# 6: Lagrangian Dynamics

- Page ID
- 9599

- 6.1: Introduction to Lagrangian Dynamics
- The algebraic Lagrange mechanics approach is based on the concept of scalar energies which circumvents many difficulties in handling constraint forces and many-body systems.

- 6.2: Newtonian plausibility argument for Lagrangian mechanics
- Insight into the physics underlying Lagrange mechanics is given by showing the direct relationship between Newtonian and Lagrangian mechanics. The variational approaches to classical mechanics exploit the first-order spatial integral of the force, which equals the work done between the initial and final conditions.

- 6.3: Lagrange Equations from d’Alembert’s Principle
- The Principle of Virtual Work provides a basis for a rigorous derivation of Lagrangian mechanics.

- 6.4: Lagrange equations from Hamilton’s Principle
- Lagrange equations from Hamilton's action principle underlies Lagrangian mechanics.

- 6.5: Constrained Systems
- The motion for systems subject to constraints is difficult to calculate using Newtonian mechanics because all the unknown constraint forces must be included explicitly with the active forces in order to determine the equations of motion. Lagrangian mechanics avoids these difficulties by allowing selection of independent generalized coordinates that incorporate the correlated motion induced by the constraint forces. This allows the constraint forces acting on the system to be ignored.

- 6.6: Applying the Euler-Lagrange equations to classical mechanics
- d’Alembert’s principle of virtual work is used to derive the Euler-Lagrange equations, which also satisfy Hamilton’s Principle, and the Newtonian plausibility argument.

Thumbnail: Joseph-Louis Lagrange