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Lagrangian mechanics intro
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L is kinetic minus potential energy; motion extremises the action.
L = T − UThe Lagrangian L = T − U depends on coordinates and velocities. The equations of motion follow from the Euler–Lagrange equation, which works in any coordinates, ideal for constrained systems.
For a pendulum L = ½mL²θ̇² + mgL cosθ.
L is T − U, not T + U.
In more detail
The Lagrangian is the kinetic energy minus the potential energy. Its equations of motion, derived from the principle of least action, give the same results as Newton's laws in many problems.
Real-world uses
- Engineers use Lagrangian mechanics to model robot arms with several joints.
- Physicists use it to derive equations of motion for complex constrained systems.
- Robotics engineers write equations of motion for linked arms using the Lagrangian, which handles joint angles directly.
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Frequently asked questions
Why use the Lagrangian instead of Newton's laws?
It handles constraints and coordinates systematically, which can make complex problems easier.
Does it change the physics?
No. It is an equivalent formulation of classical mechanics.
Why use the Lagrangian instead of Newton's laws?
It avoids working out constraint forces, and the same method works in any coordinate system.
Related topics in Work, Energy & Power
Principle of virtual work · Work–energy with variable mass · Hamiltonian as total energy · Euler–Lagrange equations · Principle of least action · Noether's theorem