I am interested in knowing whether something along the lines of the "diagonal variation" defined below has been studied before. In spirit, the basic idea is that it is a kind of generalisation of expressions of the form $$ \int_a^b \left. \frac{\partial F(\tau,t)}{\partial \tau} \right|_{\tau\,=\,t} \, dt $$ to situations where $F$ is not smooth.
After presenting my formal definition of "diagonal variation", I will explain my original motivation for being interested in it; but I am curious as to whether the idea has already found application in pure and/or applied mathematics.
Let $a,b \in \mathbb{R}$ with $a<b$.
A sample of $[a,b]$ is a finite list $\,\mathcal{S}=\big([s_i,t_i]\big)_{i=1}^N\,$ of intervals $[s_i,t_i]$ with $\,a \leq s_i < t_i \leq b$.
A uniform fine-sampling of $[a,b]$ is a sequence $\,(\mathcal{S}^{(n)})_{n \in \mathbb{N}}\,$ of samples $\,\mathcal{S}^{(n)}=\big([s_i^{(n)},t_i^{(n)}]\big)_{i=1}^{N^{(n)}}\,$ of $[a,b]$ with the properties that $$ \max_{i \in \{1,\ldots,N^{(n)}\}} (t_i^{(n)} - s_i^{(n)}) \to 0 \qquad \text{as } n \to \infty $$ and for all $s,t$ with $\,a \leq s < t \leq b$, $$ \frac{\sum_{i \in J_{s,t,n}} (t_i^{(n)} - s_i^{(n)})}{\sum_{i=1}^{N^{(n)}} (t_i^{(n)} - s_i^{(n)})} \ \to \ \frac{t-s}{b-a} \qquad \text{as } n \to \infty $$ where $\,J_{s,t,n}:=\Big\{i \in \{1,\ldots,N^{(n)}\} : [s_i^{(n)},t_i^{(n)}] \subset [s,t] \Big\}$.
Remark. The most canonical example of a sample of $[a,b]$ is a partition of $[a,b]$, and it is not hard to show that a sequence of partitions of $[a,b]$ is a uniform fine-sampling of $[a,b]$ if and only if the mesh tends to $0$. [In fact, we could choose to weaken the definitions that follow by only considering sequences of partitions; but it's hard to see a "physical" justification for only considering perfectly adjacent non-overlapping intervals in those definitions.]
Now suppose we have a function $F \colon [a,b] \times [a,b] \to \mathbb{R}$.
Given a sample $\,\mathcal{S}=\big([s_i,t_i]\big)_{i=1}^N\,$ of $[a,b]$ and a value $\lambda \in [0,1]$, we define $$ V_\lambda^{a,b}(F,\mathcal{S}) \, = \, (b-a)\frac{\sum_{i=1}^N (F(t_i,u_{\lambda,i}) - F(s_i,u_{\lambda,i}))}{\sum_{i=1}^N (t_i-s_i)} $$ where $\,u_{\lambda,i}:=(1-\lambda)s_i+\lambda t_i$.
Remark. In the case that $\mathcal{S}$ is a partition, the sum in the denominator is simply equal to $b-a$ and so the definition simplifies to $$ V_\lambda^{a,b}(F,\mathcal{S}) \, = \, \sum_{i=1}^N (F(t_i,u_{\lambda,i}) - F(s_i,u_{\lambda,i})). $$
Given $\lambda \in [0,1]$, we say that $F$ admits a $\lambda$-diagonal variation on $[a,b]$ if there exists $I \in \mathbb{R}$ such that for any uniform fine-sampling $\,(\mathcal{S}^{(n)})_{n \in \mathbb{N}}\,$ of $[a,b]$, $$ V_\lambda^{a,b}(F,\mathcal{S}^{(n)}) \to I \quad \text{as } n \to \infty. $$ In this case, we refer to $I$ as the $\lambda$-diagonal variation of $F$ on $[a,b]$.
Remark. If $F$ is bounded, then I believe the requirement "$\max_{i \in \{1,\ldots,N^{(n)}\}} (t_i^{(n)} - s_i^{(n)}) \to 0$" could be dropped from the definition of a uniform fine-sampling without affecting the definition of a $\lambda$-diagonal variation.
I believe that if $F$ is $C^1$ then the $\lambda$-diagonal variation of $F$ on $[a,b]$ will just be $$ I = \int_a^b f_1(t,t) \, dt $$ where $\nabla F=:\begin{pmatrix} f_1 \\ f_2 \end{pmatrix}$, i.e. $f_1$ denotes the partial derivative of $F$ with respect to its first argument. More generally, I think we have the following:
Theorem (I think). Suppose $F$ is continuous on $[a,b] \times [a,b]$, and is also
- $C^1$ on $\{(\tau,t) : a \leq \tau \leq t \leq b\}$, with gradient $\,\nabla F =: \begin{pmatrix} f_1^- \\ f_2^+ \end{pmatrix}$
- $C^1$ on $\{(\tau,t) : a \leq t \leq \tau \leq b\}$, with gradient $\,\nabla F =: \begin{pmatrix} f_1^+ \\ f_2^- \end{pmatrix}$.
Then for every $\lambda \in [0,1]$, $F$ admits a $\lambda$-diagonal variation on $[a,b]$, the value of which is given by $$ I = \int_a^b (1-\lambda)f_1^+(t,t) + \lambda f_1^-(t,t) \, dt. $$
But of course, there may exist circumstances outside the setting of the above theorem where $F$ still admits a $\lambda$-diagonal variation.
My original motivation.
Given a rocket for which the emission rate, the emission velocity, and the net external force are all continuous functions of time, the "thrust force exerted by the fuel on the rocket" is given by the product of the emission rate and the emission velocity.
The derivation of this formula typically involves considering a transfer of momentum over an "infinitesimal time-increment". Of course, what is not so satisfying about such a derivation is that it imagines time as evolving in discrete infinitesimal increments and imagines forces as taking a value at each such time-increment, whereas in reality [by which I mean, the reality of the Newtonian model of the universe, not necessarily the reality of the universe itself], time proceeds as a continuum and the forces evolve as functions of this continuous time.
So I was interested in the question of how to define the thrust force and derive its formula, all explicitly with reference to a timeline modelled rigorously by the real line $\mathbb{R}$. The naïve approach would be to take the definition of a force as an instantaneous rate of transfer of momentum from one body of matter to another, where "instantaneous rate" is defined as a time-derivative of time-cumulative momentum transferred, and then to try to treat the "thrust force" as a force according to this definition applied directly to the idealised description of a rocket that is presupposed in the formula. But the non-smoothness that is inherent to the idealised description [where the fuel undergoes an instantaneous change in velocity] causes this approach to fail.
So I've thought of a few ways to try and understand the thrust force "rigorously":
Assume a one-dimensional setup for simplicity. Fixing a start time $a \in \mathbb{R}$, if we let $F(\tau,t)$ denote the momentum transferred over the time-interval $[a,\tau]$ from the body consisting of all the fuel whose emission time is $\leq t$ to the body consisting of all the fuel whose emission time is $>t$, then $F$ fulfils the conditions of the above theorem [for fixed $b>a$], with \begin{align*} f_1^-(t,t) &= \text{the desired formula} \\ f_1^+(t,t) &= 0. \end{align*}
[In full: Letting $m \colon [a,b] \to (0,\infty)$ [assumed to be $C^1$] be the mass of the rocket, and letting $v_e \colon [a,b] \to (-\infty,0]$ [assumed to be continuous] be the emission velocity, the above $\text{desired formula}$ is $\dot{m}(t)v_e(t)$, and we have $$ F(\tau,t) = \begin{cases} m(a)v(a) - m(\tau)v(\tau) + m(t)(v(\tau)-v(a)) - \int_a^\tau (v(s)+v_e(s))\dot{m}(s) \, ds & a \leq \tau \leq t \\ (m(a)-m(t))v(a) - \int_a^t (v(s)+v_e(s))\dot{m}(s) \, ds & a \leq t \leq \tau \end{cases} $$ where $v \colon [a,b] \to \mathbb{R}$ is the velocity of the rocket, which (due to conservation of momentum) fulfils the equation $$ \dot{v}(t) = \frac{\mathfrak{f}(t) + \dot{m}(t)v_e(t)}{m(t)}, \quad t \geq a $$ where $\mathfrak{f} \colon [a,b] \to \mathbb{R}$ [assumed to be continuous] is the net external force on the rocket.]
Incidentally, the fact that $f_1^+(t,t)$ and $f_1^-(t,t)$ disagree with each other illustrates how the naïve approach to rigorously defining the thrust force fails. Of my ideas for rigorously defining the thrust force, the one that is probably the most tenuous from a physics point-of-view and yet is also probably closest in spirit to the handwavy "infinitesimals" approach is:
Consider the "time-cumulative momentum transferred to the rocket by the fuel as it is in the process of being forced out of the rocket", which we define rigorously as the $1$-diagonal variation of $F$ on $[a,t]$; then the time-derivative of this can be what we define the thrust force to be, and the above Theorem together with the fundamental theorem of calculus then gives that this is well-defined and equal to the desired formula.
