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Euler–Lagrange equations
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Equations of motion from L, valid in any coordinates.
d/dt(∂L/∂q̇) = ∂L/∂qd/dt(∂L/∂q̇) = ∂L/∂q gives one equation per coordinate. For the pendulum it reproduces θ̈ = −(g/L) sinθ.
Applied to a free particle it gives constant velocity.
Treat q and q̇ as independent when differentiating partially.
In more detail
The Euler-Lagrange equations come from the Lagrangian by requiring the action to be stationary. They give the equations of motion for each generalised coordinate.
Real-world uses
- Deriving the equations of motion for a double pendulum.
- Describing the motion of charged particles in electromagnetic fields.
- Physicists derive the motion of a double pendulum by applying the Euler-Lagrange equations to its Lagrangian.
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Frequently asked questions
What is a generalised coordinate?
Any set of variables that describes the configuration of a system, such as angles or lengths.
Do they require forces to be known?
Not directly, because the potential determines the forces.
Are the Euler-Lagrange equations the same as Newton's second law?
They are equivalent for standard systems, but they are easier to use when constraints or unusual coordinates are involved.
Related topics in Work, Energy & Power
Work–energy with variable mass · Hamiltonian as total energy · Lagrangian mechanics intro · Principle of least action · Noether's theorem