The Principle of Least Action

(1887)

(By H. von Helmholtz)

My principle of least action is different from the original one by Maupertuis in 1744.

• It received a precise determination for the variational condition and a complete proof much later by Lagrange.

His principle should subsume the various transformed forms of that theorem by Sir W. Rowan Hamilton.

Hamilton presented 2 differential equations.

C. G. J. Jacobi later combined these into a single one that had the common source of these transformations.

• The physical assumptions with which the calculations started were not changed.

Jacobi first applied the principle of least action to only the mechanics of ponderable bodies.

• He represented the motions of a system as being either freely mobile or rigidly coupled to another mass point in a chain.

He started by assuming:

• Newton’s laws of motion and
• how one defined the phenomenon that would correspond to the action of rigidly-coupled mass points mechanically.

He showed that after Maupertius’ integral was understoodd, the validity of the law of constancy of energy must be assumed.

At first, this seems a restriction on the validity of the principle of least action.

But recent investigations have established that:

• the law of the constancy of energy is also valid in general
• the apparent restriction does not restrict anything.

However, one must know completely all of the forms that the equivalence of energy can take for a process being examined in order to include them in calculations.

But other physical processes might enter in.

• These might use energy quanta and not align simply with Newton’s laws

But these still agree with the principle of least action.

I choose one of Hamilton’s forms as the most convenient form for the principle of least action.

• This allows external forces that depend on time to act on the mechanical system, whose internal forces are only conservative ones.

The potential energy of the system is Fand the vis viva byL` then the function (viz., Hamilton’s principal function) whose time integral will be a minimum for the normal motion between end points will be:

H = F – L,

while the energy of the system will be:

E = F + L.

In this, F depends only on the coordinates. L is a homogeneous function of second degree of the velocities.

H is the one in terms of whose differential quotients Lagrange expressed the forces that act upon the moving system from the outside.

Since H plays an important role in all of the problems here, I call it kinetic potential.

Thus, they include:

• F. E. Neumann’s potential of two electrical currents
• R. Clausius’s () electrodynamical potential

J. W. Gibbs called the same thermodynamic function that I called “free energy” the force function for constant temperature.

P. Duhem (), by contrast, called that function the thermodynamic potential.

There thus exist sufficient examples for the new choice of terminology.

The principle of least action can then be expressed as follows:

The mean value of the kinetic potential that is calculated for equal time elements is a minimum for the actual path of the system (a limiting value for longer intervals, respectively), in comparison to all other neighboring paths that lead from the initial position to the final positions in the same amount of time.

For a state of rest, the kinetic potential goes to the values of the potential energy (the potential in the sense that was used up to now, respectively).

We do not need to take the mean value for them, since the values that were different while in motion will all be equal to each other here.

For a state of rest, our theorem says simply that the potential energy must be a minimum for equilibrium.

Jacobi showed that the function H can also include time explicitly with making the construction of the variation and differential equations that follow from it impossible.

I have employed it in order to add a sum

… to H, in which the pα are coordinate

The Pα mean the forces that act in the direction of the coordinates, while the latter are taken in a sense that will be discussed in more detail below.

The Pα will be considered to be given functions of time that are, however, independent of the coordinates.

In this form, the minimum principle will yield the Lagrange equations for the forces Pα, and in that way, an entire series of special investigations that are based upon Lagrange’s equations of motion will also be subsumed by the somewhat-modified principle of least action.

Where it is necessary to distinguish that modified principle from the original one, I would like to call it the law of minimum kinetic potential.

The form that Lagrange gave to the equations of motion is important because we can apply it to cases in which processes are at work that are no longer rationally resolvable in various ways.

Examples are friction, galvanic resistance, etc., and in which equilibrium must exist between the conservative forces that are included in Lagrange’s formula.