*Post by Luigi Fortunati*In the case of my animation

https://www.geogebra.org/m/mjnqb8vk

[etc.]

There are two answers to consider.

(This may be the last note I send out, through the Google portal, if it

gets through.)

[[Mod. note -- Yes, apparently google is shutting down its "google groups"

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See

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-- jt]]

First...

in the mechanics of a system whose dynamics is described by a Lagrangian

L(q,v,t) that is a function of the configuration space coordinates q =

(q^a: 0 < a <= N) and its corresponding velocities v = (v^a = dq^a/dt: 0 <

a <= N), these play the role of the "kinematic variables", with v^a =

dq^a/dt being the "kinematic equation", while the Lagrangian is

essentially a device used to generate the corresponding "dynamic

variables" - the momentum p_a = @L/@v^a and force f_a = @L/@q^a, as the

partial derivatives (using "@" to denote the partial derivative

operator) ... along with the "dynamic equation" dp_a/dt = f_a, as the

Euler-Lagrange equation.

The two sets of equations

Kinematic: dq^a/dt = v^a

Dynamic: dp_a/dt = f_a

and their corresponding set of variables provide a general framework for

the system's dynamics, while the

Constitutive Relations: f_a = @L/@q^a, p_a = @L/@v^a

fill in the gap on the detailed structure of the system being described.

If you rewrite this as a second order equation, with acceleration

components a = (a^a: 0 < a <= N), it would take the form:

dp_a/dt = f_a => f_a = m_{ab} a^b + s_{ab} v^b + @/@v^a (@L/@t)

(summation convention used here and below)

where

s_{ab} = @p_a/@q^b = @f_b/@v^a

and

m_{ab} = @p_a/@v^b = @p_b/@v^a = m_{ba}

being the *Coefficients Of Inertia*.

The remaining second order derivatives

k_{ab} = -@f_a/@q^b = -@f_b/@q^a = -k_{ba}

play the analogue of spring coefficients, particularly if the system is

near an equilibrium point.

So, in this sense, the inertia is a rank (0,2) configuration space tensor.

For a system composed of a single body, configuration space is confused

with geometric space, with N = 3 and the configuration coordinates (q^1,

q^2, q^3) being the body's spatial coordinates. In that context, the

coefficients of inertia form a 3 x 3 rank (0,2) tensor with respect to

spatial coordinates.

In a non-relativistic setting, in Cartesian coordinates, it is diagonal

and constant. For instance, a conservative system has constitutive laws of

the following form:

p_a = m delta_{ab} v^b, f_a = -@U/@q^a, U = U(q)

delta_{ab} = Kronecker delta (1 if a = b, 0 if a != b).

For non-Cartesian coordinates, it is not constant and need not be

diagonal. For cylindrical coordinates, the coefficients of inertia would

be a matrix of the form m = diag(m, m, m r^2), where r is the radial

coordinate.

In a relativistic setting, the v-dependent part of the Lagrangian has the

form

L = m v^2/(1 + sqrt(1 - A v^2)) + ..., A = 1/c^2

"sqrt" denotes the square root operation

so that

p_a = @L/@v^a = m delta_{ab} v^b/sqrt(1 - A v^2).

The coefficients of inertia then have the form

m_{ab} = M delta_{ab} + M A v_a v_b/(1 - A v^2)

v_a = delta_{ab} v^b

M = m/sqrt(1 - A v^2)

This decomposes into the "longitudinal" and "transverse" mass.

For a system with two or more bodies, the configuration space has N = 3n,

where n is the number of bodies, and the coefficients of inertia - as a

matrix - reduce to a block diagonal form consisting of n 3 x 3 blocks of

similar form as those just described.

Second ...

In Relativity, under an infinitesimal boost (by an infinitesimal boost

velocity upsilon), the total energy E and momentum p = (p_x, p_y, p_z)

transform as:

delta(E) = -upsilon.p,

delta(p) = -upsilon M

M = A E, A = 1/c^2,

().() = inner product

The transform has the following invariant:

E^2 - p^2/A

which characterizes the "rest mass" m, if it is positive, with:

E^2 - p^2/A = m^2/A^2.

The non-relativistic correspondence to this is arrived at by decomposing

the total energy E into the kinetic energy H and mass-energy mc^2 as:

E = H + m/A, i.e. M = m + A H.

The corresponding transforms, assuming m is invariant are

delta(m) = 0

delta(H) = -upsilon.p,

delta(p) = -upsilon M,

delta(M) = -A upsilon.p.

The transform has the following two invariants:

p^2 - 2 M H + A H^2 = p^2 - 2 m H - A H^2

m = M - A H

In the non-relativistic limit, this becomes:

p^2 - 2 M H = p^2 - 2 m H

m = M

and the transforms become

delta(H) = -upsilon.p,

delta(p) = -upsilon M,

delta(M) = 0.

The inertia "M" - in both the relativistic and non-relativistic settings -

is a component of the 5-vector (H, p_x, p_y, p_z, M). For relativity, the

5-vector splits into a 4-vector (E = H + M/a, p_x, p_y, p_z) and a 1-

vector (m = M - A H). The condition that the 1-vector be the norm of the

4-vector (with respect to the Minkowski metric) is:

p^2 - 2 M H + A H^2 = 0.

This can be actually be generalized to:

p^2 - 2 M H + A H^2 = -2 m U + A U^2

by allowing H to have an "internal" part, U:

H = m v^2/(1 + sqrt(1 - A v^2)) + U.

In that case, the invariant M - A H no longer coincides with the rest

mass, but has the form

M - A H = mu = m - A U,

the rest mass, itself, now decomposing into an invariant mass "mu" plus a

contribution from the internal energy:

m = mu + A U.

The non-relativistic counterpart, corresponding to setting A = 0, would

then be:

H = 1/2 m v^2 + U,

M = mu = m.

This is, strictly, *more* general than Special Relativity - the symmetry

group for the two invariants above is not the Poincare' group, but the

one-dimensional extension of it (which is where the linear invariant,

"mu", comes in). The latter is what has, as its non-relativistic limit,

the Bargmann group. The Bargmann group is the central extension of the

Galilei group and is more properly considered to be the symmetry group for

non-relativistic theory, than the Galilei group is, because Galilei

corresponds only to the case m = 0.

In Relativity, the coordinates have the following transform under

infinitesimal boosts:

delta(t) = -A upsilon.r,

delta(r) = -upsilon t,

r = (x, y, z).

Correspondingly, one also has the following transforms for the coordinate

differentials:

delta(dt) = -A upsilon.dr,

delta(dr) = -upsilon dt,

dr = (dx, dy, dz).

This leads to the invariants:

dt^2 - A dr^2 = dt^2 - A (dx^2 + dy^2 + dz^2)

E dt - p.dr

which pairs off, naturally, with the 4-vector (E, p) = (E, p_x, p_y, p_z).

There is no non-relativistic version of this, and if you attempt to create

one by replacing the total energy E with the kinetic+internal energy H as:

H dt - p.dr

then you would find that it has the following as its transform

delta(H dt - p.dr) = (M - A H) upsilon.dr = mu upsilon.dr.

It is not invariant, but strongly suggests the inclusion of another

coordinate "u" with the transform:

delta(u) = -upsilon.r,

delta(du) = -upsilon.dr,

that would make it invariant with:

delta(H dt - p.dr + mu du) = 0

This applies both relativistically and non-relativistically. In the non-

relativistic case, the extra coordinate yields a 4+1 dimensional geometry:

Bargmann geometry. The relativistic version of it has no standard name. It

is characterized by the invariants:

dx^2 + dy^2 + dz^2 + 2 dt du + A du^2,

ds = dt + A du

where "s" plays the role of an absolute time - a lingering vestige of the

absolute time of non-relativistic theory.

The inertia "m" (more accurately, "mu") then plays the role of being the

conjugate to the "u" coordinate. Requiring the dynamics of a system

described by a Lagrangian to be independent of the u-coordinate then leads

- by way of the Noether Theorem - the constancy of m.

An example of a Lagrangian for a free body would be given by the action

S = integral L dt = integral m (dr^2 + 2 dt du + A du^2)/ds

i.e.

L = m/2 (|dr/dt|^2 + 2 du/dt + A (du/dt)^2)/(1 + A du/dt)

where "m" is written in as a Lagrange multiplier.

The Euler-Lagrange equations can be reduced to:

d/dt (M dr/dt) = 0,

dm/dt = 0,

|dr/dt|^2 + 2 du/dt + A (du/dt)^2 = 0,

M = m/sqrt(1 - A |dr/dt|^2) = +/- m/(1 + A du/dt)

An alternative that includes an "internal energy" U would be

S = integral m/2 (dr^2 + 2 dt du + A du^2)/ds + U (ds - dt - A du)

written with "s" as the parameter:

S = integral

m/2 (|dr/ds|^2 + 2 dt/ds du/ds + A (du/ds)^2) ds

+ U (1 - dt/ds - A du/ds) ds

where "U" is written in as a second Lagrange multiplier.

The Euler-Lagrange equations reduce to:

dp/ds = 0, d(mu)/ds = 0, dH/ds = 0

where

p = m dr/ds = M dr/dt, where M = m dt/ds

H = -m du/ds + U = -M du/dt + U

mu = m (dt/ds + A du/ds) - A U = m - A U

dt/ds + A du/ds = 1, i.e. dt/ds (1 + A du/dt) = 1

|dr/ds|^2 + 2 dt/ds du/ds + A (du/ds)^2 = 0

i.e. (1 + A du/dt)^2 = 1 - A |dr/dt|^2

with

-m du/dt = M v^2/(1 + sqrt(1 - A |dr/dt|^2)

These work both relativistically (A = 1/c^2) and non-relativistically (A =

0) and provides a unified framework to cover all of the cases.