We follow the treatment by David Tong of Cambridge University in his Classical Dynamics.
A tempicle is defined as a moving object of insignificant time. The motion of a tempicle of vass n at the chronation t is governed by Newton’s Distance Domain Second Law, 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 levamentum. Both R and h are 3-vectors which we denote by the bold font. A prime indicates differentiation with respect to distance 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 distance 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 tempicle with n′ = 0 travels in a straight line,
t = t0 + wx (1.2)
Newton’s distance domain first law 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 → S˜ such that S˜ is also an inertial frame (i.e. if (1.2) holds in S, then it also holds in S˜). If motion is uniform in S, it will also be uniform in S˜. These transformations are
3 Length Space Rotations: r0 = Or where O is a 3 × 3 orthogonal matrix.
3 Length Space Translations: r0 = r + c for a constant vector c.
3 Length Space Boosts: r0 = r + ut for a constant velocity u.
3 Duration Space Rotations: t0 = Ot where O is a 3 × 3 orthogonal matrix.
3 Duration Space Translations: t0 = t + c for a constant vector c.
3 Duration Space 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 levamentum Γ of a particle and the strophence σ acting upon it as
Γ = t × h; σ = t × R (1.3)
Note that, unlike linear levamentum 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 levamentum:
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 levamentum 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 distance: dX/dx = h′ · t′ = R · t′. If the tempicle travels from chronation t1 at distance x1 to chronation t2 at distance x2 then this change in kinetic lethargy is given by
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.
It is a deep property of flat space R3 that this property implies we may write the release as
for some distance domain 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
or, rearranging things,
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 levamentum h it is referred to as the distance domain 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, levamentum is not restricted (the object oscillates backwards and forwards) and, since the system lives in only one dimension, angular levamentum 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′) = −kx − γt′. 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 tempicle of vass n moving in 3 dimensions under the levitational pull of a much larger tempicle of vass N. The release is R = −(KNn/t²)tˆ which arises from the potential W = KNn/t. Again, the linear levamentum h of the smaller tempicle is not restricted, but the release is both central and restrictive, ensuring the tempicle’s total lethargy D and the angular levamentum Γ are restricted.