This note summarizes the geometric
structure implied by the finite–window
Little identity
\[
L(T) = \Lambda(T)\,w(T)
\] for all observation windows
\(T>0\).
We examine the argument in two
domains:
Assume
\[
L(T)=\Lambda(T)\,w(T)
\] for all \(T>0\).
Fix any two endpoints \(0<T_1<T_2\)
and write
\[
L_i=L(T_i),\qquad
\Lambda_i=\Lambda(T_i),\qquad w_i=w(T_i)
\] with finite increments
\[
\Delta L=L_2-L_1,\qquad
\Delta\Lambda=\Lambda_2-\Lambda_1,\qquad
\Delta w = w_2-w_1.
\]
Using
\[
L_1=\Lambda_1 w_1,\qquad L_2=\Lambda_2
w_2,
\] subtracting gives the exact
identity
\[
\Delta L=\Lambda_1\,\Delta w \;+\;
w_1\,\Delta\Lambda \;+\;
\Delta\Lambda\,\Delta w.
\]
This shows that \((\Delta
L,\Delta\Lambda,\Delta w)\) cannot
vary independently.
The coupling term \(\Delta\Lambda\,\Delta
w\) reflects the curvature of the
surface
\[
L = \Lambda w.
\]
When \(T_2\) is
close to \(T_1\), the
product term is second order,
yielding
\[
\Delta L \approx \Lambda_1\,\Delta w +
w_1\,\Delta\Lambda.
\]
This is the equation of the
tangent plane to the
nonlinear surface
\[
L = \Lambda w
\] at the point \((\Lambda_1,w_1,L_1)\).
Small finite–window steps therefore move
approximately within this plane.
For smooth trajectories, taking \(T_2\to T_1\)
gives the exact differential
constraint
\[
\frac{dL}{dT}
=
w(T)\,\frac{d\Lambda}{dT}
+
\Lambda(T)\,\frac{dw}{dT}.
\]
This is the total derivative of \(L=\Lambda
w\).
In real space the geometry is therefore
curved, with linear
behavior obtained only locally via the
tangent plane.
Define the log–coordinates
\[
\ell = \log L,\qquad a =
\log\Lambda,\qquad r = \log w.
\]
Applying \(\log\) to the
finite–window identity gives
\[
\ell = a + r.
\]
This is the equation of a plane in \((a,r,\ell)\)–space.
Let
\[
P_1=(a_1,r_1,\ell_1),\qquad
P_2=(a_2,r_2,\ell_2)
\] be any two points satisfying
\(\ell=a+r\),
and define increments
\[
\Delta a=a_2-a_1,\qquad \Delta
r=r_2-r_1,\qquad \Delta\ell=\ell_2-\ell_1.
\]
Because both points lie on the plane,
we have
\[
\Delta\ell=\Delta a+\Delta r.
\]
This holds for
arbitrary finite steps:
large windows, small windows, stable or
unstable processes.
The geometry is globally linear.
Differentiating \(\ell=a+r\)
gives
\[
\frac{d\ell}{dT}
=
\frac{da}{dT}
+
\frac{dr}{dT}.
\]
Equivalently, in terms of the original
variables,
\[
\frac{1}{L}\frac{dL}{dT}
=
\frac{1}{\Lambda}\frac{d\Lambda}{dT}
+
\frac{1}{w}\frac{dw}{dT}.
\]
This expresses relative (logarithmic) changes as a simple additive decomposition, in contrast to the nonlinear real–space identity.
In log space:
Thus the curved geometry of real space collapses into a globally linear structure in log space.
In real space,
\(L=\Lambda
w\) defines a curved surface and
the increments satisfy
\[
\Delta L=\Lambda_1\,\Delta
w+w_1\,\Delta\Lambda+\Delta\Lambda\,\Delta
w,
\] with the differential
equation
\[
\frac{dL}{dT}=w\,\frac{d\Lambda}{dT}+\Lambda\,\frac{dw}{dT}.
\]
In log space,
\(\ell=a+r\)
defines a plane and all increments satisfy
the exact identity
\[
\Delta\ell=\Delta a+\Delta r,
\] with the differential
version
\[
\frac{d\ell}{dT}=\frac{da}{dT}+\frac{dr}{dT}.
\]
Real space is nonlinear, curved, and
locally constrained.
Log space is globally linear, additive,
and geometrically flat.
We assume the finite-window identity holds for every prefix length \(T>0\): \[ L(T) = \Lambda(T)\,w(T). \] Here \(L(T)\), \(\Lambda(T)\), and \(w(T)\) are functionals of the same underlying sample path: for each \(T\) they are computed deterministically from the events in \([0,T]\).
For each \(T\), define \[ f(T) := L(T),\quad g(T) := \Lambda(T),\quad h(T) := w(T). \] The finite-window identity is simply \[ f(T) = g(T)\,h(T)\quad\text{for all }T>0. \]
Assume \(f,g,h\) are differentiable at some \(T\). Then we can apply the ordinary product rule to \(f(T) = g(T)\,h(T)\): \[ \frac{df}{dT}(T) = \frac{d}{dT}\bigl(g(T)\,h(T)\bigr) = g(T)\,\frac{dh}{dT}(T) + h(T)\,\frac{dg}{dT}(T). \] Now substitute back \(f=L\), \(g=\Lambda\), and \(h=w\): \[ \frac{dL}{dT}(T) = \Lambda(T)\,\frac{dw}{dT}(T) + w(T)\,\frac{d\Lambda}{dT}(T). \]
Fix \(T\) and a small increment \(\Delta T>0\). Using the identity at \(T\) and \(T+\Delta T\): \[ L(T) = \Lambda(T)\,w(T),\qquad L(T+\Delta T) = \Lambda(T+\Delta T)\,w(T+\Delta T). \] Define finite differences \[ \Delta L = L(T+\Delta T)-L(T),\quad \Delta\Lambda = \Lambda(T+\Delta T)-\Lambda(T),\quad \Delta w = w(T+\Delta T)-w(T). \] Then \[ \Delta L = \Lambda(T)\,\Delta w + w(T)\,\Delta\Lambda + \Delta\Lambda\,\Delta w. \] Divide by \(\Delta T\): \[ \frac{\Delta L}{\Delta T} = \Lambda(T)\,\frac{\Delta w}{\Delta T} + w(T)\,\frac{\Delta\Lambda}{\Delta T} + \frac{\Delta\Lambda}{\Delta T}\,\Delta w. \] If \(L,\Lambda,w\) are differentiable at \(T\), then as \(\Delta T\to 0\) we have \(\Delta w\to 0\) and \[ \frac{\Delta L}{\Delta T}\to\frac{dL}{dT}(T),\quad \frac{\Delta\Lambda}{\Delta T}\to\frac{d\Lambda}{dT}(T),\quad \frac{\Delta w}{\Delta T}\to\frac{dw}{dT}(T), \] while the mixed term vanishes: \[ \frac{\Delta\Lambda}{\Delta T}\,\Delta w \;\longrightarrow\; 0. \] Taking the limit \(\Delta T\to 0\) yields \[ \frac{dL}{dT}(T) = \Lambda(T)\,\frac{dw}{dT}(T) + w(T)\,\frac{d\Lambda}{dT}(T), \] which is the desired governing differential equation for the co-evolution of the three functionals.
We start from the identities at two window endpoints: \[ L_1 = \Lambda_1 w_1,\qquad L_2 = \Lambda_2 w_2. \]
Define the finite differences: \[ \Delta L = L_2 - L_1,\qquad \Delta\Lambda = \Lambda_2 - \Lambda_1,\qquad \Delta w = w_2 - w_1. \]
Write the second point in terms of the first point and increments: \[ \Lambda_2 = \Lambda_1 + \Delta\Lambda,\qquad w_2 = w_1 + \Delta w. \]
Substitute into \(L_2 = \Lambda_2 w_2\): \[ L_2 = (\Lambda_1 + \Delta\Lambda)(w_1 + \Delta w). \]
Expand the product: \[ (\Lambda_1 + \Delta\Lambda)(w_1 + \Delta w) = \Lambda_1 w_1 + \Lambda_1 \Delta w + w_1 \Delta\Lambda + \Delta\Lambda\,\Delta w. \]
Subtract \(L_1 = \Lambda_1 w_1\) to obtain the finite-difference identity: \[ \Delta L = \Lambda_1\,\Delta w + w_1\,\Delta\Lambda + \Delta\Lambda\,\Delta w. \]
Let the state of the finite–window metrics at time \(T\) be represented by the vector \[ X(T) = \begin{bmatrix} L(T) \Lambda(T) w(T) \end{bmatrix}. \]
For any two window endpoints \(T_1 < T_2\), define the increment tensor \[ \Delta X = \begin{bmatrix} \Delta L \Delta\Lambda \Delta w \end{bmatrix} = \begin{bmatrix} L_2 - L_1 \Lambda_2 - \Lambda_1 w_2 - w_1 \end{bmatrix}. \]
The finite–window identity \[ L = \Lambda\,w \] can be expressed as the bilinear form \[ F(X) = X_2\,X_3 - X_1 = 0, \] with \(X_1=L\), \(X_2=\Lambda\), \(X_3=w\).
The first–order expansion (tensor form of the differential) is \[ DF(X_1)\,\Delta X = \frac{\partial F}{\partial X_1}\Delta X_1 + \frac{\partial F}{\partial X_2}\Delta X_2 + \frac{\partial F}{\partial X_3}\Delta X_3. \]
Compute the gradient: \[ \nabla F(X_1) = \begin{bmatrix} -1 w_1 \Lambda_1 \end{bmatrix}. \]
Thus the constraint on increments is \[ \nabla F(X_1)^{\!\top}\,\Delta X = -\,\Delta L + w_1\,\Delta\Lambda + \Lambda_1\,\Delta w. \]
Because both \(X_1\) and \(X_2\) satisfy \(F(X)=0\), the linearized constraint must equal the second–order remainder: \[ \nabla F(X_1)^{\!\top}\,\Delta X = -\,\Delta\Lambda\,\Delta w. \]
Rearranging yields the finite-difference tensor identity: \[ \Delta L = \Lambda_1\,\Delta w + w_1\,\Delta\Lambda + \Delta\Lambda\,\Delta w. \]
This expression decomposes the increment \(\Delta L\) into a first–order multilinear term \[ (\nabla F(X_1))^\top \Delta X = \Lambda_1\,\Delta w + w_1\,\Delta\Lambda \] and a symmetric second–order tensor correction \[ \Delta\Lambda\,\Delta w. \]
This document derives the Hamiltonian mechanics for a flow system governed by the finite-window Little’s Law. We treat the system as a dynamic particle constrained to the manifold defined by the identity.
We define the generalized coordinates \(q\) in Log Space, where the geometry is globally linear and additive.
Let the independent coordinates be: \[ q_1 = a = \log \Lambda \quad (\text{Arrival Intensity}) \] \[ q_2 = r = \log w \quad (\text{Service Intensity}) \]
The dependent coordinate is \(\ell = \log L\). The system is subject to the holonomic constraint: \[ \ell(a, r) = a + r \]
This defines a flat 2D plane in the 3D configuration space \((a, r, \ell)\).
We assume a “free particle” model where the system has inertia (resistance to infinite volatility) but no external potential (\(V=0\)) acting on it yet. The Lagrangian is the kinetic energy of the system.
Using the differential form \(\dot{\ell} = \dot{a} + \dot{r}\), the kinetic energy \(T\) is:
\[ T = \frac{1}{2}m \left( \dot{a}^2 + \dot{r}^2 + \dot{\ell}^2 \right) \]
Substituting the constraint \(\dot{\ell} = \dot{a} + \dot{r}\):
\[ \mathcal{L}(q, \dot{q}) = \frac{m}{2} \left[ \dot{a}^2 + \dot{r}^2 + (\dot{a} + \dot{r})^2 \right] \]
Expanding the square term yields the Lagrangian with an inertial coupling term:
\[ \mathcal{L} = m \left( \dot{a}^2 + \dot{r}^2 + \dot{a}\dot{r} \right) \]
Physical Interpretation: The cross-term \(m\dot{a}\dot{r}\) arises because the geometry of the finite-window identity couples the fluctuations of arrival rates and service times.
We derive the generalized momenta \(p_i = \frac{\partial \mathcal{L}}{\partial \dot{q}_i}\):
\[ p_a = \frac{\partial \mathcal{L}}{\partial \dot{a}} = m(2\dot{a} + \dot{r}) \]
\[ p_r = \frac{\partial \mathcal{L}}{\partial \dot{r}} = m(2\dot{r} + \dot{a}) \]
In matrix form \(p = M v\):
\[ \begin{bmatrix} p_a \\ p_r \end{bmatrix} = m \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} \dot{a} \\ \dot{r} \end{bmatrix} \]
The Hamiltonian \(H\) represents the total energy of the flow. It is obtained via the Legendre transform:
\[ H(q, p) = p^\top \dot{q} - \mathcal{L} \]
First, we invert the mass matrix \(M\) to express velocities in terms of momenta. The inverse of \(M\) is:
\[ M^{-1} = \frac{1}{3m} \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \]
The Hamiltonian for a quadratic kinetic energy is given by \(H = \frac{1}{2} p^\top M^{-1} p\):
\[ H(p_a, p_r) = \frac{1}{6m} \left[ \begin{matrix} p_a & p_r \end{matrix} \right] \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} p_a \\ p_r \end{bmatrix} \]
Expanding this yields:
\[ H = \frac{1}{3m} \left( p_a^2 + p_r^2 - p_a p_r \right) \]
The evolution of the system in phase space is governed by the symplectic gradients.
\[ \dot{a} = \frac{\partial H}{\partial p_a} = \frac{1}{3m}(2p_a - p_r) \] \[ \dot{r} = \frac{\partial H}{\partial p_r} = \frac{1}{3m}(2p_r - p_a) \]
This highlights the coupling effect: The rate of change of the arrival intensity (\(\dot{a}\)) is dampened by the momentum of the service duration (\(p_r\)).
Since the Hamiltonian is translationally invariant in log-space (no dependence on \(a\) or \(r\) explicitly):
\[ \dot{p}_a = -\frac{\partial H}{\partial a} = 0 \implies p_a = \text{constant} \] \[ \dot{p}_r = -\frac{\partial H}{\partial r} = 0 \implies p_r = \text{constant} \]
By mapping the finite-window identity to log-space, we have successfully formulated the system as a free particle on a constrained plane. The Hamiltonian \(H = \frac{1}{3m} (p_a^2 + p_r^2 - p_a p_r)\) reveals that the “queueing fluid” has a non-diagonal mass tensor, implying that fluctuations in arrival rates and service times are inertially entangled.
This note clarifies the relationship between presence mass from Presence Calculus and the mass parameter that appears in the Hamiltonian formulation of the finite-window identity. The result is that they coincide naturally: presence mass is the correct choice for Hamiltonian mass because both quantities encode inertia of change along the prefix window.
In Hamiltonian and Lagrangian mechanics, mass represents inertia:
For generalized coordinates \(q = (q_1, q_2)\), the kinetic energy is \[ T = \frac{1}{2}\,\dot{q}^\top M\,\dot{q}, \] where \(M\) is the mass matrix. Larger mass means slower response to forces and perturbations.
Presence mass in Presence Calculus measures:
As the window \(T\) grows, presence mass increases; consequently:
This is exactly the property of mass in mechanics.
The finite-window identity in log space, \[ \ell = a + r, \] defines a holonomic constraint on the system. The Lagrangian for a free particle moving on this constraint manifold is \[ \mathcal{L} = \frac{1}{2}m \left( \dot{a}^2 + \dot{r}^2 + \dot{\ell}^2 \right), \] with \(\dot{\ell} = \dot{a} + \dot{r}\). Substituting gives the reduced Lagrangian \[ \mathcal{L} = m\left(\dot{a}^2 + \dot{r}^2 + \dot{a}\dot{r}\right). \]
Here the parameter \(m\) governs:
These effects correspond exactly to the inertia induced by accumulated presence.
Thus, the mathematical role played by \(m\) is identical to the operational meaning of presence mass.
Under this identification:
This matches the classical mechanical interpretation of mass as resistance to acceleration.
Presence mass is Hamiltonian mass.
It determines:
This equivalence is not metaphorical. It follows directly from:
Presence mass therefore provides the canonical physical interpretation of the mass parameter in the Hamiltonian formulation of finite-window Little dynamics.