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Physics · Advanced university

Euler-Lagrange equation for a quartic potential

For L=mẋ²/2−ax⁴, the force is −4ax³. This one-dimensional model is conservative.

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Method and conditions

For L=mẋ²/2−ax⁴, the force is −4ax³. This one-dimensional model is conservative.

d/dt(∂L/∂ẋ) − ∂L/∂x = 0

Worked examples

Use SI-compatible constants in L=12ẋ²/2 − 4x⁴. Find ẍ when x=-2 m. Round the final result to 4 decimal places if needed. Answer in m/s².

Euler-Lagrange gives 12ẍ + 16x³=0. Substitute x=-2.

10.6667 m/s²

Use SI-compatible constants in L=7ẋ²/2 − 2x⁴. Find ẍ when x=2 m. Round the final result to 4 decimal places if needed. Answer in m/s².

Euler-Lagrange gives 7ẍ + 8x³=0. Substitute x=2.

-9.1429 m/s²

Try it

  1. Use SI-compatible constants in L=9ẋ²/2 − 9x⁴. Find ẍ when x=1 m. Round the final result to 4 decimal places if needed. Answer in m/s².

  2. Use SI-compatible constants in L=6ẋ²/2 − 4x⁴. Find ẍ when x=1 m. Round the final result to 4 decimal places if needed. Answer in m/s².

Check your work

Answers and marking notes
  1. -4 m/s²

    Euler-Lagrange gives 9ẍ + 36x³=0. Substitute x=1.

  2. -2.6667 m/s²

    Euler-Lagrange gives 6ẍ + 16x³=0. Substitute x=1.

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