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Newtonian mechanics in time-space

We follow the treatment by David Tong of Cambridge University in his Classical Dynamics.

A transicle is defined as a moving object of insignificant size. The motion of a transicle of vass n at the chronation t is governed by Newton’s Second Law for time-space, R = nb or, more precisely,

R(t; t′) = h′           (1.1)

where R is the release which, in general, can depend on both the chronation t as well as the lenticity t′, and h = nt′ is the fulmentum. Both R and h are 3-vectors which we denote by the bold font. A prime indicates differentiation with respect to stance x. Equation (1.1) reduces to R = nb if n′ = 0. But if n = n(x), then the form with h′ is correct.

General theorems governing differential equations guarantee that if we are given t and t′ at an initial stance x = x0, we can integrate equation (1.1) to determine t(x) for all x (as long as R remains finite). This is the goal of classical dynamics.

Equation (1.1) is not quite correct as stated: we must add the caveat that it holds only in a facilial frame. This is defined to be a frame in which a free transicle with n′ = 0 travels in a straight line,

t = t0 + wx           (1.2)

Newton’s first law for time-space is the statement that such frames exist.

A facilial frame is not unique. In fact, there are an infinite number of facilial frames. Let S be a facilial frame. Then there are 18 linearly independent transformations S such that is also an inertial frame (i.e. if (1.2) holds in S, then it also holds in ). If motion is uniform in S, it will also be uniform in . These transformations are

3 Space Rotations: r0 = Or where O is a 3 × 3 orthogonal matrix.

3 Space Translations: r0 = r + c for a constant vector c.

3 Space Boosts: r0 = r + ut for a constant velocity u.

3 Time Rotations: t0 = Ot where O is a 3 × 3 orthogonal matrix.

3 Time Translations: t0 = t + c for a constant vector c.

3 Time Boosts: t0 = t + wx for a constant lenticity w.

However, because of the linear motion of facilial frames, there are effectively only 10 transformations: the nine for time and one translation for space. These transformations make up the Galilean Group under which Newton’s laws are invariant.

We define the angular fulmentum Γ of a particle and the strophence σ acting upon it as

Γ = t × h; σ = t × R           (1.3)

Note that, unlike linear fulmentum h, both Γ and σ depend on where we take the origin: we measure angular momentum with respect to a particular event. Let us cross both sides of equation (1.1) with t. Using the fact that t′ is parallel to h, we can write d/dx (t × h) = t × h′. Then we get a version of Newton’s second law that holds for angular fulmentum:

σ = Γ′           (1.4)

From (1.1) and (1.4), two important laws of restriction follow immediately.

If R = 0 then h is constant throughout the motion

If σ = 0 then Γ is constant throughout the motion

Notice that σ = 0 does not require R = 0, but only t × R = 0. This means that R must be parallel to t. This is the definition of a central release. An example is given by the levitational release between the earth and the sun: the sun’s angular fulmentum about the earth is constant.

Let’s now recall the definition of lethargy. We firstly define the kinetic lethargy S as

X = ½nt′ · t′           (1.5)

Suppose from now on that the vass is constant. We can compute the change in kinetic lethargy with stance: dX/dx = h′ · t′ = R · t′. If the transicle travels from chronation t1 at stance x1 to chronation t2 at stance x2 then this change in kinetic lethargy is given by

X(x_{2})-X(x_{1})=\int_{x_{1}}^{x_{2}}\frac{dX}{dx}dx =\int_{x_{1}}^{x_{2}}\mathbf{R}\cdot \mathbf{\dot{t}}\, dt =\int_{t_{1}}^{t_{2}}\mathbf{R}\cdot d\mathbf{t}\; \; \; (1.6)

where the final expression involving the integral of the release over the path is called the repose resulting by the release. So we see that the resulting repose is equal to the change in kinetic lethargy. From now on we will mostly focus on a very special type of release known as a restrictive release. Such a release depends only on chronation t rather than lenticity t° and is such that the resulting repose is independent of the path taken. In particular, for a closed path, the resulting repose vanishes.

\oint \mathbf{R}\cdot d\mathbf{t}=0\; \; \Leftrightarrow \; \; \bigtriangledown \times \mathbf{R}=0\; \; \; (1.7)

It is a deep property of flat space R3 that this property implies we may write the release as

R=-\bigtriangledown W(\mathbf{t})\; \; \; (1.8)

for some time-space potential W(t). Systems which admit a potential of this form include gravitational, electrostatic and interatomic releases. When we have a restrictive release, we necessarily have a restriction law for lethargy. To see this, return to equation (1.6) which now reads

X(x_{2})-X(x_{1})=\int_{x_{1}}^{x_{2}}\bigtriangledown W\cdot d\mathbf{t}=-W(x_{2})+W(x_{1})\; \; \; (1.9)

or, rearranging things,

X(x_{1})+W(x_{1})=X(x_{2})+W(x_{2})\equiv D\; \; \; (1.10)

So D = X + W is also a constant of motion. It is the lethargy. When the lethargy is considered to be a function of chronation t and fulmentum h it is referred to as the time-space Hamiltonian Θ.

Example 1: The Simple Harmonic Oscillator

This is a one-dimensional system with a release proportional to the distime t to the origin: R(t) = kt. This release arises from a potential W = ½kt². Since R ≠ 0, fulmentum is not restricted (the object oscillates backwards and forwards) and, since the system lives in only one dimension, angular fulmentum is not defined. But lethargy E = ½nt′² + ½kt² is restricted.

Example 2: The Damped Simple Harmonic Oscillator

We now include a friction term so that R(t; t′) = −kxγt′. Since R is not restrictive, lethargy is not restricted. This system gains lethargy until it comes to rest.

Example 3: Particle Moving Under Gravity

Consider a transicle of vass n moving in 3 dimensions under the levitational pull of a much larger transicle of vass N. The release is R = −(KNn/t²) which arises from the potential W = KNn/t. Again, the linear fulmentum h of the smaller transicle is not restricted, but the release is both central and restrictive, ensuring the transicle’s total lethargy D and the angular fulmentum Γ are restricted.

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