• describes the motion and dynamics of systems using energies rather than forces
• based on the principle of stationary action: the path a physical system takes is the one for which action is stationary.

T – kinetic, V – potential

Key concepts

• The Lagrangian ($L$):
  • unlike the total energy ($T+V$) used in Hamiltonian mechanics, the Lagrangian is the difference in energies.
• Action ($A$): a quantity that defines a specific trajectory through space and time
  • integral over time of the Lagrangian along a given path: $A={\int }_{t_1}^{t_2}Ldt$.
• Principle of Stationary Action: Principle of Least Action
  • the actual trajectory a system follows is the one where the action is stationary (functional differential is zero, $\delta A=0$)
• Euler-Lagrange Equation (Lagrangian version of Newton’s second law ($F=ma$): $$\frac{d}{dt}\frac{\partial L}{\partial \dot x}=\frac{\partial L}{\partial x}$$

Advantages Over Newtonian Mechanics

• scalar quantities (less directions to track)
• generalized coordinates: freedom in choosing coordinate systems
• forces of constraint, e.g., pendulum moving in a circle
• Look at the degrees of freedom: tells how many Euler-Lagrange equations there are
• step by step: find coordinates, define the Lagrangian, apply the EL equations, solve the differential equations
• Noether's theorem -> identify the conservation laws, symmetries of the system