*Oliver Johns*

- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198567264
- eISBN:
- 9780191717987
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567264.003.0003
- Subject:
- Physics, Atomic, Laser, and Optical Physics

One attractive feature of the Lagrangian method is the ease with which it solves so-called constraint problems. This chapter presents several different ways of solving such problems, with examples of ...
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One attractive feature of the Lagrangian method is the ease with which it solves so-called constraint problems. This chapter presents several different ways of solving such problems, with examples of each. In the previous chapter, the generalised coordinates were assumed to be independent variables. However, there are problems of interest in which these coordinates are not independent, but rather are forced into particular relations by constraints. In this chapter, constraints are defined and virtual displacement is discussed, along with virtual work, form of the forces of constraint, general Lagrange equations with constraints, alternate notation for holonomic constraints, reduction of degrees of freedom, recovery of the forces of constraint, generalised energy theorem with constraints, and tractable non-holonomic constraints.Less

One attractive feature of the Lagrangian method is the ease with which it solves so-called constraint problems. This chapter presents several different ways of solving such problems, with examples of each. In the previous chapter, the generalised coordinates were assumed to be independent variables. However, there are problems of interest in which these coordinates are not independent, but rather are forced into particular relations by constraints. In this chapter, constraints are defined and virtual displacement is discussed, along with virtual work, form of the forces of constraint, general Lagrange equations with constraints, alternate notation for holonomic constraints, reduction of degrees of freedom, recovery of the forces of constraint, generalised energy theorem with constraints, and tractable non-holonomic constraints.

*Oliver Davis Johns*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780191001628
- eISBN:
- 9780191775161
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780191001628.003.0013
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter presents an extended theory in which the traditional Lagrange equations and generalised energy theorem are combined into a single set of equations that restore the symmetry of the ...
More

This chapter presents an extended theory in which the traditional Lagrange equations and generalised energy theorem are combined into a single set of equations that restore the symmetry of the mathematical system. The traditional Lagrangian methods are analogous to the ‘coordinate parametric method’ in the calculus of variations, while the extended Lagrangian theory is analogous to the recommended ‘general parametric method’. The generalised energy theorem of traditional Lagrangian theory, which had been a separate equation analogous to the ‘second form’ of the Euler-Lagrange equations, gets restored to its proper place as just another of the extended Lagrange equations, which now form a complete set of equations appropriate to the problem.Less

This chapter presents an extended theory in which the traditional Lagrange equations and generalised energy theorem are combined into a single set of equations that restore the symmetry of the mathematical system. The traditional Lagrangian methods are analogous to the ‘coordinate parametric method’ in the calculus of variations, while the extended Lagrangian theory is analogous to the recommended ‘general parametric method’. The generalised energy theorem of traditional Lagrangian theory, which had been a separate equation analogous to the ‘second form’ of the Euler-Lagrange equations, gets restored to its proper place as just another of the extended Lagrange equations, which now form a complete set of equations appropriate to the problem.

*Oliver Davis Johns*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780191001628
- eISBN:
- 9780191775161
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780191001628.003.0014
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter uses the traditional Hamilton equations as the basis for an extended Hamiltonian theory in which time is treated as a coordinate. The traditional Hamilton equations, including the ...
More

This chapter uses the traditional Hamilton equations as the basis for an extended Hamiltonian theory in which time is treated as a coordinate. The traditional Hamilton equations, including the Hamiltonian form of the generalised energy theorem, will be combined into one set of extended Hamilton equations. The extended Hamilton theory developed in the chapter is of fundamental importance for the more advanced topics in mechanics. It is used to write the relativistically covariant Hamiltonian, which is then used to derive the Klein-Gordon equation of relativistic quantum mechanics. The extended Hamilton equations also provide the basis for the discussion of canonical transformations. The objective of extended Hamiltonian theory is to write the equations of motion in terms of an extended set of phase-space variables.Less

This chapter uses the traditional Hamilton equations as the basis for an extended Hamiltonian theory in which time is treated as a coordinate. The traditional Hamilton equations, including the Hamiltonian form of the generalised energy theorem, will be combined into one set of extended Hamilton equations. The extended Hamilton theory developed in the chapter is of fundamental importance for the more advanced topics in mechanics. It is used to write the relativistically covariant Hamiltonian, which is then used to derive the Klein-Gordon equation of relativistic quantum mechanics. The extended Hamilton equations also provide the basis for the discussion of canonical transformations. The objective of extended Hamiltonian theory is to write the equations of motion in terms of an extended set of phase-space variables.