Méchanique analitique¶
Lagrange, J. -. (1788). Méchanique analitique. Mécanique analytique.
Cited by¶
10 citations across 9 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Conjugate Variables
- in classical mechanics and the Lagrangian-momentum pairing
This sourceEstablishes the Lagrangian formulation and generalized coordinates from which the conjugate (canonical) momentum p = ∂L/∂q̇ is defined — the Lagrangian-momentum pairing cited for field theory.
- in classical mechanics and the Lagrangian-momentum pairing
- Conservation Laws
- Lagrange's generalized coordinates
This sourceRecasts mechanics in terms of generalized coordinates and a variational principle; cyclic (ignorable) coordinates yield conserved generalized momenta, supporting the claim that symmetries encode directly into the equations of motion via cyclic-coordinate conservation laws.
Supported in partVerified against the work's full text
The Mécanique analytique does reduce a system to a few independent variables (generalized coordinates) in its Statics; the supplied text stops before the dynamics and cyclic-coordinate material.
“on peut toujours réduire les variables qui déterminent la position de ces points dans l'espace”
- Lagrange's generalized coordinates
- Constraint
- (4) The deeper abstraction is that **constraint reasoning is the structural prerequisite for all disciplined decision-making under restriction: Lagrange's 1788 Mécanique Analytique
This sourceMultiplier technique originates in Lagrange's 1760s–70s calculus-of-variations memoirs. Historical treatment: Fraser, "Lagrange's Analytical Mathematics, Its Cartesian Origins and Reception in Comte's Positive Philosophy." Studies in History and Philosophy of Science 21, no. 2 (1990): 243–256; Goldstine, A History of the Calculus of Variations from the 17th through the 19th Century (Springer, 1980).
- (4) The deeper abstraction is that **constraint reasoning is the structural prerequisite for all disciplined decision-making under restriction: Lagrange's 1788 Mécanique Analytique
- Degrees of Freedom
- The construct originates in Lagrange's generalized-coordinate framework
This sourceHistorical treatment: Fraser, "Lagrange's Analytical Mathematics, Its Cartesian Origins and Reception in Comte's Positive Philosophy"90024-3). Studies in History and Philosophy of Science 21, no. 2 (1990): 243–256; Goldstine, A History of the Calculus of Variations from the 17th through the 19th Century (Springer, 1980).
Supported in partVerified against the work's full text
Lagrange reduces coordinates to independent variables by eliminating one per condition and allows any coordinates (art. 11-12), which is the generalized-coordinate method, though DOF is not named or claimed as originating here.
“on peut toujours réduire les variables qui déterminent la position de ces points dans l'espace, à un petit nombre de variables indé- pendantes, en éliminant, au moyen des équations de condition données par la nature du système, autant de variables qu'il y a de conditions”
- The construct originates in Lagrange's generalized-coordinate framework
This sourceDevelops the principle that for a system with k holonomic constraints and 3N particle coordinates, the number of independent generalized coordinates equals 3N − k; these q_i become the natural degrees of freedom for Lagrangian mechanics.
Supported in partVerified against the work's full text
Lagrange reduces position coordinates to independent variables by eliminating one per constraint equation, the generalized-coordinate method; the text makes no claim about origins.
“on peut toujours réduire les variables qui déterminent la position de ces points dans l'espace, à un petit nombre de variables indé- pendantes, en éliminant, au moyen des équations de condition données par la nature du système, autant de variables qu'il y a de conditions”
- The construct originates in Lagrange's generalized-coordinate framework
- Duality
- … cross-side inference: once the pairing is explicit and the preserved structure is named, theorems proved on one side immediately imply dual theorems on the other (doubling the payoff of each result), optimization problems that are hard in primal form can be recast and solved in dual form (where the Lagrangian dual
This sourceMultiplier technique originates in Lagrange's 1760s–70s calculus-of-variations memoirs. Historical treatment: Fraser, "Lagrange's Analytical Mathematics, Its Cartesian Origins and Reception in Comte's Positive Philosophy." Studies in History and Philosophy of Science 21, no. 2 (1990): 243–256; Goldstine, A History of the Calculus of Variations from the 17th through the 19th Century (Springer, 1980).
- … cross-side inference: once the pairing is explicit and the preserved structure is named, theorems proved on one side immediately imply dual theorems on the other (doubling the payoff of each result), optimization problems that are hard in primal form can be recast and solved in dual form (where the Lagrangian dual
- Noether's Theorem
- The variational structure underlying Noether's theorem roots in Lagrange (1788)
This sourceMultiplier technique originates in Lagrange's 1760s–70s calculus-of-variations memoirs. Historical treatment: Fraser, "Lagrange's Analytical Mathematics, Its Cartesian Origins and Reception in Comte's Positive Philosophy." Studies in History and Philosophy of Science 21, no. 2 (1990): 243–256; Goldstine, A History of the Calculus of Variations from the 17th through the 19th Century (Springer, 1980).
- The variational structure underlying Noether's theorem roots in Lagrange (1788)
- Oscillation
- Lagrange's 1788 Mécanique analytique
This sourceMultiplier technique originates in Lagrange's 1760s–70s calculus-of-variations memoirs. Historical treatment: Fraser, "Lagrange's Analytical Mathematics, Its Cartesian Origins and Reception in Comte's Positive Philosophy." Studies in History and Philosophy of Science 21, no. 2 (1990): 243–256; Goldstine, A History of the Calculus of Variations from the 17th through the 19th Century (Springer, 1980).
Supported in partVerified against the work's full text
Lagrange's Statics generalises the choice of coordinates and his preface promises unified formulae, but the dynamics and small-oscillation sections are missing from the text.
“rien n'o- blige dans cette méthode à se servir de coordonnées rectangles plutôt que d'autres lignes ou quantités relatives aux lieux des corps”
- Lagrange's 1788 Mécanique analytique
- Perturbation
- Reference state. A baseline — equilibrium configuration, exact solvable limit, well-understood operating point, current trajectory — is specified. Small departure. The perturbation is small in a specifiable sense (amplitude as fraction of baseline, energy as fraction of characteristic energy, probability as distance from typical realization). Linearized or expanded response. System response is analyzed as a series expansion in the perturbation amplitude — linear term, quadratic correction, higher orders — or as an ensemble of small-perturbation responses.
This sourceMultiplier technique originates in Lagrange's 1760s–70s calculus-of-variations memoirs. Historical treatment: Fraser, "Lagrange's Analytical Mathematics, Its Cartesian Origins and Reception in Comte's Positive Philosophy." Studies in History and Philosophy of Science 21, no. 2 (1990): 243–256; Goldstine, A History of the Calculus of Variations from the 17th through the 19th Century (Springer, 1980).
Supported in partVerified against the work's full text
Lagrange expands about an equilibrium state in a series in small displacements and keeps only second-order terms; the general perturbation method itself lies beyond the cached text.
“et réduisant en série, suivant les dimensions des quantités très-petites”
- Reference state. A baseline — equilibrium configuration, exact solvable limit, well-understood operating point, current trajectory — is specified. Small departure. The perturbation is small in a specifiable sense (amplitude as fraction of baseline, energy as fraction of characteristic energy, probability as distance from typical realization). Linearized or expanded response. System response is analyzed as a series expansion in the perturbation amplitude — linear term, quadratic correction, higher orders — or as an ensemble of small-perturbation responses.
- Principle of Least Action
- The principle of least action (more precisely, the principle of stationary action) is the foundational variational principle of classical and quantum physics, which states that the actual trajectory of a physical system between specified initial and final configurations is the one that makes a particular integral functional — the action S = ∫L dt (where L is the Lagrangian, the difference between kinetic and potential energy in the simplest case) — stationary with respect to small variations of the trajectory.
This sourceMultiplier technique originates in Lagrange's 1760s–70s calculus-of-variations memoirs. Historical treatment: Fraser, "Lagrange's Analytical Mathematics, Its Cartesian Origins and Reception in Comte's Positive Philosophy." Studies in History and Philosophy of Science 21, no. 2 (1990): 243–256; Goldstine, A History of the Calculus of Variations from the 17th through the 19th Century (Springer, 1980).
Supported in partVerified against the work's full text
Lagrange's text supports the multiplier treatment of constraints (equations de condition); the symmetry/conserved-charge (Noether) half is not in this source.
“Mais ces réductions pouvant être embarrassantes, il conviendra de les éviter par le moyen de la métbode des multiplicateurs”
- The principle of least action (more precisely, the principle of stationary action) is the foundational variational principle of classical and quantum physics, which states that the actual trajectory of a physical system between specified initial and final configurations is the one that makes a particular integral functional — the action S = ∫L dt (where L is the Lagrangian, the difference between kinetic and potential energy in the simplest case) — stationary with respect to small variations of the trajectory.
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