Lewe 1906 – The Sudden Fixations of a Rigid Body
A clear reading for curious minds
Original Documents located here
Lewe, 1906 is pure rigid-body dynamics. He asks what happens when one or more points (or a whole line) of a moving rigid body are suddenly forced to stop or to take on a new prescribed velocity. The forces that appear are impulsive — infinitely large for an infinitely short time — so they change velocities instantly but leave the positions of every point unchanged at the instant of the impulse.
The whole paper is built on vector analysis. Lewe systematically answers three questions:
- What is the new motion of the body after the sudden constraint?
- What impulsive forces (and moments) act during the constraint?
- How does the kinetic energy change?
He works throughout with the linear momentum of the centre of mass and the angular momentum about a chosen point, both written in vector form. Because the impulses act only for an infinitesimal time, ordinary finite forces can be ignored.
Core physical ideas
An impulsive force is defined as the time-integral of an ordinary force taken over an interval that shrinks to zero while the force itself becomes infinite, yet the integral stays finite. The result is a sudden jump in velocity with no simultaneous jump in position.
For a rigid body the two fundamental statements are therefore:
- The change in linear momentum of the centre of mass equals the sum of all external impulses.
- The change in angular momentum about any point equals the sum of the moments of those impulses about the same point.
When a single point O of the body is suddenly fixed, the new motion can only be a pure rotation about some axis through O. The angular-momentum equation about O then determines the new angular-velocity vector completely. The calculation is simplest when the coordinate axes are the principal axes of inertia at O; the three components of the new angular velocity then decouple.
When a whole straight line (or two distinct points) is suddenly fixed, the body is forced to rotate about that line. The component of angular momentum along the line is conserved (because the impulsive reactions have no moment about the line). The magnitude of the new angular velocity follows at once from the moment of inertia about the fixed line.
Any more general sudden change of velocity of one or two points can be reduced to a pure fixation problem by viewing the motion relative to a suitably chosen moving frame. The Coriolis theorem, proved afresh in vector language, guarantees that the angular-momentum balance retains the same form in the moving frame, so the fixation formulae already derived can be applied directly.
Kinetic energy
Lewe re-derives, in compact vector form, the classical statements about the loss of kinetic energy in an impulsive constraint (Thomson–Tait, Carnot, Appell). He also obtains a short generalisation that separates the kinetic energy of the relative motion from the kinetic energy of the motion of the reference frame. The proofs are deliberately short; the vector notation makes the cancellations transparent.
What the paper does not do
It does not treat deformable bodies, friction, or any continuum model of elasticity. It stays strictly inside the dynamics of a single rigid body subjected to impulsive constraints. The methods are those of classical analytical mechanics written in the then-modern language of vectors.
This is the physics Lewe presented in 1906: a clean, vector-based account of how a rigid body responds when points or lines on it are suddenly fixed.
A brief walk-through of Lewe’s method (1906)
Showing each step the way he made it transparent
Lewe’s procedure is deliberately sequential. At every stage he states what is known, what is unknown, and which vector equation supplies the missing piece.
Step 1 – Define the impulsive force
An ordinary force P acting for a very short time Δt produces an impulse
G = integral of P dt
taken while Δt → 0 and P → ∞, yet G remains finite.
The impulse changes velocity but leaves position unchanged at the instant it acts.
Step 2 – Write the two global balance laws for the rigid body
- Change in linear momentum of the centre of mass = sum of all external impulses.
- Change in angular momentum about any chosen point O = sum of the moments of those impulses about O.
Because the time interval is infinitesimal, ordinary finite forces drop out.
Step 3 – Express the velocity field of a rigid body
Before the impulse the velocity of any point P is
v = v_S + [u × r]
where v_S is the velocity of the centre of mass, u is the angular-velocity vector, and r is the position vector from the centre of mass to P.
After the impulse the same form holds with new values v_S’ and u’.
Step 4 – Sudden fixation of a single point O
The new motion can only be a pure rotation about some axis through O, so v_O’ = 0 and
v’ = [u’ × r’]
(r’ measured from O).
The angular-momentum balance about O now becomes an equation for the single unknown vector u’.
When the axes are chosen as the principal axes of inertia at O the three component equations separate and u’ is obtained at once.
Step 5 – Sudden fixation of a straight line (or of two points)
The body is forced to rotate about that line.
The impulsive reactions have no moment about the line, so the component of angular momentum along the line is conserved.
The new angular speed is therefore simply
|u’| = (old angular momentum along the line) / (moment of inertia about the line).
Step 6 – Reduce a general velocity change to a fixation problem
If one or two points are suddenly given new non-zero velocities, introduce a moving frame that already possesses those velocities.
In that frame the points are instantaneously at rest, so the problem reduces to an ordinary fixation.
The Coriolis theorem (proved in vector form) guarantees that the angular-momentum balance keeps the same appearance in the moving frame.
Solve the fixation problem relative to the moving frame, then transform back to obtain the absolute motion.
Step 7 – Kinetic energy
Once the new velocities are known, form the kinetic energy before and after the impulse.
Vector identities immediately recover the classical statements (Thomson–Tait, Carnot, Appell) that the loss of kinetic energy equals the kinetic energy of the relative motion, and supply a short generalisation of those statements.
At every step the unknowns are isolated, the relevant vector equation is written, and the solution is read off. That is the transparency Lewe practised in 1906.
ace-consultancy.uk 