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Stationarity under allowed variations
The first variation vanishes; its sign at second order is a separate issue.
2
The equation is 3qddot+6q=0, so a=2.
∂L/∂qdot
Euler-Lagrange gives zero derivative of the conjugate momentum.
12
p=∂L/∂qdot=4qdot=12.
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.
They need not be Cartesian distances
They parameterize independent configurations under constraints.
For L(q,qdot,t), stationarity of S=∫L dt with fixed endpoints yields d/dt(∂L/∂qdot)−∂L/∂q=0.
Advanced universityFor L(q,qdot,t), stationarity of S=∫L dt with fixed endpoints yields d/dt(∂L/∂qdot)−∂L/∂q=0.
Advanced universityFor a regular Lagrangian, p=∂L/∂qdot and H=p qdot−L after expressing qdot in terms of q,p,t.
Advanced universityFor a regular Lagrangian, p=∂L/∂qdot and H=p qdot−L after expressing qdot in terms of q,p,t.
Advanced universityFor L=mẋ²/2−ax⁴, the force is −4ax³. This one-dimensional model is conservative.
Advanced universityFor L=mẋ²/2−ax⁴, the force is −4ax³. This one-dimensional model is conservative.
Advanced universityFor H=p²/(2m)+V(q), momentum is canonical and the potential has no momentum dependence.
Advanced universityFor H=p²/(2m)+V(q), momentum is canonical and the potential has no momentum dependence.
Advanced universityUse the declared order; reversing f and g reverses the bracket sign.
Advanced universityUse the declared order; reversing f and g reverses the bracket sign.
Advanced universityConnect rotation, oscillations, central forces and continuous wave models.
Work with Gauss laws, electrical field models, thermodynamic laws and electromagnetic waves.
Compare relativistic measurement with quantum states, the Schrödinger equation and uncertainty.
Use Lagrangian and Hamiltonian methods, then interpret canonical ensembles.
Explore perturbation theory, scattering, variational quantum estimates and a conceptual bridge to general relativity.