# 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.

Course: [Analytical and statistical mechanics](https://studybrix.com/en/courses/analytical-and-statistical-mechanics/index.md)
Level: advanced
Prerequisites: Calculus of variations and Newtonian mechanics.

## 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.

## Practice questions

### 1. What is the usual variational condition on action?

A. Stationarity under allowed variations
B. Always zero action
C. Always maximum kinetic energy
D. Always a strict global minimum

Answer: Stationarity under allowed variations

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

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

Answer: 2

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

### 3. If ∂L/∂q=0, what is conserved?

A. ∂L/∂qdot
B. L in every driven system
C. Time itself
D. q in every trajectory

Answer: ∂L/∂qdot

Explanation: Euler-Lagrange gives zero derivative of the conjugate momentum.

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

Answer: 12

Explanation: p=∂L/∂qdot=4qdot=12.

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

Sample answer / marking guidance: The velocity derivative is mqdot, its time derivative is mqddot, and the coordinate derivative is −kq. Hence mqddot+kq=0.

Explanation: Credit both partial derivatives and their signs.

### 6. Which statement about generalized coordinates is correct?

A. There must be exactly three
B. They need not be Cartesian distances
C. They are always momenta
D. They must all be angles

Answer: They need not be Cartesian distances

Explanation: They parameterize independent configurations under constraints.

### 7. Why are endpoint variations set to zero in the standard fixed-endpoint derivation?

Sample answer / marking guidance: Integration by parts produces a boundary term involving δq. Fixed endpoints make that term vanish while leaving interior variations arbitrary.

Explanation: Credit the boundary term and interior arbitrariness.

### 8. Why must friction not be silently inserted into a conservative potential?

Sample answer / marking guidance: Generic friction depends on motion and dissipates energy rather than deriving from a position-only conservative potential. A generalized force or appropriate dissipative formulation is needed.

Explanation: Credit the physical and variational distinction.

## Further reading

- https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/
- https://openstax.org/details/books/university-physics-volume-3

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.