‘Symmetry’ implies ‘redundant coordinate’ implies ‘cyclic coordinates in your Lagrangian / Hamiltonian’ implies ‘conservation of conjugate momentum’
And because the action principle (where the true system trajectory extremizes your action, i.e. integral of Lagrangian) works in various dynamical systems, the above argument works in non-physical dynamical systems.
Thus conserved quantities usually exist in a given dynamical system.
mmm, but why does the action principle hold in such a wide variety of systems though? (like how you get entropy by postulating something to be maximized in an equilibrium setting)
‘Symmetry’ implies ‘redundant coordinate’ implies ‘cyclic coordinates in your Lagrangian / Hamiltonian’ implies ‘conservation of conjugate momentum’
And because the action principle (where the true system trajectory extremizes your action, i.e. integral of Lagrangian) works in various dynamical systems, the above argument works in non-physical dynamical systems.
Thus conserved quantities usually exist in a given dynamical system.
mmm, but why does the action principle hold in such a wide variety of systems though? (like how you get entropy by postulating something to be maximized in an equilibrium setting)