Questions: Lagrangian Mechanics: Foundations and Applications

Question 1

A bead is constrained to slide on a frictionless wire shaped as a parabola. How does the Lagrangian approach handle the constraint force that keeps the bead on the wire?

• A. The constraint force is computed first using Newton's second law, then substituted into the Lagrangian
• B. The constraint force appears as an extra term in the Euler-Lagrange equation with a Lagrange multiplier
• C. The constraint is absorbed by choosing a single generalized coordinate (position along the wire); T and V are written in this coordinate, and the constraint force never appears
• D. The Lagrangian method cannot handle constraints — you must first convert to Cartesian coordinates

Answer: (C)

This is the core practical advantage of the Lagrangian formulation. Instead of explicitly computing the normal force that keeps the bead on the wire, you simply parametrize the bead's position by a single number s (arc length along the wire). Every position of the bead satisfying the constraint corresponds to a value of s; the constraint is encoded in the choice of coordinates, not as a separate equation. Writing T and V in terms of s and applying the Euler-Lagrange equation yields the equation of motion directly — the constraint force never enters the calculation, because you never needed it.

Question 2

If the Lagrangian L = T − V for a rotating system does not depend explicitly on the rotation angle θ (θ is a cyclic coordinate), what can you immediately conclude?

• A. The kinetic energy T is constant throughout the motion
• B. The total energy T + V is conserved
• C. The generalized momentum p_θ = ∂L/∂θ̇ — which corresponds to angular momentum — is conserved
• D. The rotation angle θ must be constant, so the system is not rotating

Answer: (C)

This is Noether's theorem applied directly. A cyclic (ignorable) coordinate is one that does not appear explicitly in L — only its time derivative θ̇ appears. The Euler-Lagrange equation then reads d/dt(∂L/∂θ̇) = ∂L/∂θ = 0, which means the generalized momentum p_θ = ∂L/∂θ̇ is constant. For rotation angle, p_θ is the angular momentum — so independence of θ means angular momentum conservation. This is why Noether's theorem is so powerful: symmetry of L (here, rotational symmetry) directly implies conservation laws without any force analysis.

Question 3

The Euler-Lagrange equations derived from L = T − V yield the same equations of motion as Newton's second law F = ma for the same physical system.

• T. True
• F. False

Answer: True

Lagrangian mechanics is not a different physical theory — it is a reformulation of classical mechanics. Both Newton's laws and the Euler-Lagrange equations describe the same physics; they are mathematically equivalent for conservative systems. The Lagrangian formulation has practical advantages for constrained and multi-body systems, but it does not predict different trajectories. You can verify this by deriving equations of motion for a simple pendulum using both methods: both yield the same θ̈ + (g/L)sin θ = 0.

Question 4

To use Lagrangian mechanics on a constrained system, you is expected to first solve for most constraint forces, then eliminate them from the equations of motion.

• T. True
• F. False

Answer: False

This describes the Newtonian approach, not the Lagrangian one. The Lagrangian method's key advantage is precisely that you *never* compute constraint forces. Instead, you choose generalized coordinates that automatically satisfy the constraints — by construction, any values of the generalized coordinates correspond to configurations that obey all constraints. Writing T and V in these coordinates and applying the Euler-Lagrange equations yields the complete equations of motion without constraint forces ever appearing. For a system with k constraints, you use (3N − k) generalized coordinates instead of 3N Cartesian coordinates.

Question 5

How does Noether's theorem reveal the connection between the form of the Lagrangian and conservation laws, and why is this more systematic than identifying conservation laws in Newtonian mechanics?

Model answer: Noether's theorem states that for every continuous symmetry of the Lagrangian, there is a corresponding conserved quantity. Specifically, if L does not depend explicitly on a generalized coordinate q_i (a cyclic coordinate), then the generalized momentum p_i = ∂L/∂q̇_i is conserved. This means: no dependence on position → linear momentum conservation; no dependence on rotation angle → angular momentum conservation; no explicit time dependence → energy conservation. The systematic part is that you identify conservation laws directly from the structure of L, by inspection. In Newtonian mechanics, conservation laws must be discovered through algebraic manipulation of force equations — they are not immediately visible from F = ma. The Lagrangian approach makes symmetry the starting point, so conservation laws are derived conclusions rather than discovered surprises.

This is why physicists value the Lagrangian (and Hamiltonian) formulations even for problems that could in principle be solved with Newton's laws. Writing down L forces you to identify the symmetries of the system, which immediately tells you what is conserved — which in turn constrains and simplifies the solution. Conservation laws that would require pages of force analysis in the Newtonian framework emerge in one line from the Lagrangian.