A simple Lagrangian

Classical Mechanics Level 3

Consider a particle initially at rest at time t = 0 t=0 and coordinate x = 0 x = 0 . The particle's motion is given by the following Lagrangian:

L = 1 2 ( d x d t ) 2 + 2 x L = \frac{1}{2} \left(\frac{dx}{dt}\right)^2+2x

Evaluate the integral S = ∫ 0 3 L d t . \int \limits_{0}^{3}L\, dt.


The answer is 36.

Harmonic Oscillator by harmonic oscillator , 19, Japan

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1 solution

From the Euler-Lagrange equation , we find the equation of motion to be:

d 2 x d t 2 \frac{d^2x}{dt^2} - 2 = 0.

Solving the above equation with the initial conditions, we get x = t 2 x = t^2 .

The integral then becomes ∫ 0 3 4 t 2 d t = 36 \int_0^3 4t^2dt = 36 .

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