Questions: Lagrangian Mechanics (Introduction)

Question 1

A bead slides along a frictionless wire bent into an arbitrary curve. Using Newton's laws requires calculating the normal force at every point. Using Lagrangian mechanics with arc length along the wire as the generalized coordinate:

• A. Also requires calculating the normal force, because all forces must be accounted for in any mechanics approach
• B. Eliminates the need to calculate the normal force — it is a constraint force that does no work and disappears when the constraint is encoded in the generalized coordinate
• C. Cannot be used because the bead's motion is constrained to a curve
• D. Requires more equations than Newton's approach because generalized coordinates are more complex to define

Answer: (B)

Constraint forces (normal forces, string tension) do no work and don't appear in T − V when you choose generalized coordinates that automatically satisfy the constraint. With arc length as the single generalized coordinate, the bead is always on the wire by definition — the constraint is built into the coordinate system. The Euler-Lagrange equation yields the equation of motion directly, with the constraint force never appearing. This is one of the primary practical advantages of the Lagrangian framework.

Question 2

The Lagrangian of a system is found to not depend on position x — it only depends on velocity ẋ and time. By Noether's theorem, what is conserved?

• A. Energy, because any system with a well-defined Lagrangian conserves energy
• B. Angular momentum, because spatial independence always implies rotational symmetry
• C. Linear momentum in the x-direction, because translational symmetry in x implies momentum conservation
• D. Nothing — Noether's theorem applies only to time symmetry, not spatial symmetry

Answer: (C)

Noether's theorem states that each continuous symmetry of the Lagrangian corresponds to a conserved quantity. If L doesn't depend on x, the system is invariant under translation in x — and the conserved quantity is linear momentum in the x-direction. Time-translation symmetry → energy; spatial translation symmetry → momentum; rotational symmetry → angular momentum. Option A confuses time-translation symmetry (which conserves energy) with any arbitrary Lagrangian.

Question 3

The principle of stationary action states that nature usually takes the path that minimizes the action integral.

• T. True
• F. False

Answer: False

Hamilton's principle states that nature takes the path for which the action is stationary — meaning the first variation is zero — not necessarily minimized. The action can be a minimum, a maximum, or a saddle point depending on the system and boundary conditions. 'Principle of least action' is a historical misnomer that persisted; the mathematically correct statement is stationary action. The distinction matters in field theory and for paths near caustics.

Question 4

Noether's theorem implies that if a system's Lagrangian has time-translation symmetry (it does not explicitly depend on time), then the system's total energy is conserved.

• T. True
• F. False

Answer: True

Time-translation symmetry — the Lagrangian has the same form regardless of when you observe the system — corresponds via Noether's theorem to conservation of energy. This gives conservation of energy not as an independent empirical fact but as a logical consequence of time symmetry. Systems whose Lagrangians explicitly depend on time (e.g., driven oscillators with time-varying external forces) break time-translation symmetry and do not conserve energy.

Question 5

Why does using generalized coordinates make the Lagrangian approach easier to apply to constrained systems than Newton's force-based approach?

Model answer: In Newton's approach, constraints generate unknown forces (normal forces, tension) that must be solved for simultaneously with the equations of motion — introducing extra unknowns and equations. In the Lagrangian approach, you choose generalized coordinates that automatically satisfy the constraints, encoding them in the coordinate system itself. Since constraint forces do no work, they don't appear in T − V, and the Euler-Lagrange equations yield the equations of motion directly in terms of unconstrained degrees of freedom, with constraint forces never entering the calculation.

A pendulum illustrates this perfectly. Newton requires resolving tension along and perpendicular to the rod, setting up two coupled equations, and eliminating the tension. Lagrange uses θ alone as the generalized coordinate (the rod length is constant by constraint), writes T and V in terms of θ and θ̇, and the Euler-Lagrange equation immediately gives mℓ²θ̈ + mgℓ sin θ = 0 — the correct equation of motion with no mention of tension.