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Lagrange's equation and Lagrangian and derived in 1788. It is different from Newtonian mechanics view because Newton emphasizes the external force acts on the body. But the Lagrange's. view is that the internal energy (kinetic or potential, etc) associated with the body.

Hamilton principle is stated in 1834. It is a variational and minimal (minimization) principle.

Since Lagrange equation 1788, and Hamilton principle 1834 are nearly 50 years apart,

how and what does Lagrange rely on (as which logical thought behind) to derive his Lagrange's equation and Lagrangian, without relying on Hamilton variational principle? How is it possible?

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  • $\begingroup$ Related: physics.stackexchange.com/q/78138/2451 $\endgroup$
    – Qmechanic ♦
    May 1, 2021 at 21:08
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    $\begingroup$ I don't really understand what this question is asking for, but if you are interested in the historical development, this question would be more appropriate for History of Science and Mathematics $\endgroup$
    – ACuriousMind ♦
    May 1, 2021 at 21:42
  • $\begingroup$ Question is: how does Lagrange derive his Lagrange's equation and Lagrangian, without relying on Hamilton variational principle? (since variational principle is credited to Hamilton, after 50 years of Lagrange's equation.) I am asking the Lagrange way of physics thinking. (not about history) $\endgroup$
    – ann marie cœur
    May 1, 2021 at 22:38
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    $\begingroup$ I think you should look into d'Alembert's principle. You don't need to use Hamilton's principle to derive the Lagrangian, it's just typically more elegant. $\endgroup$
    – Wihtedeka
    May 1, 2021 at 22:45
  • $\begingroup$ Wihtedeka - thanks so much! $\endgroup$
    – ann marie cœur
    May 1, 2021 at 23:15

2 Answers 2

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Lagrangian and Euler-Lagrange equations come from the Newton's second law plus the requirement that the force is a function of position. In order to be valid for generalized coordinates $q_i$ and $\dot q_i$ the D'Alembert principle is also necessary.

$\delta(\int {L}dt) = 0$ results from that equations, and not the other way around.

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  • $\begingroup$ Thanks - but how is that differed and in contrast with Hamilton principle 1834? Could you add more details. +1 $\endgroup$
    – ann marie cœur
    May 1, 2021 at 21:02
  • $\begingroup$ Also I thought Lagrange tries to avoid the force into his formulation. I think only Newton uses force. Lagrange does not mention force except connecting with Newton result $\endgroup$
    – ann marie cœur
    May 1, 2021 at 21:03
  • $\begingroup$ I don't know the historical details, but I think that in order to talk only in terms of $V(x)$ it is necessary to be sure that non-conservative forces are absent. In a certain way, only that kind of forces must be treated as such. $\endgroup$
    – Claudio Saspinski
    May 1, 2021 at 21:14
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The most important innovation introduced by Lagrange was the systematic use of generalized coordinates.

In retrospect: Lagrange used the work-energy theorem, except not in the modern form. The form that Lagrange used was, as already stated in another answer, d'Alembert's virtual work.

Looking back: from my perspective d'Alembert's virtual work looks like an obfuscated version of the work-energy theorem.

Be it as it may: Lagrange was able to create a formulation of classical mechanics (Lagrangian mechanics) that is very expressive and versatile. So I'm guessing that for Lagrange there was no particular incentive to clarify the concept of d'Alembert's virtual work.


Force is frame-independent. On the other hand: the kinetic energy that is attributed to an object is frame dependent. Evaluation in terms of kinetic energy and potential energy is inherently in terms of change of energy; it is change of energy that is frame-independent.

There is no such thing as an inherent zero point of potential energy; what can be defined is difference of potential energy. Physicists capitalize on that by placing the zero point of potential energy at whatever level happens to be practical to perform calculations.

Lagrangian mechanics does not work with kinetic/potential energy itself. To evaluate physics taking place lagrangian mechanics works with the derivative of kinetic/potential energy.



William Rowan Hamilton had made significant contributions in the field of geometric optics. Geometric optics is treating all of optics in terms of statements in the same form as Fermat's least time.

Hamilton next set out to find out whether it is possible to restate classical mechanics in a variational form, analogous to Fermat's least time.

Hamilton's stationary action is mathematically equivalent to the work-energy theorem.


Other than the concept of Hamilton's stationary action: Hamilton also laid the basis of the concept of describing physics taking place in terms of motion in phase space. There are classes of problems that were solved because of insights gained from examining motion in phase space.

Just for completeness:
Hamiltonian mechanics and Hamilton's stationary action do not have overlap; they are independent concepts.

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  • $\begingroup$ Thanks - I will think about it. +1 $\endgroup$
    – ann marie cœur
    May 1, 2021 at 22:42
  • $\begingroup$ So what is Lagrange's own principle? that is different from Hamilton's principle to derive Lagrange's equation $\endgroup$
    – ann marie cœur
    May 1, 2021 at 22:44
  • $\begingroup$ @anniemarieheart Lagrange arrived at the form of Lagrangian mechanics by application of d'Alembert's virtual work. (Of course, d'Alembert virtual work corroborates $F=ma$, otherwise Lagrange wouldn't have used it.) I assert that d'Alembert's virtual work is an unnecessarily complicated way of stating the work-energy theorem, making d'Alembert's virtual work a very opaque instrument. Physics SE contributor Claudio Saspinsky recently gave a derivation that arrives at the same form as the EL-equation, but starting from d'Alembert's virtual work $\endgroup$
    – Cleonis
    May 2, 2021 at 5:12

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