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Physics · Advanced university

Euler-Lagrange equations and constraints

For L(q,qdot,t), stationarity of S=∫L dt with fixed endpoints yields d/dt(∂L/∂qdot)−∂L/∂q=0.

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Prerequisites

Calculus of variations and Newtonian mechanics.

Study notes

Action

For L(q,qdot,t), stationarity of S=∫L dt with fixed endpoints yields d/dt(∂L/∂qdot)−∂L/∂q=0.

Coordinates

Generalized coordinates describe independent degrees of freedom. Holonomic constraints can reduce their number when handled consistently.

Cyclic coordinate

If L has no dependence on a coordinate, its conjugate momentum is conserved, provided the Euler-Lagrange assumptions apply.

Caution

Stationary action is not always a minimum. Nonconservative forces may require generalized forces or an extended formulation.

Worked examples

What is the usual variational condition on action?

The first variation vanishes; its sign at second order is a separate issue.

Stationarity under allowed variations

For L=3qdot²/2−6q²/2, find the coefficient a in qddot=−a q.

The equation is 3qddot+6q=0, so a=2.

2

Check your understanding

  1. If ∂L/∂q=0, what is conserved?

    1. ∂L/∂qdot
    2. L in every driven system
    3. Time itself
    4. q in every trajectory

  2. For L=2qdot² and qdot=3, find conjugate momentum.

  3. Derive the harmonic-oscillator equation from L=mqdot²/2−kq²/2.

  4. Which statement about generalized coordinates is correct?

    1. There must be exactly three
    2. They need not be Cartesian distances
    3. They are always momenta
    4. They must all be angles

Answers and marking guidance

Answers and marking notes
  1. ∂L/∂qdot

    Euler-Lagrange gives zero derivative of the conjugate momentum.

  2. 12

    p=∂L/∂qdot=4qdot=12.

  3. The velocity derivative is mqdot, its time derivative is mqddot, and the coordinate derivative is −kq. Hence mqddot+kq=0.

    Credit both partial derivatives and their signs.

  4. They need not be Cartesian distances

    They parameterize independent configurations under constraints.

Further reading and review references

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