Post by Luigi FortunatiIn the case of my animation
https://www.geogebra.org/m/mjnqb8vk
[etc.]
There are two answers to consider.
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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.