Mechanics of particles and systems Definition

Classical mechanics

Builds on Euclidean space Ordinary differential equation Picard–Lindelöf theorem

Definition

A Newtonian system of $N$ point particles consists of the following data.

Masses $m_1, \dots, m_N > 0$ .
For each $i$ , a position $x_i(t) \in \R^3$ at time $t$ , where $\R^3$ is three-dimensional Euclidean space.
For each $i$ , a force law $F_i(t, x, \dot x) \in \R^3$ , a function of the time $t$ , of all positions $x = (x_1, \dots, x_N)$ and of all velocities $\dot x = (\dot x_1, \dots, \dot x_N)$ .

A motion of the system is a twice differentiable map $t \mapsto (x_1(t), \dots, x_N(t))$ on an interval that solves the ordinary differential equations $m_i \ddot x_i(t) = F_i\big(t, x(t), \dot x(t)\big), \qquad i = 1, \dots, N.$ Classical mechanics, in its Newtonian form, is the study of such systems and their motions.

Remarks

Example: one particle of mass $m$ under the constant force $F = (0, 0, -mg)$ with $g > 0$ . The map $x(t) = (t, 0, -g t^2/2)$ is a motion, since $m \ddot x = (0, 0, -mg) = F$ . Non-example: the map $x(t) = (t, 0, 0)$ is not a motion of this system, since $m \ddot x = 0 \ne F$ . If the force laws are locally Lipschitz, the Picard–Lindelöf theorem shows that the initial positions and velocities determine the motion on some interval. A dot over a symbol denotes the derivative with respect to time.

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Differential equation concept
Euclidean space object
Function concept
Interval concept
Ordinary differential equation concept
Ordinary differential equation definition
Lipschitz condition definition
Solution of an ordinary differential equation definition
Initial value problem definition
Picard–Lindelöf hypotheses definition
Picard operator definition
Integral equation form of an initial value problem statement
An iterate of the Picard operator is a contraction statement
Picard–Lindelöf theorem statement
Classical mechanics definition

Where this leads

96 pages build on this one.

Tag	`SH7J0X3NCC`
URI	`m2://SH7J0X3NCC`
Kind	definition
Writing	W0 stub
MSC	70A05, 70F10
Defines	Classical mechanics
Uses	Euclidean space, Ordinary differential equation, Picard–Lindelöf theorem
Changed	3 November 2026

Classical mechanics

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