# Conservation laws in Newtonian and Hamiltonian (symplectic) mechanics

• mma
In summary, conservation laws in Newtonian mechanics are derived from Newton's laws and the assumption of central forces in an isolated system. In Hamiltonian mechanics, conservation laws are closely related to symmetries, which are one-parameter Lie groups with a symplectic group action that preserves the Hamiltonian. However, the Hamiltonian description is less general than Newtonian mechanics because it does not account for dissipative forces. Attempts have been made to achieve the same generality through symmetry considerations, but the standard symplectic Hamiltonian description is still more limited than Newton's laws. This is because the most general form of the Noether theorem only requires the variation of the action to remain invariant, which automatically accounts for the non-uniqueness of the

#### mma

In Newtonian mechanics, conservation laws of momentum and angular momentum for an isolated system follow from Newton's laws plus the assumption that all forces are central. This picture tells nothing about symmetries.

In contrast, in Hamiltonian mechanics, conservation laws are tightly connected to symmetries. A symmetry is a one-parameter Lie group with a symplectic group action on phase space that preserves Hamiltonian (a Hamiltonian-preserving symplectic flow), and the infinitesimal generator of this flow is a conserved quantity.

However, the standard symplectic Hamiltonian description is less general than Newton's law because dissipative forces aren't incuded. So, it seems that Newton's conservation laws are more general then the Hamiltonian ones. Is there any attempt for achieving the same generality by symmetry considerations, as Newtonian description have?

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Why do you think the Hamiltonian description is less general? On a fundamental level, it's the complete description of Newtonian mechanics, which is a mathematically closed system (in contradistinction to relativistic point-particle mechanics, which is not as complete, but that's another story).

BTW: The most general form of the quoted Noether theorem (according to which each (global) one-parameter Lie symmetry defines a conserved quantity and vice versa) makes the much weaker assumption that only the variation of the action must stay invariant. This exhausts automatically the fact that for the dynamics of a given system the Hamiltonian is not unique, but there are many equivalent Hamiltonians describing the same system.

Why do you have to choose at all? If you buy a screwdriver does that mean you have to stop using your hammer?

vanhees71 and Dale
vanhees71 said:
On a fundamental level, it's the complete description of Newtonian mechanics
This is a mathematical statement. Can you prove it?

That's a physical statement, and of course you can't prove it.