Classical Mechanics PHYS 2006 Tim Freegarde 2020 21

Classical Mechanics PHYS 2006 Tim Freegarde 2020 -21 29 Lagrangian mechanics

Lagrangian mechanics RECIP E write the system Lagrangian 1. kinetic potential 2. write the Euler-Lagrange equation 3. substitute (1) into (2), treating and as independent variables J-L Lagrange (1736 -1813) 2

Lagrangian mechanics RECIP E write the system Lagrangian 1. • Motion in a non-uniform potential kinetic potential 2. write the Euler-Lagrange equation 3. substitute (1) into (2), treating and as independent variables NEWTON’S 2 ND LAW 3

Lagrangian mechanics RECIP E write the system Lagrangian 1. • 1 -D harmonic oscillator kinetic potential 2. write the Euler-Lagrange equation 3. substitute (1) into (2), treating and as independent variables 4

Lagrangian mechanics: springy pendulum EQUIVALENT TO RESOLVING RADIALLY ALONG SPRING EQUIVALENT TO EQUATING TORQUE ABOUT PIVOT TO RATE OF CHANGE OF ANGULAR MOMENTUM 5

Lagrangian mechanics RECIP E write the system Lagrangian 1. • Springy pendulum kinetic potential 2. write the Euler-Lagrange equations 3. substitute (1) into (2), treating and as independent variables 6

Lagrangian mechanics RECIP E write the system Lagrangian 1. • 3 -D harmonic oscillator kinetic potential 2. write the Euler-Lagrange equation 3. substitute (1) into (2), treating and as independent variables 7

Principle of least action (stationary action) 8

Calculus of variations i. e. a straight line 9

Fermat’s principle of least time S S a P b P 0 x L x • light rays follow the path of least time between two points Pierre de Fermat (1601 -1665) Principle of least action • classical trajectory is path of least action J-L Lagrange (1736 -1813) 10

Path integral in quantum mechanics • all possible contributions to the quantum wavefunction are combined, with phases depending on the action • contributions interfere, favouring those around the path of least variation • the quantum mechanical trajectory is path of least action Richard Feynman (1918 -1988) 11

Classical Mechanics PHYS 2006 Tim Freegarde