Physics · Advanced university
Euler-Lagrange equations and constraints · revision notes
For L(q,qdot,t), stationarity of S=∫L dt with fixed endpoints yields d/dt(∂L/∂qdot)−∂L/∂q=0.
Key ideas and conditions
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.
Explain without looking
- Why are endpoint variations set to zero in the standard fixed-endpoint derivation?
- Why must friction not be silently inserted into a conservative potential?
Further reading and review references
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