Mathematical methods of classical mechanics, part 2

Part 1 is here.

C Measures of motion

A motion in RN is a differentiable mapping x: I → RN, where I is an interval on the real axis.

The derivative

\dot{\mathbf{x}}(r_0)=\left .\frac{d\mathbf{x}}{dr} \right |_{r=r_0}=\lim_{h \to 0}\frac{\mathbf{x}(r_0+h)-\mathbf{x}(r_0)}{h}\in \mathbb{R}^N

is called the velocity vector at the point r0I.

The second derivative

\ddot{\mathbf{x}}(r_0)=\left .\frac{d^2\mathbf{x}}{dr^2} \right |_{r=r_0}

is called the acceleration vector at the point r0.

We will assume that the functions we encounter are continuously differentiable as many times as necessary. In the future, unless otherwise stated, mappings, functions, etc. are understood to be differentiable mappings, functions, etc. The image of a mapping x: I → RN is called a trajectory or curve in RN.

We now define a mechanical system of n points moving in three-dimensional euclidean space.

Let x: R → R3 be a motion in R3. The graph of this mapping is a curve in R x R3.

A curve in galilean space which appears in some (and therefore every) galilean coordinate system as the graph of a motion, is called a world line.

A motion of a system of n points gives, in galilean space, n world lines. In a galilean coordinate system they are described by n mappings xi: R → R3, i = 1, … , n.

The direct product of n copies of R3 is called the configuration space of the system of n points. Our n mappings xi: R → R3, define one mapping

x: R → RN such that N = 3n

of the time axis into the configuration space. Such a mapping is also called a motion of a system of n points in the galilean coordinate system on R × R3.

D Newton’s equations

According to Newton’s principle of determinacy (Section 1C) all motions of a system are uniquely determined by their initial positions (x(r0) ∈ RN) and initial velocities (x.(r0) ∈ RN).

In particular, the initial positions and velocities determine the acceleration. In other words, there is a function F: RN x RN x R → RN such that

(1)\mathbf{\ddot{x}}=(\mathbf{x, \dot{x}},r).

Newton used Equation (1) as the basis of mechanics. It is called Newton’s equation.

By the theorem of existence and uniqueness of solutions to ordinary differential equations, the function F and the initial conditions x(r0) and x.(r0) uniquely determine a motion.

For each specific mechanical system the form of the function F is determined experimentally. From the mathematical point of view the form of F for each system constitutes the definition of that system.

E Constraints imposed by the principle of relativity

Galileo’s principle of relativity states that in physical space-time there is a selected galilean structure (“the class of inertial coordinate systems”) having the following property.

If we subject the world lines of all the points of any mechanical system to one and the same galilean transformation, we obtain world lines of the same system (with new initial conditions).

This imposes a series of conditions on the form of the right-hand side of Newton’s equation written in an inertial coordinate system: Equation (1) must be invariant with respect to the group of galilean transformations.

From invariance under passage to a uniformly moving coordinate system (which does not change x: or xixk, but adds to each x.i a fixed vector v) it follows that the right-hand side of Equation (1) in an inertial system of coordinates can depend only on the relative velocities

\ddot{\mathbf{x}}=\mathbf{f}_i({\mathbf{x}_j-\mathbf{x}_k,\dot{\mathbf{x}}_j-\dot{\mathbf{x}}_k}),\; \; i,j,k=1,...,n.

PROBLEM. Is the class of inertial systems unique?
ANSWER. No. Other classes can be obtained if one changes the units of length and time or the direction of time.