Hamiltonian Function
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A Hamiltonian Function is a function that ...
- Context:
- It can range from being a Linear Hamiltonian Function to being Polynomial Hamiltonian Function.
- See: Classical Mechanics, Statistical Mechanics, Quantum Mechanics.
References
2016
- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Hamiltonian_mechanics#Mathematical_formalism Retrieved:2016-9-9.
- Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics. It uses a different mathematical formalism, providing a more abstract understanding of the theory. Historically, it was an important reformulation of classical mechanics, which later contributed to the formulation of statistical mechanics and quantum mechanics.
Hamiltonian mechanics was first formulated by William Rowan Hamilton in 1833, starting from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788.
- Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics. It uses a different mathematical formalism, providing a more abstract understanding of the theory. Historically, it was an important reformulation of classical mechanics, which later contributed to the formulation of statistical mechanics and quantum mechanics.
- https://www.technologyreview.com/s/602344/the-extraordinary-link-between-deep-neural-networks-and-the-nature-of-the-universe/?utm_campaign=socialflow
- QUOTE: To put this in perspective, consider the order of a polynomial function, which is the size of its highest exponent. So a quadratic equation like y=x2 has order 2, the equation y=x24 has order 24, and so on. Obviously, the number of orders is infinite and yet only a tiny subset of polynomials appear in the laws of physics. “For reasons that are still not fully understood, our universe can be accurately described by polynomial Hamiltonians of low order,” say Lin and Tegmark. Typically, the polynomials that describe laws of physics have orders ranging from 2 to 4.
The laws of physics have other important properties. For example, they are usually symmetrical when it comes to rotation and translation.
- QUOTE: To put this in perspective, consider the order of a polynomial function, which is the size of its highest exponent. So a quadratic equation like y=x2 has order 2, the equation y=x24 has order 24, and so on. Obviously, the number of orders is infinite and yet only a tiny subset of polynomials appear in the laws of physics. “For reasons that are still not fully understood, our universe can be accurately described by polynomial Hamiltonians of low order,” say Lin and Tegmark. Typically, the polynomials that describe laws of physics have orders ranging from 2 to 4.
2005
- (Mei et al., 2005) ⇒ Shengwei Mei, Tielong Shen, Wei Hu, Qiang Lu, and L . Sun. (2005). “Robust H∞ Control of a Hamiltonian System with Uncertainty and Its Application to a Multi-machine Power System." IEE Proceedings-Control Theory and Applications 152, no. 2 doi:10.1049/ip-cta:20041121
- ABSTRACT: … The proposed method is used to create a Hamiltonian-like model with uncertainty which is able to describe power system dynamics on a full scale. Consequently a decentralised nonlinear robust H/sub / spl infin// control law can be produced for a multi-machine power system using the Hamiltonian function. Simulations performed on a six-machine system verify that the proposed excitation control can cope with large disturbances and can enhance the transient stability of the power system more effectively than other types of controllers.