Lagrangian Mechanics

• Energy based
• Practical problems with Newtonian mechanics led to development of Lagrangian mechanics Example of problem that would be nearly impossible to solve with newtonian mechanics

Lagrange’s equation of motion

aka Euler-Lagrange Equation

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}-\frac{\partial L}{\partial x}=0$$

5 simple steps to Lagrangian mechanics

1. Find the kinetic energy (T) $T=\frac{1}{2}m{v}^2$

2. Find the potential energy $U=mgh$ for gravity

3. Calculate Lagrangian $L=T-U$

4. Calculate Euler-Lagrange Equation $\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}-\frac{\partial L}{\partial x}=0$

5. Solve Diff. eq (optional)

   See 2024-01-23 Lagrangian Mechanics for examples. See HW3 Paul Lea and HW4 Paul Lea for more (probably incorrect ) examples