# A Conservation laws in Newtonian and Hamiltonian (symplectic) mechanics

#### mma

Summary
Must we choose between generality and mathematical elegance?
In Newtonian mechanics, conservation laws of momemtum 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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#### vanhees71

Gold Member
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.

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

#### mma

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

#### vanhees71

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

"Conservation laws in Newtonian and Hamiltonian (symplectic) mechanics"

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