It's neither inferior, nor superior. It's equivalent, for those forces that can be expressed as gradients of a potential. For forces that can't, the vatiational principle can, still, be applied, but, now one needs a more general drcomposition of the force, in terms, also, of the vector potential. This was discovered by Clebsch and Monge (independently) in the 19th century. Cf. for instance, Sommerfeld's textbook.

Уважаемый Афксенти,

Ваше утверждение о превосходстве энергетического подхода над ньютоновским требует уточнения. Формула F=−∇U не является более фундаментальной, чем F=ma, поскольку энергия всегда определяется через силу и перемещение: E=∫F dr.

Таким образом, сила выступает аргументом энергии, а не наоборот.

Например, закон Кулона F=kq2r2 непосредственно связывает взаимодействие материальных зарядов с расстоянием между ними. На заряд в потенциальном поле действует кулоновская сила, которая позволяет строго вычислить как величину взаимодействия, так и энергетическое состояние системы. Используя этот закон, можно определять также относительное положение объектов в системе координат. Это демонстрирует, что именно сила является исходным понятием, а энергетическое описание производно от неё.

Лагранже и гамильтонов формализмы безусловно ценны как математический аппарат, однако они представляют собой более абстрактное описание и не раскрывают механизма взаимодействия. Их основное преимущество заключается в удобстве математической обработки задач, но физически фундаментальным остаётся ньютоновский закон, который напрямую связывает действие силы и движение материи.

“Is potential-based def of force superiorvto to Newtonian?.... Force is a Response to Gradients

I.e.force arises from potential gradients (per classical mechanics: F= -nabla U

- This force causes acceleration F= m{a}....”

- “Newtonian” F=ma indeed only describes motion of a body under action of some force F, making at that no explanations – for/by what reason/way does this “F” appear? However, F= -nabla U really differs from the first Eq. only in that in this case the question above is as “for/by what reason/way do these “F” , “nabla” and “U” appear?”; the other difference in that F=ma isn’t applicable at statics, when some forces evidently act - though a=0, really is inessential.

Really the questions above can be, and practically for sure are, really rigorously scientifically answered only in the Shevchenko-Tokarevsky’s Planck scale informational physical model, see Preprint Planck scale informational physical model and fundamental pr...

, section 6. “Mediation of the fundamental forces in complex systems”; briefly:

- really utmost general force that acts on some body equation is F=dP/dt, where P is transmitted to the body momentum;

- at that the momentum P is [specifically] transmitted to given “irradiated” body at/by actions of some fundamental Nature [Gravity, Electric, Nuclear/Strong] Forces mediators that are radiated by other bodies’ particles, which [the mediators] transmit elementary momentums p=ћ/r, ћ is the reduced Planck constant, r is distance between “radiating” and “irradiated body’s” particles.

The number, N, of p that act on irradiated body in one second is dP/dt, i.e the force F=Np.

Besides two important in this case points else:

- first one: as that is rigorously shown in the 3 Forces models in section 6 above, Matter is rather simply logically organized informational system, and so really on Planck scale all these Forces act by the same scheme, including the all Forces mediators transmit the same elementary specific momentums ±p; and

- the Forces mediators aren’t particles, and so don’t carry/don’t transmit “kinematical” momentums and energy - in fundamental contrast to the mainstream theories of the Forces, where classical Forces fields [which are intensive flows of the mediators] have energy, in QFTs the “virtual particles-mediators” also carry “kinematical” momentums/energy; what evidently violates the energy conservation law.

The real mediators momentums p above are specific – the momentums cause in irradiatred particles appearance of the “kinematical” elementary momentums p, and corresponding real elementary portion of energy, which is released from the irradiated particle’s own energy E=mc2.

More see the linked paper.

Cheers

Уважаемый Сергей.

Вы утверждаете, что предельная общая сила — это F=dP/dt . Но это лишь математическое жонглирование без физического смысла. В Ньютоновской механике импульс — это производная величина, определяемая через силу и время:

P=F⋅t.

Таким образом, сила первична, а импульс вторичен. Создавать видимость, будто импульс вызывает силу, — полное противоречие физике.

Работа и энергия системы также определяются только через действие силы:

E=∫F dr.

Поэтому любые рассуждения, в которых импульс или энергия подаются как первичные источники силы, являются методологической ошибкой.

Это заблуждение было введено квантовой механикой из-за неверного понимания терминов классической механики. Получается парадокс: отвергая механику Ньютона, они в то же время цепляются к её терминам, но искажают их смысл.

P.S. Сторонники квантовой механики также жонглируя математическими выражениями без физического смысла довели саму теорию до абсурда. Если вас интересуют факты, вы можете проверить их предсказания с экспериментом по визуализации атома водорода в 2013 г. под руководством А. Стадольны. Результаты опубликованы в работах:

• Реальный радиус атома водорода: новые теоретические и экспериментальные подходы (март 2025). DOI: 10.13140/RG.2.2.18580.00644
• Валидация детерминированных и квантовых моделей радиусов электронных орбит в атоме водорода с помощью фотоионизационной микроскопии (март 2025). DOI: 10.13140/RG.2.2.33076.56961

По результатам этих исследований:

• Квантовая механика предсказала радиус возбуждённого атома водорода ≈ 900 Å против экспериментальных 1,4 Å — адекватность ≈ 64 000% (полный провал).
• Детерминистическая теория атома (ДТА, Баходира Турсунбаева) предсказала 1,333 Å, что даёт адекватность ≈ 4,8% (попадание в эксперимент в пределах погрешности).

Dear Bakhodir,

Newton indeed very essentially grounded mechanics in his 3 laws , including the F=ma, [and the law “an action of force F of a body on an other body is equal to the force -F the other body acts on the first body with which”, what includes the static case] on the 1690s level. However the force F isn’t universal parameter of interactions.

So at further mechanics development really utmost fundamental physical laws that at interactions in closed systems of bodies total systems’ a scalar energy, E, and momentum vector, P, conserve, were discovered; including it was rigorously shown that F=dP/dt.

At that in simple cases, when interactions aren’t too short, F=mais well applicable – and in principle is measurable, however at particles interactions that happen lesser than in ~ 10-20 s, the parameter “force” losses a sense, and description and analysis of the interactions first of all balances of energies and momentums are considered.

Cheers

Dear Sergey,

You argue that “force is not a universal parameter” and that “momentum and energy are more fundamental.” This is a methodological mistake.

In Newtonian mechanics, force is primary – it is force that sets matter into motion and performs work:

E=∫F dr, P=F⋅t.

Momentum and energy are derivative quantities, consequences of the action of force.

Формул F = dP/dt не раскрывает "сверхфундаментальный закон"; это просто переписанная форма P=F⋅t . Это математическое преобразование, но не новый физический принцип.

The conservation laws of energy and momentum only record the results of forces acting in a closed system, they do not replace the concept of force itself. To claim that in “rare interactions” the force “loses its meaning” is to confuse the limitations of measurement techniques with the fundamental laws of nature.

P.S. Supporters of quantum mechanics have similarly juggled mathematical expressions without physical meaning, pushing the theory into absurdity. Experimental verification confirms this. The direct visualization of the hydrogen atom (A. Stodolna, 2013) and subsequent works (The Real Radius of the Hydrogen Atom: New Theoretical and Experimental Approaches, March 2025, DOI: 10.13140/RG.2.2.18580.00644; Validation of Deterministic and Quantum Models of Electron Orbit Radii in the Hydrogen Atom Using Photoionization Microscopy, March 2025, DOI: 10.13140/RG.2.2.33076.56961) have clearly shown:

• quantum mechanical predictions do not match reality (discrepancies up to 64,000%),
• whereas the Deterministic Theory of the Atom (Bahodir Tursunbaev) gives results consistent with experiment within the margin of error (~5%).

The facts speak for themselves.

Dear Bakhodir,

- you seems don’t read SS posts above attentively enough.

Again, say, this

“.....in Newtonian mechanics, force is primary – it is force that sets matter into motion and performs work:

E=∫F dr,..”

- even in Newtonian mechanics isn’t complete force F definition - say, at statics there are no dr and no E, but force as F=dP/dt really acts – and, since in this case the force F really adds nothing useful to what is in dP/dt already, its use is senseless, only momentums P can be used at elaborations of such – and really all other – physical tasks.

As to rest – sorry, but I comment as a rule only conventional physics and don’t comment alternative approaches.

Cheers

Dear Sergey,

The expression dP/dt is often presented as if it reveals the very essence of how force arises — but this is a complete misconception. Momentum was already known in Newton’s time, yet no mechanic ever elevated it to such a status, precisely because they understood its derivative nature. In classical mechanics, force is the cause, momentum the consequence:

P=F⋅t

To invert this relationship and claim that dP/dtdP/dtdP/dt explains the origin of force is not physics — it is merely algebraic wordplay without physical meaning.

Recent SS posts in https://www.researchgate.net/post/Since_atoms_are_made_of_charged_particles_why_should_we_exclude_the_possibility_that_gravity_comes_from_atomic_charges_and_polarization_effects

- are relevant to this thread question.

Cheers