Force

In General > s.a. anomalous acceleration; physical theories; physics teaching; symplectic structures.
* Idea: The time rate of change of momentum, or the spatial rate of change of energy.
* History: There is a tradition of trying to avoid the use of the concept (e.g., H Hertz, quantum mechanics, general relativity below).
* Measurement: 2000, The smallest measured forces are femtonewton forces between DNA molecules [@ news pn(00)may]; 2006, Measurement of attonewton forces [@ Arlett et al NanoLett(06)] and practical piconewton force measurement for cell biology [@ news pn(06)dec]; 2011, Measurements down to the yoctonewton (yN) level [@ news nn(11)apr].
@ General references: Jammer 57; Manchak PhSc(09)jul [in Cartesian physics]; Lange AJP(11)apr [historical controversy over the force composition law].
@ Measurement: Norte et al PRL(16), Reinhardt et al PRX(16) + Teufel Phy(16) [using trampoline resonators to sense extremely weak forces]; Jain et al PRL(16) + Safavi-Naeini Phy(16) [force sensor affected by momentum kicks of individual photons]; Ivanov PRA(16)-a1606 [precision beyond the standard quantum limit].
@ In special relativity: Coleman NCB(06) [and gravity]; Kholmetskii & Yarman EJP(07) [resolution of apparent paradoxes]; Muñoz Díaz a1206 [in Lagrangian mechanics].

Types and Examples > s.a. Centrifugal Force; friction; newtonian gravity [tidal]; Lorentz Force; self-force; sound [acoustic force].
* Pseudoforces: Fictitious forces that arise when one uses non-inertial frames of reference; Examples are the Coriolis force (velocity-dependent), and, in Einstein's view, the force of gravity.
@ Entropic forces: Sokolov EJP(10) [nature, statistical mechanics]; > s.a. emergent gravity; formulations of electromagnetism.
@ Other special types: Dolgov PRP(99) [long-range]; Lynden-Bell Obs(00)ap [preserving the magnitude of a vector]; Vallejo mp/06 [inertial].

In Curved Spacetime / General Relativity > s.a. geodesic [deviation]; orbits of test bodies and gravitating objects.
* Idea: Given a world-line with velocity ξ^a and curvature/acceleration ξ^m∇_m ξ^a, the force on it is defined by

F^a := m ξ^m∇_m ξ^a = m^–1 p^m ∇_m p^m ;

This force cannot be given as a vector field and is strongly observer-dependent, so one often tries to avoid its use.
* Maximum force or power: The principle according to which no force can exceed the "gravitational vacuum tension" F_max = \(\tau\) = c⁴/4G = 3.0 × 10⁴³ N, and no power can exceed P_max = c⁵/4G = 9.1 × 10⁵¹ W; The values can be obtained easily from Planck units (up to the factor 4), but physically force or power corresponds to momentum flow, and the maximum can be attained only through horizons; Numerically, \(\tau \approx 3.026\times 10^{43}\)N.
@ General references: De Felice MNRAS(91); Qadir & Sharif NCB(92)gq/07; Qadir & Zafarullah NCB(96); Arminjon AoM(96)gq/06 [and Newton's second law]; Ridgely EJP(10) [derivation and examples].
@ Inertial: Prasanna gq/01; Foertsch et al CQG(03)gq [photon surfaces]; Jonsson CQG(06)-a0708 [general formalism, and optical geometry].
@ Centrifugal: Prasanna MPLA(02)gq/01 [and Coriolis]; Hasse & Perlick GRG(02)gq/01 [and lensing].
@ Maximum force or tension: Schiller IJTP(05)phy/06 [general relativity from maximum power or force]; Gibbons FP(02)ht [maximal tension = c⁴/4G]; Barrow & Gibbons MNRAS(14)-a1408 [maximum force, examples and ramifications]; Dąbrowski & Gohar PLB(15)-a1504 [theories in which it does not hold]; Bolotin & Cherkaskiy a1507 [and the holographic principle]; > s.a. deformed uncertainty principle.
@ Related topics: Semerák NCB(95) [decomposition]; Price GRG(04)gq [normal to 4-velocity].

In Quantum Theory > s.a. interaction.
@ References: Mullin & Blaylock AJP(03)dec [particle statistics and "exchange force"].

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