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# Classical dynamics

The following presents the spatio-temporal and temporo-spatial versions of Newton’s laws based on the book Classical Dynamics of Particles and Systems by Thornton and Marion (Fifth Edition, 2008).

2.2 Newton’s [Spatio-temporal] Laws

We begin by simply stating in conventional form Newton’s laws of mechanics:

I. A body remains at [spatially] rest or in uniform motion unless acted upon by a force.

II. A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force.

III. If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

To demonstrate the significance of Newton’s Third Law, let us paraphrase it in the following way, which incorporates the appropriate definition of mass:

III′. If two bodies constitute an ideal, isolated [spatial] system, then the accelerations of these bodies are always in opposite directions, and the ratio of the magnitudes of the accelerations is constant. This constant ratio is the inverse ratio of the masses of the bodies.

With this statement, we can give a practical definition of mass and therefore give precise meaning to the equations summarizing Newtonian dynamics. For two [spatially] isolated bodies, 1 and 2, the Third Law states that

F1 = −F2          (2.3)

Using the definition of force as given by the Second Law, we have

dp1/dt = −dp2/dt          (2.4a)

or, with constant masses,

m1 (dv1/dt) = m2 (−dv2/dt)          (2.4b)

and, because acceleration is the time derivative of velocity,

m1 (a1) = m2 (−a2)          (2.4c)

Hence,

m2 / m1 = −a1a2           (2.5)

where the negative sign indicates only that the two acceleration vectors are oppositely directed. Mass is taken to be a positive quantity.

We can always select, say, m1, as the unit mass. Then, by comparing the ratio of accelerations when m1 is allowed to interact with any other body, we can determine the mass of the other body.

2.4 The Equations of Motion for a Particle

Newton’s equation F = dp/dt can be expressed alternatively as

F = d/dt (mv) = m dv/dt = mr″          (2.7)

if we assume that the mass m does not vary with time. This is a second-order differential equation that may be integrated to find r = r(t) if the function F is known. Specifying the initial values of r and r′ = v then allows us to evaluate the two arbitrary constants of integration. We then determine the motion of a particle by the force function F and the initial values of position r and velocity v.

The force F may be a function of any combination of position, velocity, and time and is generally denoted as F(r, v, t). For a given dynamic system, we normally want to know r and v as a function of time. Solving Equation 2.7 will help us do this by solving for r″. Applying Equation 2.7 to physical situations is an important part of mechanics.

Compare this with:

2.2 Newton’s Temporo-spatial Laws

We begin by simply stating in conventional form Newton’s temporo-spatial laws of mechanics:

I. A body remains [temporally] at rest or in uniform motion unless acted upon by a release.

II. A body acted upon by a release moves in such a manner that the space rate of change of fulmentum equals the release.

III. If two bodies exert releases on each other, these releases are equal in magnitude and opposite in direction.

To demonstrate the significance of Newton’s Third Temporo-spatial Law, let us paraphrase it in the following way, which incorporates the appropriate definition of vass:

III′. If two bodies constitute an ideal, isolated [temporal] system, then the retardations of these bodies are always in opposite directions, and the ratio of the magnitudes of the retardations is constant. This constant ratio is the inverse ratio of the vasses of the bodies.

With this statement, we can give a practical definition of vass and therefore give precise meaning to the equations summarizing Newtonian temporo-spatial dynamics. For two [temporally] isolated bodies, 1 and 2, the Third Law states that

R1 = −R2          (2.3)

Using the definition of release as given by the Second Law, we have

dq1/dx = −dq2/dx          (2.4a)

or, with constant vasses,

n1 (dw1/dx) = n2 (−dw2/dx)          (2.4b)

and, because retardation is the time derivative of lenticity,

n1 (b1) = n2 (−b2)          (2.4c)

Hence,

n2 / n1 = −b1b2           (2.5)

where the negative sign indicates only that the two retardation vectors are oppositely directed. Vass is taken to be a positive quantity.

We can always select, say, n1, as the unit vass. Then, by comparing the ratio of retardation when n1 is allowed to interact with any other body, we can determine the vass of the other body.

2.4 The Equation of Motion for a Tempicle

Newton’s temporo-spatial equation R = dq/dx can be expressed alternatively as

R = d/dx (nw) = n dw/dx = ns″          (2.7)

if we assume that the vass n does not vary with stance. This is a second-order differential equation that may be integrated to find s = s(x) if the function R is known. Specifying the initial values of s and s′ = w then allows us to evaluate the two arbitrary constants of integration. We then determine the motion of a tempicle by the release function R and the initial values of chronation s and lenticity w.

The release R may be a function of any combination of chronation, lenticity, and stance and is generally denoted as R(s, w, x). For a given dynamic system, we normally want to know s and w as a function of stance. Solving Equation 2.7 will help us do this by solving for s″. Applying Equation 2.7 to physical situations is an important part of mechanics.