Euler-Lagrange Equations
The Euler-Lagrange equations are the differential equations of motion that follow from requiring the action to be stationary, one equation for each generalized coordinate.
Definition
The Euler-Lagrange equations are the set of second-order differential equations, obtained by setting the variation of the action to zero, that govern the time evolution of each generalized coordinate of a mechanical system.
Scope
This topic covers the derivation of the Euler-Lagrange equations from Hamilton's principle, their form for systems of generalized coordinates, the definition of generalized (canonical) momenta, the treatment of cyclic coordinates yielding conserved momenta, and their extension to systems with constraints via Lagrange multipliers. They are the central computational output of Lagrangian mechanics.
Core questions
- How do the Euler-Lagrange equations follow from the stationary-action condition?
- What is a generalized momentum, and when is it conserved?
- How are constraints incorporated through Lagrange multipliers?
Key concepts
- Generalized coordinates and velocities
- Generalized (canonical) momentum
- Cyclic (ignorable) coordinates
- Lagrange multipliers for constraints
- Equivalence with Newton's second law
Key theories
- Euler-Lagrange equations of motion
- Demanding stationary action gives, for each generalized coordinate, an equation equating the time derivative of the generalized momentum to the generalized force derived from the Lagrangian.
- Cyclic coordinates and conserved momenta
- When the Lagrangian does not depend on a particular coordinate, the corresponding generalized momentum is conserved, giving a direct route to constants of motion.
Clinical relevance
Because they generate equations of motion directly from energies in any convenient coordinates, the Euler-Lagrange equations are the standard derivation tool in robotics, aerospace multibody dynamics, and control engineering, where Cartesian force balances would be cumbersome.
History
Euler derived the central equation of the calculus of variations in the 1740s, and Lagrange generalized it and applied it systematically to mechanics in his 1788 Mécanique analytique, giving the equations their joint name. Their reinterpretation through Hamilton's principle in the nineteenth century clarified their variational origin.
Key figures
- Leonhard Euler
- Joseph-Louis Lagrange
- William Rowan Hamilton
Related topics
Seminal works
- goldstein2002
- arnold1989
Frequently asked questions
- Are the Euler-Lagrange equations equivalent to Newton's laws?
- Yes, for systems both describe. In Cartesian coordinates with the Lagrangian T − V they reproduce Newton's second law exactly, but they are valid in any generalized coordinates and handle constraints more cleanly.
- What is a generalized momentum?
- It is the derivative of the Lagrangian with respect to a generalized velocity; for ordinary Cartesian coordinates it reduces to the usual linear momentum, but for an angular coordinate it is an angular momentum.