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The stationary-action criterion and the potential-part Euler-Lagrange equation

Physical scope

The Euler-Lagrange equation of the mechanical Lagrangian of a declared potential presentation: for a C^2 curve of the model on a fixed interval, stationarity of the action, the vanishing of the Euler-Lagrange expression and the pointwise equation mass-operator acceleration equals minus the potential gradient are equivalent. The equivalence identifies the variational critical-point condition with the potential-part equation on the interval, and with the full model equation when d is identically zero; it is conditional on the adopted force branch and the declared presentation, and it asserts no solution existence, no minimisation and no empirical content.

Statement

Adopt the force model with potential presentation $(U, d)$ and the mechanical Lagrangian $L = K - U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with masses $m_{1}, \dots, m_{N} > 0$ , open configuration domain $Q \subseteq R^{3 N}$ , time window $I_{0}$ , mass operator $M$ and $C^{2}$ potential $U : I_{0} \times Q \to R$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets). Let $a < b$ with $[a, b] \subseteq I_{0}$ , and let $x : [a, b] \to Q$ be a $C^{2}$ admissible curve (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations). Then the following three conditions are equivalent:

$x$ is stationary for the action $S$ of $L$ ;
$E_{L} (x) (t) = 0$ for every $t \in [a, b]$ ;
$M x^{''} (t) = - \nabla_{x} U (t, x (t))$ for every $t \in [a, b]$ .

Condition 3 is the model's equation of motion of the potential branch (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iv)) written along the curve for the presentation $(U, d)$ with the potential part alone: the residual forces $d$ do not appear, and the statement is therefore a statement about the potential part of the force map. No existence of curves satisfying these conditions is asserted.

Facts & Assumptions

Given: The model data, presentation and curve of the statement.

[F1]

The chart instance and first variation (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, The first variation of a time-dependent action on a finite interval: admissibility, the integral formula and continuity of the Euler-Lagrange coefficient): $(O, L)$ with $O = I_{0} \times Q \times R^{3 N}$ is a time-dependent Lagrangian chart over $[a, b] \times Q$ , and the notions of admissible curve, variation, action, first variation and stationarity are those of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations instantiated at it.

[F2]

The stationarity theorem (Stationarity of the time-dependent action is equivalent to the vanishing of the Euler-Lagrange expression): a $C^{2}$ admissible curve $q$ of a time-dependent Lagrangian chart is stationary for the action if and only if its Euler-Lagrange expression vanishes identically on the interval.

[F3]

The Euler-Lagrange expression in model terms (The first variation of the mechanical action and the Euler-Lagrange expression in model terms): for the curve $x$ one has $E_{L} (x) (t) = - (M x^{''} (t) + \nabla_{x} U (t, x (t)))$ for every $t \in [a, b]$ .

[F4]

The model's equation of motion (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iv), Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator): a motion of the model with force map $F$ satisfies $m_{i} x_{i}^{''} (t) = F_{i} (t, x (t), x^{'} (t))$ on its interval, equivalently $M x^{''} (t) = F (t, x (t), x^{'} (t))$ ; for the presentation $(U, d)$ of the statement the potential part of $F$ is $- \nabla_{x} U$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (a)).

Proof

technique · direct; combine the mathematical stationarity characterisation with the model-terms Euler-Lagrange identity and read the result through the equation of motion

1.1F1F2given

Conditions 1 and 2 are equivalent: $x$ is a $C^{2}$ admissible curve of the chart instance of [F1], so the stationarity theorem [F2] applied to that chart gives that $x$ is stationary for $S$ if and only if $E_{L} (x) (t) = 0$ for every $t \in [a, b]$ .

1.2F3algebra

Conditions 2 and 3 are equivalent: by [F3], at every $t \in [a, b]$ the value $E_{L} (x) (t)$ is the negative of $M x^{''} (t) + \nabla_{x} U (t, x (t))$ ; a vector-valued function on $[a, b]$ vanishes identically if and only if its negative does.

1.3F4given

Reading condition 3: by [F4], for the presentation $(U, d)$ the potential part of the model's force map is $- \nabla_{x} U$ , so condition 3 says that $x$ satisfies the model's equation of motion with the declared force map replaced by its potential part, i.e. the equation of the global potential model obtained from the presentation by discarding $d$ .

2.1step 1.1step 1.2step 1.3∎

Steps 1.1 and 1.2 establish the equivalence of conditions 1-3; step 1.3 records its reading as the model's equation of motion of the potential part. No existence, minimisation or empirical claim is made.

Remarks

Residual forces are excluded here. Condition 3 uses $- \nabla_{x} U$ only. When $d \equiv̸ 0$ the model's full equation of motion is $M x^{''} = F = - \nabla_{x} U + d$ , and the additional term $d$ appears on the right of condition 3; the variational reading of the full equation requires generalized forces and is treated separately on this page.
The equation is an identity of curves, not a solution statement. The equivalence is between three conditions on one given $C^{2}$ curve. Whether such curves exist through given data, and how they relate to the maximal motions of the force model, is the content of the following theorem.

The stationary-action criterion and the potential-part Euler-Lagrange equation

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources