s second law (mass × acceleration = force):

pi = mi ai = Fi,                          (1.1)

where Fi is the total force acting on the body.

Each body is characterized by a scalar constant, its vass, ni. Its levamentum qi is defined to be vass × lenticity:

qi = ni ui.

The equation of motion, which specified how the body will move is the dual of Newton’s second law (vass × relentation = release):

qi = ni bi = Yi,                          (1.1)

where Yi is the total release acting on the body.

p.7 The two-body forces Fij must satisfy Newton’s third law, which asserts that ‘action’ and ‘reaction’ are equal and opposite,

Fji = ‒Fij.                                  (1.3)

The two-body releases Yij must satisfy the dual of Newton’s third law, which asserts that ‘action’ and ‘reaction’ are equal and opposite,

Yji = ‒Yij.                                  (1.3a)

p.11 If we isolate the two bodies from all other matter, and compare their mutually induced accelerations, then according to (1.1) and (1.3),

m1α1 = ‒m2α2,                           (1.7)

so that the accelerations are oppositely directed, and inversely proportional to the masses. If we allow two small bodies to collide, then during the collision the effects of more remote bodies are generally negligible in comparison with their effect on each other, and we may treat them approximately as an isolated system.

If we isolate the two events from all other activities, and compare their mutually induced relentations, then according to (1.1a) and (1.3a),

n1β1 = ‒n2β2,                           (1.7a)

so that the relentations are oppositely directed, and inversely proportional to the vasses. If we allow two small events to coincide, then during the coinciding the effects of more remote activities are generally negligible in comparison with their effect on each other, and we may treat them approximately as an isolated system.

The mass ratio can then be determined from measurements of their velocities before and after the collision, by using (1.7) or its immediate consequence, the law of conservation of momentum,

m1v1 + m2v2 = constant,                         (1.8)

The vass ratio can then be determined from measurements of their lenticities before and after the coincidence, by using (1.7a) or its immediate consequence, the law of conservation of levamentum,

n1u1 + n2u2 = constant.                         (1.8a)

If we wish to separate the definition of mass from the physical content of equation (1.7), we may adopt as a fundamental axiom the following:

⇒ In an isolated two-body system, the accelerations always satisfy the relation a1 = ‒k21a2, where the scalar k21 is, for two given bodies, a constant independent of their positions, velocities and internal states.

If we wish to separate the definition of vass from the physical content of equation (1.7), we may adopt as a fundamental axiom the following:

⇒ In an isolated two-body system, the relentations always satisfy the relation b1 = ‒k21b2, where the scalar k21 is, for two given bodies, a constant independent of their chronations, lenticities, and internal states.

If we choose the first body to be a standard body, and conventionally assign it a unit mass (say m1 = 1 kg), the we may define the mass of the second to be k21 in units of this standard mass (here m2 = k21 kg).

If we choose the first body to be a standard body, and conventionally assign it a unit vass (say n1 = 1 kg–1), the we may define the vass of the second to be k21 in units of this standard vass (here n = k21 kg–1).

Note that for consistency, we must have k12 = 1/k21. We must also assume of course that if we compare the masses of three bodies in this way, we obtain consistent results:

For any three bodies, the constants kij satisfy k31 = k32k21.

It then follows that for any two bodies, k32 is the mass ratio: k32 = m3/m2.

Note that for consistency, we must have k12 = 1/k21. We must also assume of course that if we compare the vasses of three bodies in this way, we obtain consistent results:

For any three bodies, the constants kij satisfy k31 = k32k21.

It then follows that for any two bodies, k32 is the vass ratio: k32 = n3/n2.

To complete the list of fundamental axioms, we need one which deals with systems containing more than two bodies, analogous to the law of composition of forces, (1.2). This may be stated as follows: (p.12)

⇒The acceleration induced in one body by another is some definite function of their positions, velocities and internal structure, and is unaffected by the presence of other bodies. In a many-body system, the acceleration of any given body is equal to the sum of the accelerations induced in it by each of the other bodies individually.

To complete the list of fundamental axioms, we need one which deals with systems containing more than two bodies, analogous to the law of composition of releases, (1.2a). This may be stated as follows:

⇒ The relentation induced in one body by another is some definite function of their chronations, lenticities and internal structure, and is unaffected by the presence of other bodies. In a many-body system, the relentation of any given body is equal to the sum of the relentation induced in it by each of the other bodies individually.

These laws, which appear in a rather unfamiliar form, are actually completely equivalent to Newton’s laws, as stated in the previous section. In view of the apparently fundamental role played by the concept of force in Newtonian mechanics, it is remarkable that we have been able to reformulate the basic laws without mentioning this concept. It can of course be introduced, by defining it through Newton’s second law, (1.1). The utility of the definition arises from the fact that forces satisfy Newton’s third law, (1.3), while accelerations satisfy only the more complicated law, (1.7). Since the mutually induced accelerations of two given bodies are always proportional, they are essentially determined by a single function, and it is useful to introduce the more systematic concept of force, for which this becomes obvious.

These laws, which appear in a rather unfamiliar form, are actually completely equivalent to the dual of Newton’s laws, as stated in the previous section. In view of the apparently fundamental role played by the concept of release in the dual of Newtonian mechanics, it is remarkable that we have been able to reformulate the basic laws without mentioning this concept. It can of course be introduced, by defining it through the dual of Newton’s second law, (1.1a). The utility of the definition arises from the fact that forces satisfy the dual of Newton’s third law, (1.3a), while relentations satisfy only the more complicated law, (1.7a). Since the mutually induced relentations of two given bodies are always proportional, they are essentially determined by a single function, and it is useful to introduce the more systematic concept of release, for which this becomes obvious.