Jan Modersitzki
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198528418
- eISBN:
- 9780191713583
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528418.003.0008
- Subject:
- Mathematics, Applied Mathematics
This chapter introduces non-parametric registrations. The idea behind this type of registration is to come up with an appropriate measure both for the similarity as well as for the likelihood of a ...
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This chapter introduces non-parametric registrations. The idea behind this type of registration is to come up with an appropriate measure both for the similarity as well as for the likelihood of a non-parametric transformation. It sets up a general framework for the consideration of different registration techniques, which is based on a variational formulation of the registration problem; the numerical schemes to be considered are based on the Euler-Lagrange equations which characterize a minimizer.Less
This chapter introduces non-parametric registrations. The idea behind this type of registration is to come up with an appropriate measure both for the similarity as well as for the likelihood of a non-parametric transformation. It sets up a general framework for the consideration of different registration techniques, which is based on a variational formulation of the registration problem; the numerical schemes to be considered are based on the Euler-Lagrange equations which characterize a minimizer.
Jan Modersitzki
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198528418
- eISBN:
- 9780191713583
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528418.003.0009
- Subject:
- Mathematics, Applied Mathematics
This chapter introduces elastic registration using the general framework introduced in Chapter 8. A physical motivation for this particular regularizer is given. The two images are viewed as two ...
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This chapter introduces elastic registration using the general framework introduced in Chapter 8. A physical motivation for this particular regularizer is given. The two images are viewed as two different observations of an elastic body, one before and one after deformation. The displacement (or transformation) of the elastic body is derived using a linear elasticity model. In order to introduce different boundary conditions and for later reference, the eigenvalues and eigenfunctions of the Navier-Lame operator are computed. It is shown that the Navier-Lame equations can be derived from the general framework. The continuous Euler-Lagrange equations are discretized and the discretized partial differential operator can be diagonalized using Fast-Fourier type techniques. The Moore-Penrose pseudo inverse of the discrete Navier-Lamé operator is explicitly given. Registration results for academic as well as real life problems are presented.Less
This chapter introduces elastic registration using the general framework introduced in Chapter 8. A physical motivation for this particular regularizer is given. The two images are viewed as two different observations of an elastic body, one before and one after deformation. The displacement (or transformation) of the elastic body is derived using a linear elasticity model. In order to introduce different boundary conditions and for later reference, the eigenvalues and eigenfunctions of the Navier-Lame operator are computed. It is shown that the Navier-Lame equations can be derived from the general framework. The continuous Euler-Lagrange equations are discretized and the discretized partial differential operator can be diagonalized using Fast-Fourier type techniques. The Moore-Penrose pseudo inverse of the discrete Navier-Lamé operator is explicitly given. Registration results for academic as well as real life problems are presented.
Peter Mann
- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198822370
- eISBN:
- 9780191861253
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198822370.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This crucial chapter focuses on the stationary action principle. It introduces Lagrangian mechanics, using first-order variational calculus to derive the Euler–Lagrange equation, and the inverse ...
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This crucial chapter focuses on the stationary action principle. It introduces Lagrangian mechanics, using first-order variational calculus to derive the Euler–Lagrange equation, and the inverse problem is described. The chapter then considers the Ostrogradsky equation and discusses the properties of the extrema using the second-order variation to the action. It then discusses the difference between action functions (of Dirichlet boundary conditions) and action functionals of the extremal path. The different types of boundary conditions (Dirichlet vs Neumann) are elucidated. Topics discussed include Hessian conditions, Douglas’s theorem, the Jacobi last multiplier, Helmholtz conditions, Noether-type variation and Frenet–Serret frames, as well as concepts such as on shell and off shell. Actions of non-continuous extremals are examined using Weierstrass–Erdmann corner conditions, and the action principle is written in the most general form as the Hamilton–Suslov principle. Important applications of the Euler–Lagrange formulation are highlighted, including protein folding.Less
This crucial chapter focuses on the stationary action principle. It introduces Lagrangian mechanics, using first-order variational calculus to derive the Euler–Lagrange equation, and the inverse problem is described. The chapter then considers the Ostrogradsky equation and discusses the properties of the extrema using the second-order variation to the action. It then discusses the difference between action functions (of Dirichlet boundary conditions) and action functionals of the extremal path. The different types of boundary conditions (Dirichlet vs Neumann) are elucidated. Topics discussed include Hessian conditions, Douglas’s theorem, the Jacobi last multiplier, Helmholtz conditions, Noether-type variation and Frenet–Serret frames, as well as concepts such as on shell and off shell. Actions of non-continuous extremals are examined using Weierstrass–Erdmann corner conditions, and the action principle is written in the most general form as the Hamilton–Suslov principle. Important applications of the Euler–Lagrange formulation are highlighted, including protein folding.
David D. Nolte
- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198844624
- eISBN:
- 9780191880216
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198844624.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Dynamical systems follow trajectories for which the mechanical action integrated along the trajectory is an extremum. The action is defined as the time average of the difference between kinetic and ...
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Dynamical systems follow trajectories for which the mechanical action integrated along the trajectory is an extremum. The action is defined as the time average of the difference between kinetic and potential energies, which is also the time average of the Lagrangian. Once a Lagrangian has been defined for a system, the Euler equations of variational calculus lead to the Euler–Lagrange equations of dynamics. This chapter explores applications of Lagrangians and the use of Lagrange’s undetermined multipliers. Conservation laws, central forces, and the virial theorem are developed and explained.Less
Dynamical systems follow trajectories for which the mechanical action integrated along the trajectory is an extremum. The action is defined as the time average of the difference between kinetic and potential energies, which is also the time average of the Lagrangian. Once a Lagrangian has been defined for a system, the Euler equations of variational calculus lead to the Euler–Lagrange equations of dynamics. This chapter explores applications of Lagrangians and the use of Lagrange’s undetermined multipliers. Conservation laws, central forces, and the virial theorem are developed and explained.
Dusa McDuff and Dietmar Salamon
- Published in print:
- 2017
- Published Online:
- June 2017
- ISBN:
- 9780198794899
- eISBN:
- 9780191836411
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198794899.003.0002
- Subject:
- Mathematics, Analysis, Geometry / Topology
The first chapter develops the basic concepts of symplectic topology from the vantage point of classical mechanics. It starts with an introduction to the Euler–Lagrange equation and shows how the ...
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The first chapter develops the basic concepts of symplectic topology from the vantage point of classical mechanics. It starts with an introduction to the Euler–Lagrange equation and shows how the Legendre transformation leads to Hamilton’s equations, symplectic forms, symplectomorphisms, and the symplectic action. It ends with a brief overview of some modern results in the subject on the symplectic topology of Euclidean space, such as the Weinstein conjecture and the Gromov nonsqueezing theorem.Less
The first chapter develops the basic concepts of symplectic topology from the vantage point of classical mechanics. It starts with an introduction to the Euler–Lagrange equation and shows how the Legendre transformation leads to Hamilton’s equations, symplectic forms, symplectomorphisms, and the symplectic action. It ends with a brief overview of some modern results in the subject on the symplectic topology of Euclidean space, such as the Weinstein conjecture and the Gromov nonsqueezing theorem.
Peter Mann
- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198822370
- eISBN:
- 9780191861253
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198822370.003.0035
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter presents the general formulation of the calculus of variations as applied to mechanics, relativity and field theories. The calculus of variations is a common mathematical technique used ...
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This chapter presents the general formulation of the calculus of variations as applied to mechanics, relativity and field theories. The calculus of variations is a common mathematical technique used throughout classical mechanics. First developed by Euler to determine the shortest paths between fixed points along a surface, it was applied by Lagrange to mechanical problems in analytical mechanics. The variational problems in the chapter have been simplified for ease of understanding upon first introduction, in order to give a general mathematical framework. This chapter takes a relaxed approach to explain how the Euler–Lagrange equation is derived using this method. It also discusses first integrals. The chapter closes by defining the functional derivative, which is used in classical field theory.Less
This chapter presents the general formulation of the calculus of variations as applied to mechanics, relativity and field theories. The calculus of variations is a common mathematical technique used throughout classical mechanics. First developed by Euler to determine the shortest paths between fixed points along a surface, it was applied by Lagrange to mechanical problems in analytical mechanics. The variational problems in the chapter have been simplified for ease of understanding upon first introduction, in order to give a general mathematical framework. This chapter takes a relaxed approach to explain how the Euler–Lagrange equation is derived using this method. It also discusses first integrals. The chapter closes by defining the functional derivative, which is used in classical field theory.
Nathalie Deruelle and Jean-Philippe Uzan
- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198786399
- eISBN:
- 9780191828669
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198786399.003.0009
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter gives a brief overview of Hamiltonian mechanics. The complexity of the Newtonian equations of motion for N interacting bodies led to the development in the late 18th and early 19th ...
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This chapter gives a brief overview of Hamiltonian mechanics. The complexity of the Newtonian equations of motion for N interacting bodies led to the development in the late 18th and early 19th centuries of a formalism that reduces these equations to first-order differential equations. This formalism is known as Hamiltonian mechanics. This chapter shows how, given a Lagrangian and having constructed the corresponding Hamiltonian, Hamilton’s equations amount to simply a rewriting of the Euler–Lagrange equations. The feature that makes the Hamiltonian formulation superior is that the dimension of the phase space is double that of the configuration space, so that in addition to point transformations, it is possible to perform more general transformations in order to simplify solving the equations of motion.Less
This chapter gives a brief overview of Hamiltonian mechanics. The complexity of the Newtonian equations of motion for N interacting bodies led to the development in the late 18th and early 19th centuries of a formalism that reduces these equations to first-order differential equations. This formalism is known as Hamiltonian mechanics. This chapter shows how, given a Lagrangian and having constructed the corresponding Hamiltonian, Hamilton’s equations amount to simply a rewriting of the Euler–Lagrange equations. The feature that makes the Hamiltonian formulation superior is that the dimension of the phase space is double that of the configuration space, so that in addition to point transformations, it is possible to perform more general transformations in order to simplify solving the equations of motion.
Peter Mann
- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198822370
- eISBN:
- 9780191861253
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198822370.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter discusses point transformations in Lagrangian mechanics. Sometimes, when solving problems, it is useful to change coordinates in velocity phase space to better suit and simplify the ...
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This chapter discusses point transformations in Lagrangian mechanics. Sometimes, when solving problems, it is useful to change coordinates in velocity phase space to better suit and simplify the system at hand; this is a requirement of any physical theory. This change is often motivated by some experimentally observed physicality of the system or may highlight new conserved quantities that might have been overlooked using the old description. In the Newtonian formalism, it was a bit of a hassle to change coordinates and the equations of motion will look quite different. In this chapter, point transformations in Lagrangian mechanics are developed and the Euler–Lagrange equation is found to be covariant. The chapter discusses coordinate transformations, parametrisation invariance and the Jacobian of the transform. Re-parametrisations are also included.Less
This chapter discusses point transformations in Lagrangian mechanics. Sometimes, when solving problems, it is useful to change coordinates in velocity phase space to better suit and simplify the system at hand; this is a requirement of any physical theory. This change is often motivated by some experimentally observed physicality of the system or may highlight new conserved quantities that might have been overlooked using the old description. In the Newtonian formalism, it was a bit of a hassle to change coordinates and the equations of motion will look quite different. In this chapter, point transformations in Lagrangian mechanics are developed and the Euler–Lagrange equation is found to be covariant. The chapter discusses coordinate transformations, parametrisation invariance and the Jacobian of the transform. Re-parametrisations are also included.
Laurent Baulieu, John Iliopoulos, and Roland Sénéor
- Published in print:
- 2017
- Published Online:
- May 2017
- ISBN:
- 9780198788393
- eISBN:
- 9780191830310
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198788393.003.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics
General introduction with a review of the principles of Hamiltonian and Lagrangian mechanics. The connection between symmetries and conservation laws, with a presentation of Noether’s theorem, is ...
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General introduction with a review of the principles of Hamiltonian and Lagrangian mechanics. The connection between symmetries and conservation laws, with a presentation of Noether’s theorem, is included.Less
General introduction with a review of the principles of Hamiltonian and Lagrangian mechanics. The connection between symmetries and conservation laws, with a presentation of Noether’s theorem, is included.
Jennifer Coopersmith
- Published in print:
- 2017
- Published Online:
- June 2017
- ISBN:
- 9780198743040
- eISBN:
- 9780191802966
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198743040.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, History of Physics
The link between mathematics and physics is explained, and how the concepts “coordinates,” “generalized coordinates,” “time,” and “space” have evolved, starting with Galileo. It is also shown that ...
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The link between mathematics and physics is explained, and how the concepts “coordinates,” “generalized coordinates,” “time,” and “space” have evolved, starting with Galileo. It is also shown that “degrees of freedom” is a slippery but crucial idea. The important developments in “space research”, from Pythagoras to Riemann, are sketched. This is followed by the motivations for finding a flat region of “space”, and for Riemann’s invariant interval. A careful explanation of the three ways of taking an infinitesimal step (actual, virtual, and imperfect) is given. The programme of the Calculus of Variations is described and how this requires a virtual variation of a whole path, a path taken between fixed end-states. This then culminates in the Euler-Lagrange Equations or the Lagrange Equations of Motion. Along the way, the ideas virtual displacement and extremum are explained.Less
The link between mathematics and physics is explained, and how the concepts “coordinates,” “generalized coordinates,” “time,” and “space” have evolved, starting with Galileo. It is also shown that “degrees of freedom” is a slippery but crucial idea. The important developments in “space research”, from Pythagoras to Riemann, are sketched. This is followed by the motivations for finding a flat region of “space”, and for Riemann’s invariant interval. A careful explanation of the three ways of taking an infinitesimal step (actual, virtual, and imperfect) is given. The programme of the Calculus of Variations is described and how this requires a virtual variation of a whole path, a path taken between fixed end-states. This then culminates in the Euler-Lagrange Equations or the Lagrange Equations of Motion. Along the way, the ideas virtual displacement and extremum are explained.
George Jaroszkiewicz
- Published in print:
- 2016
- Published Online:
- January 2016
- ISBN:
- 9780198718062
- eISBN:
- 9780191787553
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198718062.003.0014
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter discusses Newton’s concept of absolute time and its use in classical mechanics. It defines clocks and our freedom to change them, that is, reparametrization of time. The chapter goes on ...
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This chapter discusses Newton’s concept of absolute time and its use in classical mechanics. It defines clocks and our freedom to change them, that is, reparametrization of time. The chapter goes on to discuss Aristotelian and Galilean–Newtonian space-times. It explains how Newton’s first law of motion is required to correlate successive copies of space. It discusses the differences between the Newtonian paradigm of motion and the use of Lagrangians in the Calculus of Variations that gives the Euler–Lagrange equations of motion in advanced mechanics. The chapter concludes with a discussion of phase space, Hamiltonians, Poisson brackets, and the role of the Hamiltonian as a generator of translation in time.Less
This chapter discusses Newton’s concept of absolute time and its use in classical mechanics. It defines clocks and our freedom to change them, that is, reparametrization of time. The chapter goes on to discuss Aristotelian and Galilean–Newtonian space-times. It explains how Newton’s first law of motion is required to correlate successive copies of space. It discusses the differences between the Newtonian paradigm of motion and the use of Lagrangians in the Calculus of Variations that gives the Euler–Lagrange equations of motion in advanced mechanics. The chapter concludes with a discussion of phase space, Hamiltonians, Poisson brackets, and the role of the Hamiltonian as a generator of translation in time.
