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member 11137
Does someone here knows something about how tensor of curvature (Riemann) and the hamilton operator associated with a particle are connected ? Makes this question sense ? Thanks
Riemann and Hamilton are two influential mathematicians who made significant contributions to the fields of geometry and mechanics, respectively. Riemann's work on differential geometry laid the foundation for Einstein's theory of general relativity, while Hamilton's formulation of Hamiltonian mechanics revolutionized classical physics.
Riemann and Hamilton's ideas have had a profound impact on modern mathematics. Riemann's work on manifolds and curvature led to the development of differential geometry, which has become an indispensable tool in many areas of mathematics, including topology, differential equations, and physics. Hamilton's formulation of mechanics using his eponymous equations has been extended to quantum mechanics and has also found applications in other areas such as control theory and optimization.
Riemann's hypothesis is a conjecture about the distribution of prime numbers, which has remained unsolved since it was proposed in 1859. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line with real part equal to 1/2. If proven, the Riemann hypothesis would have far-reaching consequences in number theory and other areas of mathematics, including cryptography.
Hamilton's principle, also known as the principle of least action, states that the path or trajectory taken by a physical system between two points in time is the one that minimizes the action integral. This principle is used in mechanics to derive the equations of motion and has been extended to other fields, such as optics and quantum mechanics, where it is applied to the wave function.
The Hamiltonian and Riemannian formalisms are two different mathematical approaches used to study mechanics and geometry, respectively. The Hamiltonian formalism is based on Hamilton's equations of motion, which describe the evolution of a physical system in terms of its energy, while the Riemannian formalism is based on Riemann's notion of curvature and is used to study the intrinsic properties of smooth surfaces and higher-dimensional spaces.