Lagrange equations
Objective. Derive motion from kinetic and potential energy.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Lagrange's equations derive equations of motion directly from energy, bypassing explicit force analysis. Define the Lagrangian L = T − V, kinetic minus potential energy, in terms of generalised coordinates qᵢ and velocities q̇ᵢ; then each coordinate obeys d/dt(∂L/∂q̇ᵢ) − ∂L/∂qᵢ = 0. Example: for a pendulum with L = ½ml²θ̇² − mgl(1−cosθ), computing ∂L/∂θ̇ = ml²θ̇, its time derivative ml²θ̈, and ∂L/∂θ = −mgl sinθ gives ml²θ̈ + mgl sinθ = 0, the pendulum equation. This method handles constrained and curvilinear motion without tracking constraint forces.
Worked example
A particle of mass m moves on a line with Lagrangian L = ½mẋ² − ½kx². Use Lagrange's equation to find the equation of motion.
- 1Compute ∂L/∂ẋ = mẋ
- 2Differentiate in time: d/dt(mẋ) = mẍ
- 3Compute ∂L/∂x = −kx
- 4Apply Lagrange's equation: mẍ − (−kx) = 0, giving mẍ + kx = 0
Answer: mẍ + kx = 0
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Practice
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Start practisingCommon mistake. Students differentiate L with respect to q̇ᵢ and q̇ᵢ alone, forgetting the explicit time derivative d/dt in the first term, which is essential whenever ∂L/∂q̇ᵢ depends on q itself.
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