EQ-EULER-LAGRANGE · Variational Mechanics
Euler–Lagrange Equation (1D)
-dL_dq + d_dt_dL_dqdot = 0
Variables
variable
dL_dq
generalized force ∂L/∂q
- Object
- point_particle
- Property
- LagrangianPartial
- Context
- inertial_frame
- Constraint
- with_respect_to_coordinate
variable
d_dt_dL_dqdot
time derivative of the generalized momentum ∂L/∂q̇
- Object
- point_particle
- Property
- GeneralizedMomentum
- Context
- inertial_frame
- Constraint
- time_derivative
Axioms
algebraic classical conservative_force deterministic linear non_relativistic stationary_action variational
Assumptions
- L is twice-differentiable in q, q̇, and t
- Action S = ∫L dt is stationary at the trajectory
- Endpoints of the path are fixed (Hamilton's principle)
- No explicit constraint forces (unconstrained system)
Derivation
- Hamilton's principle: δS = δ∫L dt = 0 on fixed-endpoint variations
- Integration by parts + endpoint vanishing leaves d/dt(∂L/∂q̇) − ∂L/∂q = 0 as the Euler–Lagrange equation
- Goldstein, Classical Mechanics, 3rd ed., §2.3
References
- Goldstein, Classical Mechanics, 3rd ed., §2.3 Eq. (2.13)
- Landau & Lifshitz, Mechanics, §2, Eq. (2.6)
- Lanczos, The Variational Principles of Mechanics, 4th ed., Ch. III