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From Archimedes and Euclid to Hamilton and Poincaré

Symplectic maps of $${\mathbb R} ^{2n}$$ are basic objects of Hamiltonian mechanics, and the time $$t$$ map of a Hamiltonian system’s position-momentum pair is symplectic. Symplectic maps of $${\mathbb R} ^2$$ are the area-preserving ones.

I recently realized that the Archimedian law of the lever amounts to an area-preservation property of a simple map of $${\mathbb R} ^2$$, as described next. Afterwards, I will reference an analogy between the Archimedian lever on the one hand and Hamiltonian mechanics on the other.

Figure 1. The left finger pushes with force f; the right finger is being pushed with force F.
Figure 1 shows a seesaw in equilibrium, pressed at both ends. Archimedes’ law of the lever gives the condition for the equilibrium, $$fl=FL,$$ i.e.

$F= \frac{l}{L} f.$

Furthermore, $$S = \frac{L}{l} s,$$ according to Euclid. To summarize, we have the “Euclid-Archimedes map,” $$(s,f)\mapsto (S,F)$$, given by

$\left\{ \begin{array}{l} S= \lambda s \\ F = \frac{1}{\lambda} f, \end{array} \right. \tag1$

where $$\lambda = L/l$$. This map is clearly area-preserving, but for a reason deeper than the explicit form $$(1)$$. Indeed, let us cyclically move the two fingers in Figure 1 so that the positions and the forces return to their original values. We end up doing zero work:

$\oint f\,ds+ \oint (-F)\,dS=0, \tag2$

The minus sign is due to the fact that the right finger presses with force $$-F$$. During the cyclic motion, the point $$(s,f)$$ describes a closed curve $$\gamma$$ in the plane, while the point $$(S,F)= \varphi (s,f)$$ describes the image curve $$\varphi (\gamma )$$. Therefore, the zero-work condition $$(2)$$ amounts to the equality of areas inside $$\gamma$$ and $$\varphi (\gamma )$$. Incidentally, $$(2)$$ is a compact way of saying that the lever is not a perpetual motion machine.

If the board can flex, as in Figure 2, then the map $$(s,f)\mapsto (S,F)$$ is no longer given by $$(1)$$, but is still area-preserving; the above proof applies without change.

Seesaw and Hamiltonian Dynamics

Figure 2. The hinge at O has a spring trying to keep the board straight.
Remarkably, the Hamiltonian flow is symplectic for the same reason that the “seesaw map” $$\varphi$$ is area-preserving.1 To make sense of the last sentence, I must specify the analogy between the seesaw in Figure 2 on the one hand and a Hamiltonian system on the other. The following explanation outlines this analogy (a full discussion can be found in [1]). Consider a mechanical system with the Lagrangian $$L,$$ depending on generalized position and velocity. Let us fix two points $$(0,q)$$ and $$(T,Q)$$ in time-space and define the action $$A(q, Q) = \int_{0}^{T} L(r(t),\dot r(t)) \,dt$$, with the integration occurring over the minimizer $$r(t)$$ of the integral subject to $$r(0)=q$$,  $$r(T)=Q$$ (we assume this minimized is unique and depends smoothly on $$q,Q$$). For any (admissible) $$T$$, the momenta $$p,P$$ at times $$t=0$$ and $$t=T$$ are given by

$P = A_Q(q,Q), \ \ p =- A_q(q, Q). \tag3$

This can be taken as the definition of the momentum, or related (in a one-line calculation) to the more standard definition, as explained on page 261 of [1].

Returning now to the seesaw of Figure 2, let $$U(s,S)$$ be the potential energy; then

$F= U_S(s,S), \ \ f=- U_s(s,S). \tag4$

A comparison between $$(3)$$ and $$(4)$$ shows that the action and the momenta $$(A, \ p, \ P)$$ are close analogs of the potential energy and the forces $$(U, \ f, \ F)$$. The proof of the symplectic character of the time $$T$$ map $$(q,p) \mapsto (Q, P)$$ for arbitrary $$T$$ becomes a verbatim copy of the area preservation’s proof of the “seesaw map” $$(s,f)\mapsto (S, F)$$.

If the spring in Figure 2 dissipates energy under deformations, then $$(2)$$ becomes

$\oint f\,ds+ \oint (-F)\,dS=W>0, \tag5$

where $$W$$ is the heat dissipated in the spring $$x$$; $$(5)$$ suggests that the area decreased by $$W$$. However, the map $$\varphi:= (s, f) \mapsto (S,F)$$ depends only on the static properties of the spring and thus must be area-preserving; there is no difference between a dissipating and a non-dissipating spring in a static state. Resolution of this paradox is left as a puzzle for interested readers and may (or may not) be discussed in the next column.

All figures are provided by the author.

Acknowledgments: The work from which these columns are drawn is partially supported by NSF grant DMS-9704554.

References
[1] Levi, M. (2014). Classical Mechanics with Calculus of Variations and Optimal Control: an Intuitive Introduction. Student Mathematical Library, vol. 69. American Mathematical Society.

Mark Levi (levi@math.psu.edu) is a professor of mathematics at the Pennsylvania State University.