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\begin{document}

\title{Where is Force in Modern Physics?}


\author{Isaac Newton,  Jean d'Alembert, Charles Coulomb, and Leonard Euler}
\affiliation{Department of Physics and Astronomy, University
of Delaware, Newark, DE 19716-2570, USA}

\begin{abstract}
We reexamine critically the current status of the concept of ``force'' which is conspicuously
absent from modern physics despite the fact that we have been demonstrating its importance since 1687.
\end{abstract}

\pacs{01.55.+b, 01.65.+g}


\maketitle

\section{Introduction} \label{sec:intro}

We have recently gathered at the University of Delaware to reexamine the developments in modern physics
over the past century. It is quite appalling to find that the most important discovery in physics of our times---Newton's  second law~\cite{newton}---has vanished from  quantum mechanics, general relativity, standard model of elementary particles, ... This fundamental law of physics essentially introduces the notion of force $\vec{F}$
\begin{equation} \label{eq:newton}
\vec{F}=m\vec{a},
\end{equation}
by relating it to the mass of a body $m$ and its acceleration $\vec{a}$. While this has been a {\em sine qua non} tool in the study of various basic problems of classical mechanics [we remind the reader that d'Alembert (constraint and contact forces), Coulomb (friction), and Euler (rigid, elastic, and fluid bodies) made fundamental contributions to what we now comprehend in the culture of force], the present generation of physicist is more likely to use Lagrange's or Hamilton's reformulations of the laws of classical  mechanics. Instead of ``forces'', these approaches employ
scalar functions, such as the Hamiltonian $H=T+V$ (which is at the core of quantization procedures
in non-relativistic quantum mechanics) or  the  Lagrangian $L=T-V$ (which plays an important role
in path integral quantization methods of quantum field theory),
\begin{eqnarray}
H & = & T+V=\frac{p^2}{2m} + V(q) \label{eq:hamilton}, \\
L & = & T-V = \frac{1}{2} m \dot{q}^2 - V(q) \label{eq:lagrange},
\end{eqnarray}
where $T$ and $V$ are the kinetic and potential energy of a conservative system, respectively.
Note that the two functions $H(p,q)$ in Eq.~(\ref{eq:hamilton}) and $L(q,\dot{q})$ in Eq.~(\ref{eq:lagrange}) are related to each other via the Legendre transformation, $H(p,q)=p \dot{q} - L(q,\dot{q})$.

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\begin{figure}
\includegraphics[scale=0.6,angle=-30]{newton}
\caption{Sir Isaac Newton in the happy days before his scientific edifice, founded on the {\em four} fundamental physical laws (from the zeroth~\cite{wilczek} to the third~\cite{newton}) has been thoroughly deconstructed in the XX century.}\label{fig:newton}
\end{figure}
%

\section{Wilczek on Force}\label{sec:wilczek}

It is quite enlightening to find recent account of these preposterous developments summarized by F. Wilczek (Nobel Prize in Physics 2004)~\cite{wilczek}: ``Newton's second law of motion, $F = ma$, is the soul of classical mechanics. Like other souls, it is insubstantial. The right-hand side is the product of two terms with profound meanings. Acceleration is a purely kinematical concept, defined in terms of space and time. Mass quite directly reflects basic measurable properties of bodies (weights, recoil velocities). The left-hand side, on the other hand, has no independent meaning. Yet clearly Newton's second law is full of meaning, by the highest standard: It proves itself useful in demanding situations. Splendid, unlikely looking bridges, like the Erasmus Bridge (known as the Swan of Rotterdam), do bear their loads; spacecraft do reach Saturn. The paradox deepens when we consider force from the perspective of modern physics. In fact, the concept of force is conspicuously absent from our most advanced formulations of the basic laws. It doesn't appear in the Schr\" odinger's equation~\cite{so_force}, or in any reasonable formulation of quantum field theory, or in the foundations of general relativity...''


\section{Conclusions} \label{sec:conclusion}
If $\vec{F} = m\vec{a}$ is formally empty, microscopically obscure, and maybe even morally suspect, what's
the source of its undeniable power? The answers might lie in Ref.~\cite{newton} or recent scrutiny~\cite{wilczek}.

\begin{acknowledgments}
We are grateful to A. Einstein for valuable insights.
\end{acknowledgments}

%********************references***********************************
\begin{thebibliography}{1}

\bibitem{newton} Sir~I.~Newton, {\em The Principia: Mathematical Principles of Natural Philosophy} (University of California Press, Berkeley, 1999).

\bibitem{wilczek} F.~Wilczek, {\em Fantastic Realities: 49 Mind Journeys And a Trip to Stockholm} (World Scientific, Singapore, 2006).

\bibitem{so_force} B.~K.~Nikoli\' c, L.~P.~Z\^ arbo, and S.~Welack, Phys. Rev. B {\bf 72}, 075335 (2005).

\end{thebibliography}

\end{document}
