Maupertuis' principle

In classical mechanics, Maupertuis' principle (named after Pierre Louis Maupertuis), is that the path followed by a physical system is the one of least length (with a suitable interpretation of path and length). It is a special case of the more generally stated principle of least action. Using the calculus of variations, it results in an integral equation formulation of the equations of motion for the system.

Mathematical formulation

Maupertuis' principle states that the true path of a system described by N generalized coordinates \mathbf{q} = \left( q_{1}, q_{2}, \ldots, q_{N} \right) between two specified states \mathbf{q}_{1} and \mathbf{q}_{2} is an extremum (i.e., a stationary point, a minimum, maximum or saddle point) of the abbreviated action functional

\mathcal{S}_{0}[\mathbf{q}(t)] \ \stackrel{\mathrm{def}}{=}\ \int \mathbf{p} \cdot d\mathbf{q}

where \mathbf{p} = \left( p_{1}, p_{2}, \ldots, p_{N} \right) are the conjugate momenta of the generalized coordinates, defined by the equation

p_{k} \ \stackrel{\mathrm{def}}{=}\ \frac{\partial L}{\partial\dot{q}_{k}}

where L(\mathbf{q},\dot{\mathbf{q}},t) is the Lagrangian function for the system. In other words, any first-order perturbation of the path results in (at most) second-order changes in \mathcal{S}_{0}. Note that the abbreviated action \mathcal{S}_{0} is not a function, but a functional, i.e., something that takes as its input a function (in this case, the path between the two specified states) and returns a single number, a scalar.

Jacobi's formulation

For many systems, the kinetic energy T is quadratic in the generalized velocities \dot{\mathbf{q}}

T = \frac{1}{2} \frac{d\mathbf{q}}{dt} \cdot \mathbf{M} \cdot \frac{d\mathbf{q}}{dt}

although the mass tensor \mathbf{M} may be a complicated function of the generalized coordinates \mathbf{q}. For such systems, a simple relation relates the kinetic energy, the generalized momenta and the generalized velocities

2 T = \mathbf{p} \cdot \dot{\mathbf{q}}

provided that the potential energy V(\mathbf{q}) does not involve the generalized velocities. By defining a normalized distance or metric ds in the space of generalized coordinates

ds^{2} = d\mathbf{q} \cdot \mathbf{M} \cdot d\mathbf{q}

one may immediately recognize the mass tensor as a metric tensor. The kinetic energy may be written in a massless form

T = \frac{1}{2} \left( \frac{ds}{dt} \right)^{2}

or, equivalently,

2 T dt = \mathbf{p} \cdot d\mathbf{q} = \sqrt{2 T} \ ds.

Hence, the abbreviated action can be written

\mathcal{S}_{0} \ \stackrel{\mathrm{def}}{=}\ \int \mathbf{p} \cdot d\mathbf{q} = \int ds \sqrt{2}\sqrt{E_{tot} - V(\mathbf{q})}

since the kinetic energy T = E_{tot} - V(\mathbf{q}) equals the (constant) total energy E_{tot} minus the potential energy V(\mathbf{q}). In particular, if the potential energy is a constant, then Jacobi's principle reduces to minimizing the path length s = \int ds in the space of the generalized coordinates, which is equivalent to Hertz's principle of least curvature.

Comparison with Hamilton's principle

Hamilton's principle and Maupertuis' principle are occasionally confused and both have been called the principle of least action. They differ from each other in three important ways:

Hamilton's principle uses \mathcal{S} \ \stackrel{\mathrm{def}}{=}\ \int L \, dt, the integral of the Lagrangian over time, varied between two fixed end times t_{1}, t_{2} and endpoints q_1, q_2. By contrast, Maupertuis' principle uses the abbreviated action integral over the generalized coordinates, varied along all constant energy paths ending at q_1 and q_2.
Hamilton's principle determines the trajectory \mathbf{q}(t) as a function of time, whereas Maupertuis' principle determines only the shape of the trajectory in the generalized coordinates. For example, Maupertuis' principle determines the shape of the ellipse on which a particle moves under the influence of an inverse-square central force such as gravity, but does not describe per se how the particle moves along that trajectory. (However, this time parameterization may be determined from the trajectory itself in subsequent calculations using the conservation of energy.) By contrast, Hamilton's principle directly specifies the motion along the ellipse as a function of time.
Maupertuis' principle requires that the two endpoint states q_{1} and q_{2} be given and that energy be conserved along every trajectory. By contrast, Hamilton's principle does not require the conservation of energy, but does require that the endpoint times t_{1} and t_{2} be specified as well as the endpoint states q_{1} and q_{2}.

History

Maupertuis was the first to publish a principle of least action, where he defined action as \int v \, ds, which was to be minimized over all paths connecting two specified points. However, Maupertuis applied the principle only to light, not matter (see the 1744 Maupertuis reference below). He arrived at the principle by considering Snell's law for the refraction of light, which Fermat had explained by Fermat's principle, that light follows the path of shortest time, not distance. This troubled Maupertuis, since he felt that time and distance should be on an equal footing: "why should light prefer the path of shortest time over that of distance?" Accordingly, Maupertuis asserts with no further justification the principle of least action as equivalent but more fundamental than Fermat's principle, and uses it to derive Snell's law. Maupertuis specifically states that light does not follow the same laws as material objects.

A few months later, well before Maupertuis' work appeared in print, Leonhard Euler independently defined action in its modern abbreviated form \mathcal{S}_{0} \ \stackrel{\mathrm{def}}{=}\ \int m v \, ds \ \stackrel{\mathrm{def}}{=}\ \int p \, dq and applied it to the motion of a particle, but not to light (see the 1744 Euler reference below). Euler also recognized that the principle only held when the speed was a function only of position, i.e., when the total energy was conserved. (The mass factor in the action and the requirement for energy conservation were not relevant to Maupertuis, who was concerned only with light.) Euler used this principle to derive the equations of motion of a particle in uniform motion, in a uniform and non-uniform force field, and in a central force field. Euler's approach is entirely consistent with the modern understanding of Maupertuis' principle described above, except that he insisted that the action should always be a minimum, rather than a stationary point.

Two years later, Maupertuis cites Euler's 1744 work as a "beautiful application of my principle to the motion of the planets" and goes on to apply the principle of least action to the lever problem in mechanical equilibrium and to perfectly elastic and perfectly inelastic collisions (see the 1746 publication below). Thus, Maupertuis takes credit for conceiving the principle of least action as a general principle applicable to all physical systems (not merely to light), whereas the historical evidence suggests that Euler was the one to make this intuitive leap. Notably, Maupertuis' definitions of the action and protocols for minimizing it in this paper are inconsistent with the modern approach described above. Thus, Maupertuis' published work does not contain a single example in which he used Maupertuis' principle (as presently understood).

In 1751, Maupertuis' priority for the principle of least action was challenged in print (Nova Acta Eruditorum of Leipzig) by an old acquaintance, Johann Samuel Koenig, who quoted a 1707 letter purportedly from Leibniz that described results similar to those derived by Euler in 1744. However, Maupertuis and others demanded that Koenig produce the original of the letter to authenticate its having been written by Leibniz. Koenig only had a copy and no clue as to the whereabouts of the original. Consequently, the Berlin Academy under Euler's direction declared the letter to be a forgery and that its President, Maupertuis, could continue to claim priority for having invented the principle. Koenig continued to fight for Leibniz's priority and soon luminaries such as Voltaire and the King of Prussia, Frederick II were engaged in the quarrel. However, no progress was made until the turn of the twentieth century, when other independent copies of Leibniz's letter were discovered. The present scholarly consensus seems to be that the quotations from Leibniz are indeed genuine, i.e., that he had invented Maupertuis' principle and applied it to several mechanical problems by 1707 (37 years before Maupertuis and Euler) but did not publish his findings.

See also

References

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