Each one corresponds to a symmetry of a Hamiltonian system's Lagrangian according to Noether's theorem.
The region of phase space defined by assigning one of these quantities a particular value is known as an invariant torus and its corresponding coordinates are known as action angles. The conjugate momenta resulting from the Hamilton-Jacobi equation are this type of quantity, which have a zero Poisson bracket with the system's Hamiltonian and with each other. Ones of these variables that do not depend on time explicitly are known as integrals. The energy is one of these in every Hamiltonian system; systems with the energy as their only one of these are known as non-integrable, while those with more than one are known as integrable. Each one corresponds to a symmetry of a Hamiltonian system's Lagrangian according to Noether's theorem. For 10 points, name these quantities that do not change along a system's trajectory.
The region of phase space defined by assigning one of these quantities a particular value is known as an invariant torus and its corresponding coordinates are known as action angles.
constants of the motion [prompt on conserved quantities; prompt on constants; prompt on integrals; prompt on conjugate momenta; prompt on actions]