Conservation laws in Newtonian and Hamiltonian (symplectic) mechanics

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
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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.
 
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Why do you have to choose at all? If you buy a screwdriver does that mean you have to stop using your hammer?
 
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vanhees71 said:
On a fundamental level, it's the complete description of Newtonian mechanics
This is a mathematical statement. Can you prove it?
 
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That's a physical statement, and of course you can't prove it.
 

FAQ: Conservation laws in Newtonian and Hamiltonian (symplectic) mechanics

1. What are conservation laws in Newtonian and Hamiltonian mechanics?

Conservation laws in Newtonian and Hamiltonian mechanics refer to fundamental principles that describe the behavior of physical systems. These laws state that certain quantities, such as energy, momentum, and angular momentum, remain constant in a closed system and cannot be created or destroyed.

2. What is the difference between conservation laws in Newtonian and Hamiltonian mechanics?

The main difference between conservation laws in Newtonian and Hamiltonian mechanics lies in the mathematical framework used to describe them. Newtonian mechanics uses classical mechanics and the laws of motion to describe the behavior of physical systems, while Hamiltonian mechanics uses the concept of energy and the Hamiltonian function to describe the same systems.

3. How do conservation laws apply to real-world systems?

Conservation laws apply to real-world systems by providing a framework for understanding and predicting their behavior. For example, the conservation of energy allows us to calculate the amount of energy needed to perform a specific task, while the conservation of momentum can help us understand the motion of objects in a collision.

4. Are conservation laws always applicable?

No, conservation laws are not always applicable. They are based on certain assumptions, such as the system being isolated and the absence of external forces. In reality, many systems are not isolated and are subject to external influences, making the application of conservation laws more complex.

5. What are some practical applications of conservation laws in mechanics?

Conservation laws in mechanics have numerous practical applications, including predicting the motion of planets and satellites, designing efficient engines and machines, and understanding the behavior of fluids and gases. They also play a crucial role in fields such as engineering, physics, and astronomy.

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