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Time

Time is the fundamental dimension used to describe the progression of motion and the relationship between displacement, velocity, and acceleration.
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The statement of the theorem

Let MM be the state space of the physical system, and let x:I→M\boldsymbol{x}: I \to M be the trajectory, where I=[t0,tf]⊂RI = [t_0, t_f] \subset \mathbb{R} is the time interval. Define the time parameter tt such that it induces a differentiable structure on the system's evolution. The state x(t)∈Rn\boldsymbol{x}(t) \in \boldsymbol{R}^n is governed by the system of first-order ordinary differential equations (ODEs): \begin{equation} \frac{d\boldsymbol{x}}{dt} = \boldsymbol{f}(\boldsymbol{x}, t) \end{equation} where f:M×I→Rn\boldsymbol{f}: M \times I \to \mathbb{R}^n is the vector field representing the instantaneous rate of change of the state. The time tt is the independent variable parameterizing the flow ρt(x0)\boldsymbol{\rho}_t(\boldsymbol{x}_0) generated by the vector field f\boldsymbol{f}, satisfying the initial value problem x(t0)=x0\boldsymbol{x}(t_0) = \boldsymbol{x}_0. Furthermore, the time elapsed Δt\Delta t between two states x(t1)\boldsymbol{x}(t_1) and x(t2)\boldsymbol{x}(t_2) is defined by the integral of the unit time measure: \begin{equation} \Delta t = \int_{t_1}^{t_2} dt \end{equation} such that t2>t1t_2 > t_1 implies Δt>0\Delta t > 0.