Physlib

Physlib.ClassicalMechanics.PointParticle.Basic

Point particles

This module defines point particles together with their motion relative to a reference frame. A `Particle` has a constant positive mass and a position over time. Velocity and acceleration are derived from that position rather than stored as independent data.

The trajectory is part of the particle's description, but need not be given by an explicit solution formula. Particle values can be considered subject to conditions on their positions and on the forces acting on them. A particular mechanical model can therefore be specified by constraints on particles, with the existence of particles satisfying those constraints established separately.

A particle by itself carries no equation of motion or assumption about which forces act on it. It is a constituent from which systems can be assembled, rather than a specification of an isolated or unconstrained one-particle system. Newton's laws are imposed when particles and forces are assembled in `ClassicalMechanics.PointParticle.NewtonianSystem`.

Position and its first time derivative are required to be differentiable when the frame is inertial. This ensures that the velocity and acceleration used in Newtonian systems are genuine derivatives. The trajectories in this definition are defined for all `Time`.

6 declarations

definition

The set of positive real numbers R+\mathbb{R}^+

The notation R+\mathbb{R}^+ denotes the set of strictly positive real numbers, defined as the subtype of real numbers x∈Rx \in \mathbb{R} such that x>0x > 0.

instance

Inertial Frame   ⟹  \implies Position is Differentiable

For any particle pp and reference frame ff, if ff is an inertial frame, then the position function pos:Time→f.Vector\text{pos} : \text{Time} \to f.\text{Vector} of the particle is differentiable with respect to time. Here, Time\text{Time} and f.Vectorf.\text{Vector} are treated as real vector spaces.

definition

Velocity of a particle v=drdt\mathbf{v} = \frac{d\mathbf{r}}{dt}

Given a particle in a reference frame ff with a position function r:Time→f.Vector\mathbf{r} : \text{Time} \to f.\text{Vector}, the velocity of the particle is the function v:Time→f.Vector\mathbf{v} : \text{Time} \to f.\text{Vector} defined as the time derivative of the position function: v(t)=drdt \mathbf{v}(t) = \frac{d\mathbf{r}}{dt} This definition is valid provided that the position function r\mathbf{r} is differentiable with respect to time.

instance

Inertial Frame   ⟹  \implies Velocity is Differentiable

For any particle and reference frame ff, if ff is an inertial frame, then the velocity function v:Time→f.Vector\mathbf{v} : \text{Time} \to f.\text{Vector} of the particle is differentiable with respect to time.

definition

Acceleration of a particle a=dvdt\mathbf{a} = \frac{d\mathbf{v}}{dt}

Given a particle in a reference frame ff with a velocity function v:Time→f.Vector\mathbf{v} : \text{Time} \to f.\text{Vector}, the acceleration of the particle is the function a:Time→f.Vector\mathbf{a} : \text{Time} \to f.\text{Vector} defined as the time derivative of the velocity: a(t)=dvdt \mathbf{a}(t) = \frac{d\mathbf{v}}{dt} This definition is valid provided that both the position function r\mathbf{r} and the velocity function v\mathbf{v} of the particle are differentiable with respect to time.

definition

Particle's position in Space d\text{Space } d at time tt

For a particle moving relative to a reference frame, this function computes its position in the dd-dimensional flat Euclidean space Space d\text{Space } d at a given time tt. The position is defined as the point in Space d\text{Space } d resulting from the additive action of the displacement vector (corresponding to the particle's internal coordinate position pos(t)\text{pos}(t)) on the frame's origin point origin(t)\text{origin}(t): pointInSpace(t)=dispEquivt(pos(t))+vorigin(t) \text{pointInSpace}(t) = \text{dispEquiv}_t(\text{pos}(t)) +_{\text{v}} \text{origin}(t) where dispEquivt\text{dispEquiv}_t is the linear equivalence that maps the frame's coordinate vectors to physical displacements at time tt.