The Hamiltonian

%23+The+Hamiltonian%3A+A+Restatement+of+Mechanics%0A%0AThe+Lagrangian+lives+in+configuration+space.+L%28q%2C+%5Cdot%7Bq%7D%2C+t%29.+You+feed+it+positions+and+velocities%2C+it+spits+out+equations+of+motion.+Elegant%2C+sure.+But+there+is+a+different+way+to+think+about+mechanics+%E2%80%94+one+that+rewrites+everything+in+terms+of+positions+and+momenta+instead.+This+is+the+Hamiltonian+formalism.%0A%0AThe+Legendre+transform+is+the+door.+Given+a+Lagrangian+L%28q%2C+%5Cdot%7Bq%7D%2C+t%29%2C+define+the+canonical+momentum%3A%0A%0Ap_i+%3D+%5Cfrac%7B%5Cpartial+L%7D%7B%5Cpartial+%5Cdot%7Bq%7D_i%7D%0A%0AThis+is+not+a+suggestion.+It%27s+a+definition.+The+momentum+is+what+the+Lagrangian+tells+you+momentum+is.+In+simple+cases%2C+p+%3D+m%5Cdot%7Bq%7D.+In+complex+ones+%E2%80%94+generalized+coordinates%2C+curved+spaces%2C+constraints+%E2%80%94+the+momentum+can+look+nothing+like+mv.+That%27s+fine.+The+formalism+handles+it.%0A%0ANow+define+the+Hamiltonian%3A%0A%0AH%28q%2C+p%2C+t%29+%3D+%5Csum_i+p_i+%5Cdot%7Bq%7D_i+-+L%28q%2C+%5Cdot%7Bq%7D%2C+t%29%0A%0ABut+wait+%E2%80%94+H+must+be+a+function+of+q+and+p%2C+not+q+and+%5Cdot%7Bq%7D.+So+you+solve+the+momentum+definition+for+%5Cdot%7Bq%7D%28q%2C+p%29+and+substitute.+The+velocities+are+eliminated.+What+remains+is+H+in+the+natural+variables%3A+positions+and+momenta.%0A%0AIn+most+physical+systems%2C+H+is+the+total+energy.+H+%3D+T+%2B+V.+The+kinetic+energy+expressed+in+terms+of+momenta%2C+plus+the+potential+energy+expressed+in+terms+of+positions.+This+is+not+always+true+%E2%80%94+there+are+exotic+Lagrangians+where+H+differs+from+T+%2B+V+%E2%80%94+but+for+the+vast+majority+of+systems+you%27ll+encounter%2C+H+is+the+energy+function.+And+that+fact+makes+the+Hamiltonian+physically+meaningful+in+a+way+the+Lagrangian+sometimes+isn%27t.%0A%0AThe+Hamilton+equations+follow%3A%0A%0A%5Cdot%7Bq%7D_i+%3D+%5Cfrac%7B%5Cpartial+H%7D%7B%5Cpartial+p_i%7D%0A%0A%5Cdot%7Bp%7D_i+%3D+-%5Cfrac%7B%5Cpartial+H%7D%7B%5Cpartial+q_i%7D%0A%0AThey+are+2n+first-order+equations%2C+where+n+is+the+number+of+degrees+of+freedom.+Compare+this+to+the+single+second-order+Euler-Lagrange+equation+per+degree+of+freedom.+The+Hamiltonian+form+doubles+the+equations+but+halves+the+derivatives.+This+trade-off+is+not+gratuitous.+First-order+equations+are+structurally+different+beasts.+They+open+the+door+to+phase+space%2C+to+conservation+theorems%2C+to+quantization.%0A%0APhase+space+is+the+space+of+all+%28q%2C+p%29+pairs.+For+a+single+particle+in+three+dimensions%2C+it%27s+six-dimensional.+Every+point+in+phase+space+represents+a+complete+state+of+the+system+%E2%80%94+all+positions+and+all+momenta+specified+simultaneously.+The+system%27s+evolution+traces+a+trajectory+through+this+space.+This+geometric+picture+replaces+the+more+abstract+notion+of+a+path+in+configuration+space.%0A%0AThe+symmetry+between+q+and+p+in+Hamilton%27s+equations+is+striking.+%5Cdot%7Bq%7D+comes+from+differentiating+H+with+respect+to+p.+%5Cdot%7Bp%7D+comes+from+differentiating+H+with+respect+to+q+%28with+a+minus+sign%29.+This+symmetry+is+not+cosmetic+%E2%80%94+it%27s+the+reason+the+Hamiltonian+formalism+is+the+natural+bridge+to+quantum+mechanics.+In+quantum+mechanics%2C+q+and+p+become+operators+with+the+commutation+relation+%5Bq%2C+p%5D+%3D+i%5Chbar.+The+classical+Poisson+bracket+%7Bq%2C+p%7D+%3D+1+is+the+direct+precursor.+You+cannot+get+this+insight+from+the+Lagrangian+formulation.%0A%0AAnother+gift+of+the+Hamiltonian+approach%3A+Noether%27s+theorem+still+works%2C+but+the+bookkeeping+is+cleaner.+Symmetries+in+q+translate+directly+to+conserved+momenta.+If+H+does+not+depend+on+q_i+%28q_i+is+cyclic%29%2C+then+%5Cdot%7Bp%7D_i+%3D+0+and+p_i+is+conserved.+The+connection+between+symmetry+and+conservation+is+immediate+and+algebraic+rather+than+variational.%0A%0ASeparable+Hamiltonians+lead+to+Hamilton-Jacobi+theory%2C+which+transforms+mechanics+into+a+single+partial+differential+equation+for+a+function+S%28q%2C+t%29.+The+solution+S+is+called+the+action%2C+and+its+gradients+give+the+canonical+momenta.+This+is+the+most+powerful+formalism+in+classical+mechanics+%E2%80%94+and+it+only+exists+because+the+Hamiltonian+framework+re-expressed+the+problem+in+the+right+variables.%0A%0AThe+Hamiltonian+is+not+just+a+reformulation.+It+is+a+different+lens.+And+through+that+lens%2C+you+see+structures+that+were+invisible+before%3A+the+geometry+of+phase+space%2C+the+algebra+of+Poisson+brackets%2C+the+path+from+classical+to+quantum.+It+rewrites+mechanics+in+terms+of+positions+and+momenta%2C+yes%2C+but+it+also+rewrites+your+understanding+of+what+mechanics+is+about.