Constraints, variational equations and cyclic coordinates
| English | Français |
|---|---|
| holonomic constraint | holonomic constraint |
| cyclic coordinate | cyclic coordinate |
A decision before an answer
- A pendulum tension changes throughout a swing, but its angular equation can be derived without solving for tension.
- Your goal: Derive Euler–Lagrange motion after imposing a holonomic constraint.
Choose independent coordinates
- A constraint removes an independent coordinate before you form the kinetic energy. For a fixed-length pendulum choose θ from the downward vertical: x=l sinθ and y=−l cosθ. Differentiating both coordinates gives v²=l²θdot², not l² sin²θ θdot².
- Thus T=ml²θdot²/2 and V=mgl(1−cosθ), with zero potential at the bottom. The fixed length is a holonomic constraint: it is a relation among coordinates and possibly time. A rolling velocity constraint needs its own analysis; do not automatically treat every constraint as a coordinate substitution.
For a pendulum angle θ measured from downward vertical, which kinetic energy uses both Cartesian velocity components?
Adding xdot² and ydot² gives l²θdot².
Derive the equation
- For ideal constraints and the usual conservative system, the stationary-action equation is d/dt(∂L/∂qdot)−∂L/∂q=0 with L=T−V. The variations vanish at the two time endpoints. A pendulum gives ∂L/∂θdot=ml²θdot and ∂L/∂θ=−mgl sinθ, so ml²θddot+mgl sinθ=0.
- The momentum derivative acts on every time-dependent factor: for a varying radius, d(mr²θdot)/dt contains 2mr rdot θdot. The variational method does not mean mechanical energy is conserved in a time-dependent system.
In L=m(rdot²+r²θdot²)/2−V(r), which statement follows from the cyclic angle?
∂L/∂θ=0 gives d(mr²θdot)/dt=0; radius and angular speed may change.
Find cyclic momentum
- For planar central motion, L=m(rdot²+r²θdot²)/2−V(r). The angle θ is cyclic because L has no explicit θ dependence, even though L depends on θdot. Its canonical momentum pθ=mr²θdot is constant. The radial equation is m rddot=mrθdot²−dV/dr.
- The first term is part of the coordinate acceleration, not a new outward real force in an inertial frame. Conservation of pθ makes angular speed increase as r decreases; constant angular speed is a different, externally driven situation.
With l=2 m and g=10 m/s², θddot=−5 sinθ. At θ=0.10 rad the exact acceleration is −0.4992 rad/s², close to −0.500 from linearisation. The small-angle period is 2π/sqrt(5)=2.810 s. For a central-force orbit, shrinking r from 3 m to 1.5 m while pθ is conserved multiplies θdot by 4, not 2.
For V=9q²/2 and local inertia M=1, small-oscillation angular frequency is ____ rad/s.
V″=9 and sqrt(9/1)=3.
Linearise with conditions
- Near a stable equilibrium q0, expand V to second order and use a constant local inertia M: V≈V(q0)+V″(q0)(q−q0)²/2. Then the small-displacement frequency is sqrt(V″/M). For a pendulum sinθ≈θ in radians, so ω=sqrt(g/l); the approximation requires small amplitude.
- A negative V″ gives instability, not an oscillation with an imaginary measurable frequency. Check the full nonlinear equation first, then state the approximation and compare units: V″/M must have units of inverse time squared.
A cyclic coordinate is absent from L itself; its velocity need not be absent. Do not discard the time derivative of a changing r² factor.
Which answer fits this case?
Derive Euler–Lagrange motion after imposing a holonomic constraint
The small-angle pendulum approximation is unchanged if the angle is inserted numerically in degrees.
sinθ≈θ requires radians; a degree value needs conversion.
Keep the distinctions
- holonomic constraint 完整约束 — A constraint expressible as a relation among coordinates and possibly time.
- cyclic coordinate 循环坐标 — A coordinate absent explicitly from the Lagrangian, with conserved canonical momentum under the Euler–Lagrange equation.
- Derive Euler–Lagrange motion after imposing a holonomic constraint.
- Use a cyclic coordinate to identify conserved canonical momentum.
- Linearise a stable equilibrium and check the resulting frequency.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.