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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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