Constraint & Extremum

Projection / capability. Optimize whole paths or constrained choices and understand exactly which rules define the optimum.

Question this mathematics answers. Among admissible alternatives, which one is stationary/optimal under the declared objective and constraints?

Equation family

• Action functional: S[q]=∫ L(q,q̇,t)dt
• Euler–Lagrange: d/dt(∂L/∂q̇)−∂L/∂q=0
• Constrained extremum: ∇f=λ_c∇g

Symbols. q(t) = candidate path; L = Lagrangian; S = scalar functional of the whole path; f = objective; g = constraint; λ_c = Lagrange multiplier.

Assumptions. Variations respect declared endpoint/regularity conditions. Lagrange-multiplier condition assumes differentiability and regular constraint geometry near the candidate optimum.

Where the inputs come from. Objective/constraint definitions come from the problem owner or physical theory; they are not chosen by the equation.

Short derivation / justification

For q→q+εδq with fixed endpoint variations, differentiate S with respect to ε, integrate the δq̇ term by parts, and require first variation zero for arbitrary δq. This yields Euler–Lagrange. For g=const, feasible directions are tangent to the constraint; at an optimum ∇f has no tangent component, so it is parallel to ∇g.

Worked example

Maximize f=x+2y on x²+y²=1. ∇f=(1,2) must be parallel to (2x,2y), so optimum direction is proportional to (1,2): (x,y)=(1,2)/√5.

Cadet or real-world application. Trajectory design, constrained allocation, mechanical paths, parameter fitting.

What to look for. Stationary does not always mean global minimum; inspect second-order/global structure as needed.

Common misinterpretation. “The mathematically optimal solution chose the objective for us.”

Break / domain boundary. Optimization solves the declared problem; it does not grant authority to declare the objective, constraints, or moral value.