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Hamilton-Jacobi Therory

Oliver Davis Johns

in Analytical Mechanics for Relativity and Quantum Mechanics

Published in print:
2011
Published Online:
December 2013
ISBN:
9780191001628
eISBN:
9780191775161
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780191001628.003.0021
Subject:
Physics, Atomic, Laser, and Optical Physics

This chapter focuses on the Hamilton-Jacobi theory, the apotheosis of Lagrangian and Hamiltonian mechanics. Action functions encode all of the possible trajectories of a mechanical system satisfying ... More


The Hamilton-Jacobi Equation and Weak KAM Theory

Alfonso Sorrentino

in Action-minimizing Methods in Hamiltonian Dynamics (MN-50): An Introduction to Aubry-Mather Theory

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691164502
eISBN:
9781400866618
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691164502.003.0005
Subject:
Mathematics, Applied Mathematics

This chapter describes another interesting approach to the study of invariant sets provided by the so-called weak KAM theory, developed by Albert Fathi. This approach can be considered as the ... More


Hamilton–Jacobi theory

S. G. Rajeev

in Advanced Mechanics: From Euler's Determinism to Arnold's Chaos

Published in print:
2013
Published Online:
December 2013
ISBN:
9780199670857
eISBN:
9780191775154
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199670857.003.0009
Subject:
Physics, Atomic, Laser, and Optical Physics

This chapter derives the Hamilton-Jacobi equation. Euler's two centre problem is solved by using separation of variables in the elliptical co-ordinate system. The connection to the eikonal equation ... More


SOME HOMOGENIZATION PROBLEMS

Andrea Braides

in Gamma-Convergence for Beginners

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198507840
eISBN:
9780191709890
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507840.003.0004
Subject:
Mathematics, Applied Mathematics

Homogenization problems for a general class of integrals are solved by a direct approach. Different homogenization formulas are given, both in an asymptotic form and as a cell problem (in the convex ... More


Introduction to the Mathematics of Rays

John A. Adam

in Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691148373
eISBN:
9781400885404
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691148373.003.0003
Subject:
Mathematics, Applied Mathematics

This chapter provides an overview of the mathematics of rays. It begins with a discussion of the theory of geometrical optics and how it can be formulated by means of the ray equations or the ... More


Stochastic Optimal Control

Tomas Björk

in Arbitrage Theory in Continuous Time

Published in print:
1998
Published Online:
November 2003
ISBN:
9780198775188
eISBN:
9780191595981
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198775180.003.0014
Subject:
Economics and Finance, Financial Economics

This chapter gives a self‐contained introduction to optimal control of stochastic differential equations. We derive the Hamilton‐Jacobi‐Bellman equation as well as a verification theorem. The general ... More


Canonical & Gauge Transformations

Peter Mann

in Lagrangian and Hamiltonian Dynamics

Published in print:
2018
Published Online:
August 2018
ISBN:
9780198822370
eISBN:
9780191861253
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198822370.003.0018
Subject:
Physics, Theoretical, Computational, and Statistical Physics

In this chapter, the Hamilton–Jacobi formulation is discussed in two parts: from a generating function perspective and as a variational principle. The Poincaré–Cartan 1-form is derived and solutions ... More


Gravitational Scattering

John A. Adam

in Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691148373
eISBN:
9781400885404
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691148373.003.0016
Subject:
Mathematics, Applied Mathematics

This chapter discusses Kepler's laws of planetary motion in relation to gravitational scattering. Gravitational scattering occurs when the paths of particles passing near a massive body are ... More


Bohm’s Approach and Individuality

Paavo Pylkkänen, Basil J. Hiley, and Ilkka Pättiniemi

in Individuals Across the Sciences

Published in print:
2015
Published Online:
November 2015
ISBN:
9780199382514
eISBN:
9780199382538
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199382514.003.0013
Subject:
Philosophy, Philosophy of Science, Metaphysics/Epistemology

Ladyman and Ross (LR) argue that quantum objects are not individuals (or are at most weakly discernible individuals) and use this idea to ground their metaphysical view, ontic structural realism, ... More


KAM theory

S. G. Rajeev

in Advanced Mechanics: From Euler's Determinism to Arnold's Chaos

Published in print:
2013
Published Online:
December 2013
ISBN:
9780199670857
eISBN:
9780191775154
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199670857.003.0018
Subject:
Physics, Atomic, Laser, and Optical Physics

The Kolmogorov-Arnold-Moser (KAM) theorem presents limits to the notion of chaos. This is the deepest result to date on this subject. This chapter starts with a review of Newton's method for solving ... More


The Lazy Universe: An Introduction to the Principle of Least Action

Jennifer Coopersmith

Published in print:
2017
Published Online:
June 2017
ISBN:
9780198743040
eISBN:
9780191802966
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198743040.001.0001
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, History of Physics

Action and the Principle of Least Action are explained: what Action is, why the Principle of Least Action works, why it underlies all physics, and what are the insights gained into energy, space, and ... More


Continuous-Time Portfolio Choice and Pricing

Kerry E. Back

in Asset Pricing and Portfolio Choice Theory

Published in print:
2017
Published Online:
May 2017
ISBN:
9780190241148
eISBN:
9780190241179
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780190241148.003.0014
Subject:
Economics and Finance, Financial Economics

The Euler equation is defined. The static approach can be used to derive an optimal portfolio in a complete market and when the investment opportunity set is constant. In the latter case, the optimal ... More


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