The Euler-Lagrange Equations

%23+The+Euler-Lagrange+Equation%3A+The+Core+of+Mechanics%0A%0AThe+Euler-Lagrange+equation+is+the+single+most+important+equation+in+analytical+mechanics.+One+line.+All+of+classical+physics.+Light.+Quantum+fields.+General+relativity.+They+all+flow+from+it.%0A%0A%5Cfrac%7Bd%7D%7Bdt%7D%5Cfrac%7B%5Cpartial+L%7D%7B%5Cpartial+%5Cdot%7Bq%7D_i%7D+-+%5Cfrac%7B%5Cpartial+L%7D%7B%5Cpartial+q_i%7D+%3D+0%0A%0AThis+is+it.+The+equation+of+motion.+No+forces.+No+vectors.+No+free-body+diagrams.+Just+the+Lagrangian+L+and+its+derivatives%2C+and+the+trajectory+falls+out.%0A%0AThe+Lagrangian+is+a+function+of+positions+q_i%2C+velocities+%5Cdot%7Bq%7D_i%2C+and+possibly+time+t.+For+a+simple+particle+in+a+potential%2C+L+%3D+T+-+V+%3D+%5Cfrac%7B1%7D%7B2%7Dm%5Cdot%7Bx%7D%5E2+-+V%28x%29.+Plug+this+in%3A%0A%0A%5Cfrac%7B%5Cpartial+L%7D%7B%5Cpartial+%5Cdot%7Bx%7D%7D+%3D+m%5Cdot%7Bx%7D+%5Cquad%5CRightarrow%5Cquad+%5Cfrac%7Bd%7D%7Bdt%7D%28m%5Cdot%7Bx%7D%29+%3D+m%5Cddot%7Bx%7D%0A%0A%5Cfrac%7B%5Cpartial+L%7D%7B%5Cpartial+x%7D+%3D+-%5Cfrac%7BdV%7D%7Bdx%7D%0A%0ASo+m%5Cddot%7Bx%7D+%3D+-%5Cfrac%7BdV%7D%7Bdx%7D.+Newton%27s+second+law.+Recovered+from+a+single+scalar+function.%0A%0ABut+the+power+is+not+that+it+recovers+Newton.+The+power+is+that+it+generalizes.+Any+coordinate+system+works.+Polar+coordinates%2C+spherical+coordinates%2C+generalized+coordinates+constrained+by+whatever+geometry+you+please.+The+equation+looks+the+same.+You+just+write+down+T+-+V+in+your+chosen+coordinates+and+apply+the+formula.+No+need+to+draw+diagrams.+No+need+to+resolve+vectors+into+components.%0A%0AThe+equation+comes+from+a+variational+principle.+Define+the+action%3A%0A%0AS+%3D+%5Cint_%7Bt_1%7D%5E%7Bt_2%7D+L%28q%2C+%5Cdot%7Bq%7D%2C+t%29+dt%0A%0AThe+action+is+a+number+for+each+possible+path+q%28t%29+between+the+endpoints.+The+Euler-Lagrange+equation+says%3A+the+actual+path+taken+by+the+system+is+the+one+that+makes+the+action+stationary.+%5Cdelta+S+%3D+0.+The+system+does+not+%22know%22+it+is+minimizing+anything.+It+simply+obeys+the+differential+equation%2C+and+that+differential+equation+is+equivalent+to+the+stationarity+of+the+action.%0A%0AThis+equivalence+is+not+mathematical+trivia.+The+variational+formulation+is+more+fundamental+than+the+differential+formulation%2C+because+it+is+coordinate-independent.+The+action+S+is+a+scalar.+It+does+not+depend+on+how+you+describe+the+system.+The+Euler-Lagrange+equations+derived+from+it+then+automatically+work+in+any+coordinate+system.+This+is+why+the+Euler-Lagrange+equation+is+the+starting+point+for+modern+physics.%0A%0AIn+quantum+mechanics%2C+Feynman%27s+path+integral+formulation+takes+the+action+literally.+Every+possible+path+contributes+to+the+quantum+amplitude%2C+weighted+by+e%5E%7BiS%2F%5Chbar%7D.+The+classical+path+%E2%80%94+the+one+satisfying+the+Euler-Lagrange+equation+%E2%80%94+is+where+the+phases+constructively+interfere.+The+quantum+theory+selects+the+classical+trajectory+not+by+imposing+a+differential+equation+but+by+summing+over+all+paths+and+letting+the+stationary+action+principle+emerge+from+interference.+The+Euler-Lagrange+equation+is+the+classical+shadow+of+a+quantum+sum.%0A%0AIn+general+relativity%2C+the+Lagrangian+is+the+Ricci+scalar+R.+The+Euler-Lagrange+equation+applied+to+the+metric+tensor+gives+the+Einstein+field+equations.+G_%7B%5Cmu%5Cnu%7D+%3D+8%5Cpi+G+T_%7B%5Cmu%5Cnu%7D+comes+from+varying+the+Einstein-Hilbert+action+S+%3D+%5Cint+R+%5Csqrt%7B-g%7D+d%5E4x.+Geometry+itself+obeys+the+Euler-Lagrange+equation.%0A%0AIn+quantum+field+theory%2C+the+Lagrangian+density+%5Cmathcal%7BL%7D+depends+on+fields+%5Cphi+and+their+derivatives+%5Cpartial_%5Cmu%5Cphi.+The+Euler-Lagrange+equation+generalizes+to%3A%0A%0A%5Cpartial_%5Cmu+%5Cfrac%7B%5Cpartial+%5Cmathcal%7BL%7D%7D%7B%5Cpartial%28%5Cpartial_%5Cmu+%5Cphi%29%7D+-+%5Cfrac%7B%5Cpartial+%5Cmathcal%7BL%7D%7D%7B%5Cpartial+%5Cphi%7D+%3D+0%0A%0AApply+this+to+the+electromagnetic+field+and+you+get+Maxwell%27s+equations.+Apply+it+to+the+Dirac+field+and+you+get+the+Dirac+equation.+The+Standard+Model+Lagrangian+%E2%80%94+a+sum+of+terms+for+U%281%29%2C+SU%282%29%2C+and+SU%283%29+gauge+fields%2C+plus+the+Higgs+sector%2C+plus+fermion+couplings+%E2%80%94+is+written+down%2C+and+the+Euler-Lagrange+equations+give+every+equation+of+particle+physics.%0A%0AThe+Euler-Lagrange+equation+is+also+the+gateway+to+conservation+laws.+If+L+does+not+depend+explicitly+on+a+coordinate+q_i+%28q_i+is+cyclic%29%2C+then+%5Cfrac%7B%5Cpartial+L%7D%7B%5Cpartial+%5Cdot%7Bq%7D_i%7D+is+conserved.+This+is+the+momentum+conjugate+to+q_i%2C+and+its+conservation+follows+immediately.+Time+translation+symmetry+%E2%86%92+energy+conservation.+Spatial+translation+symmetry+%E2%86%92+momentum+conservation.+Rotation+symmetry+%E2%86%92+angular+momentum+conservation.+Noether%27s+theorem%2C+derived+from+the+action%2C+is+one+line+of+algebra+once+you+have+the+Euler-Lagrange+equation.%0A%0AThe+equation+is+simple.+It+is+also+universal.+Every+physical+theory+built+since+Lagrange+first+articulated+it+in+1788+has+been+an+application+of+this+single+formula+to+a+suitably+chosen+Lagrangian.+The+equation+is+the+master+key.