Radiofrequency thermocoagulation of trigeminal nerve assisted by nerve bundle extraction and image fusion based on Hamilton-Jacobi equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

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Geometric Singularities for Hamilton–Jacobi Equation

Progress in Differential Geometry ◽

10.2969/aspm/02210089 ◽

2018 ◽

Cited By ~ 2

Author(s):

Shyūichi Izumiya

Keyword(s):

Jacobi Equation ◽

Hamilton Jacobi Equation

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A Large Deviation, Hamilton-Jacobi Equation Approach to a Statistical Theory for Turbulence

10.21236/ada575877 ◽

2012 ◽

Author(s):

Jin Feng

Keyword(s):

Statistical Theory ◽

Jacobi Equation ◽

Large Deviation ◽

Equation Approach ◽

Hamilton Jacobi Equation

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Hamiltonian Mechanics

10.1093/oso/9780198743040.003.0007 ◽

2017 ◽

Author(s):

Jennifer Coopersmith

Keyword(s):

Angular Momentum ◽

Jacobi Equation ◽

Modern Theory ◽

Hamiltonian Mechanics ◽

Lagrangian Mechanics ◽

Canonical Transformations ◽

Canonical Variables ◽

Light Waves ◽

Hamilton Jacobi Equation ◽

Common Action

Hamilton’s genius was to understand what were the true variables of mechanics (the “p − q,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “p − q” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.

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Classification of extinction profiles for a one-dimensional diffusive Hamilton–Jacobi equation with critical absorption

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821051700049x ◽

2018 ◽

Vol 148 (3) ◽

pp. 559-574

Author(s):

Razvan Gabriel Iagar ◽

Philippe Laurençot

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Jacobi Equation ◽

Threshold Value ◽

Monotonicity Property ◽

Fast Diffusion ◽

Extinction Time ◽

One Step ◽

Hamilton Jacobi Equation

A classification of the behaviour of the solutions f(·, a) to the ordinary differential equation (|f′|p-2f′)′ + f - |f′|p-1 = 0 in (0,∞) with initial condition f(0, a) = a and f′(0, a) = 0 is provided, according to the value of the parameter a > 0 when the exponent p takes values in (1, 2). There is a threshold value a* that separates different behaviours of f(·, a): if a > a*, then f(·, a) vanishes at least once in (0,∞) and takes negative values, while f(·, a) is positive in (0,∞) and decays algebraically to zero as r→∞ if a ∊ (0, a*). At the threshold value, f(·, a*) is also positive in (0,∞) but decays exponentially fast to zero as r→∞. The proof of these results relies on a transformation to a first-order ordinary differential equation and a monotonicity property with respect to a > 0. This classification is one step in the description of the dynamics near the extinction time of a diffusive Hamilton–Jacobi equation with critical gradient absorption and fast diffusion.

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