# FIELD-THEORY ANALOGS OF THE LAGRANGE AND POINCARE INVARIANTS

## Abstract

The Lagrange differential invariant and the Poincare integral invariant of classical dynamics have as their analogs in Lagrangian field theory a differential divergence free vector'' and an integral divergence-free vector.'' The former, which is experssible as a divergence-free vector-bracket expression, may be used to derive conservation relations associated with the transformation properties of a given system. lt is not necessary that these transformations should be infinitesimal; conservation theorems are established for systems which are periodic and for systems which are invariant under spatial inversion. The differential divergence-free vector may also be used to establish reciprocity and orthogonality relations: examples discussed are Betti's reciprocal theorem of elasticity and Lorentz's reciprocal relation of electromagnetic theory. An extended form of the differential divergence-free vector allows for variation not only of the dependent variables but also of the independent variables. The integral divergence-free vector associates a conserved quantity with any closed one-parameter family of solutions of the field equations. The equation of conservation of probability'' of quantum mechanics and a classical form of the relation between the momentum and wave vectors for a plane wave in a propagating medium are derived. The theorem of classical dynamics relating a complete set of Poisson brackets tomore »

- Authors:

- Publication Date:

- Research Org.:
- Stanford Univ., Calif.

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 4806980

- NSA Number:
- NSA-16-012020

- Resource Type:
- Journal Article

- Journal Name:
- J. Math. Phys.

- Additional Journal Information:
- Journal Volume: Vol: 3; Other Information: Orig. Receipt Date: 31-DEC-62

- Country of Publication:
- Country unknown/Code not available

- Language:
- English

- Subject:
- MATHEMATICS AND COMPUTERS; DIFFERENTIAL EQUATIONS; ELECTROMAGNETIC FIELDS; FIELD THEORY; LAGRANGIAN; LORENTZ GAUGE; MATHEMATICS; MOMENTUM; QUANTUM MECHANICS; VECTORS

### Citation Formats

```
Sturrock, P A.
```*FIELD-THEORY ANALOGS OF THE LAGRANGE AND POINCARE INVARIANTS*. Country unknown/Code not available: N. p., 1962.
Web. doi:10.1063/1.1703785.

```
Sturrock, P A.
```*FIELD-THEORY ANALOGS OF THE LAGRANGE AND POINCARE INVARIANTS*. Country unknown/Code not available. https://doi.org/10.1063/1.1703785

```
Sturrock, P A. 1962.
"FIELD-THEORY ANALOGS OF THE LAGRANGE AND POINCARE INVARIANTS". Country unknown/Code not available. https://doi.org/10.1063/1.1703785.
```

```
@article{osti_4806980,
```

title = {FIELD-THEORY ANALOGS OF THE LAGRANGE AND POINCARE INVARIANTS},

author = {Sturrock, P A},

abstractNote = {The Lagrange differential invariant and the Poincare integral invariant of classical dynamics have as their analogs in Lagrangian field theory a differential divergence free vector'' and an integral divergence-free vector.'' The former, which is experssible as a divergence-free vector-bracket expression, may be used to derive conservation relations associated with the transformation properties of a given system. lt is not necessary that these transformations should be infinitesimal; conservation theorems are established for systems which are periodic and for systems which are invariant under spatial inversion. The differential divergence-free vector may also be used to establish reciprocity and orthogonality relations: examples discussed are Betti's reciprocal theorem of elasticity and Lorentz's reciprocal relation of electromagnetic theory. An extended form of the differential divergence-free vector allows for variation not only of the dependent variables but also of the independent variables. The integral divergence-free vector associates a conserved quantity with any closed one-parameter family of solutions of the field equations. The equation of conservation of probability'' of quantum mechanics and a classical form of the relation between the momentum and wave vectors for a plane wave in a propagating medium are derived. The theorem of classical dynamics relating a complete set of Poisson brackets to a complete set of Lagrange brackets cannot be extended to the present formalism. The formula that represents the obvious extension of the classical formula for the Poisson bracket can be shown not to be canonically invariant. (auth)},

doi = {10.1063/1.1703785},

url = {https://www.osti.gov/biblio/4806980},
journal = {J. Math. Phys.},

number = ,

volume = Vol: 3,

place = {Country unknown/Code not available},

year = {1962},

month = {1}

}