Foundations of Theoretical Mechanics I: The Inverse Problem in Newtonian Mechanics: Theoretical and Mathematical Physics
Autor Ruggero Maria Santillien Limba Engleză Paperback – 18 mai 2012
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|---|---|---|
| Paperback (2) | 380.17 lei 43-57 zile | |
| Springer Berlin, Heidelberg – 1984 | 380.17 lei 43-57 zile | |
| Springer – 18 mai 2012 | 453.99 lei 38-44 zile |
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Specificații
ISBN-13: 9783642867590
ISBN-10: 3642867596
Pagini: 292
Dimensiuni: 155 x 235 x 15 mm
Greutate: 0.41 kg
Ediția:Softcover reprint of the original 1st ed. 1978
Editura: Springer
Colecția Springer
Seria Theoretical and Mathematical Physics
Locul publicării:Berlin, Heidelberg, Germany
ISBN-10: 3642867596
Pagini: 292
Dimensiuni: 155 x 235 x 15 mm
Greutate: 0.41 kg
Ediția:Softcover reprint of the original 1st ed. 1978
Editura: Springer
Colecția Springer
Seria Theoretical and Mathematical Physics
Locul publicării:Berlin, Heidelberg, Germany
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Public țintă
ResearchDescriere
The
objective
of
this
monograph
is
to
present
some
methodological
foundations
of
theoretical
mechanics
that
are
recommendable
to
graduate
students
prior
to,
or
jointly
with,
the
study
of
more
advanced
topics
such
as
statistical
mechanics,
thermodynamics,
and
elementary
particle
physics.
A
program
of
this
nature
is
inevitably
centered
on
the
methodological
foundations
for
Newtonian
systems,
with
particular
reference
to
the
central
equations
of
our
theories,
that
is,
Lagrange's
and
Hamilton's
equations.
This
program,
realized
through
a
study
of
the
analytic
representations
in
terms
of
Lagrange's
and
Hamilton's
equations
of
generally
nonconservative
Newtonian
systems
(namely,
systems
with
Newtonian
forces
not
necessarily
derivable
from
a
potential
function),
falls
within
the
context
of
the
so-called
Inverse
Problem,
and
consists
of
three
major
aspects:
I.
The
study
of
the
necessary
and
sufficient
conditions
for
the
existence
of
a
Lagrangian
or
Hamiltonian
representation
of
given
equations
of
motion
with
arbitrary
forces;
1.
The
identification
of
the
methods
for
the
construction
of
a
Lagrangian
or
Hamiltonian
from
the
given
equations
of
motion;
and
3.
The
analysis
of
the
significance
of
the
underlying
methodology
for
other
aspects
of
Newtonian
Mechanics,
e.
g.
,
transformation
theory,
symmetries,
and
first
integrals
for
nonconservative
Newtonian
systems.
This
first
volume
is
devoted
to
the
foundations
of
the
Inverse
Problem,
with
particular
reference
to
aspects
I
and
2.
Cuprins
1
Elemental
Mathematics.-
1.1
Existence
theory
for
implicit
functions,
solutions,
and
derivatives
in
the
parameters.-
1.2
Calculus
of
differential
forms,
Poincaré
lemma,
and
its
converse.-
1.3
Calculus
of
variations,
action
functional,
and
admissible
variations.-
Charts:.-
1.1
A
theorem
on
the
existence,
uniqueness,
and
continuity
of
the
implicit
functions
for
Newtonian
systems.-
1.2
A
theorem
on
the
existence,
uniqueness,
and
continuity
of
a
solution
of
a
Newtonian
initial
value
problem.-
1.3
A
theorem
on
the
existence,
uniqueness,
and
continuity
of
the
derivatives
with
respect
to
parameters
of
solutions
of
Newtonian
systems.-
1.4
A
relationship
between
local
and
global
solutions
for
conservative
systems.-
1.5
Hilbert
space
approach
to
Newtonian
Mechanics.-
Examples.-
Problems.-
2
Variational
Approach
to
Self-Adjointness.-
2.1
Equations
of
motion,
admissible
paths,
variational
forms,
adjoint
systems
and
conditions
of
self-adjointness.-
2.2
Conditions
of
self-adjointness
for
fundamental
and
kinematical
forms
of
Newtonian
systems.-
2.3
Reformulation
of
the
conditions
of
self-adjointness
within
the
context
of
the
calculus
of
differential
forms.-
2.4
The
problem
of
phase
space
formulations.-
2.5
General
and
normal
forms
of
the
equations
of
motion.-
2.6
Variational
forms
of
general
and
normal
systems.-
2.7
Conditions
of
self-adjointness
for
general
and
normal
systems.-
2.8
Connection
with
self-adjointness
of
linear
operators.-
2.9
Algebraic
significance
of
the
conditions
of
self-adjointness.-
Charts:.-
2.1
Hausdorff,
second
-countable,
?-differentiable
manifolds.-
2.2
Newtonian
systems
as
vector
fields
on
manifolds.-
2.3
Symplectic
manifolds.-
2.4
Contact
manifolds.-
2.5
Geometrical
significance
of
the
conditions
of
self-adjointness.-
Examples.-
Problems.-
3
The
Fundamental
Analytic
Theorems
of
the
Inverse
Problem.-
3.1
Statement
of
the
problem.-
3.2
The
conventional
Lagrange’s
equations.-
3.3
Self-adjointness
of
the
conventional
Lagrange’s
equations.-
3.4
The
concept
of
analytic
representation
in
configuration
space.-
3.5
The
fundamental
analytic
theorem
for
configuration
space
formulations.-
3.6
A
method
for
the
construction
of
a
Lagrangian
from
the
equations
of
motion.-
3.7
The
implications
of
nonconservative
forces
for
the
structure
of
a
Lagrangian.-
3.8
Direct
and
inverse
Legendre
transforms
for
conventional
analytic
representations.-
3.9
The
conventional
Hamilton’s
equations.-
3.10
Self-adjointness
of
the
conventional
Hamilton’s
equations.-
3.11
The
concept
of
analytic
representation
in
phase
space.-
3.12
The
fundamental
analytic
theorem
for
phase
space
formulations
and
a
method
for
the
independent
construction
of
a
Hamiltonian.-
Charts.-
3.1
The
controversy
on
the
representation
of
nonconservative
Newtonian
systems
with
the
conventional
Hamilton’s
principle.-
3.2
The
arena
of
applicability
of
Hamilton’s
principle.-
3.3
Generalization
of
Hamilton’s
principle
to
include
the
integrability
conditions
for
the
existence
of
a
Lagrangian.-
3.4
Generalization
of
Hamilton’s
principle
to
include
Lagrange’s
equations
and
their
equations
of
variation.-
3.5
Generalization
of
Hamilton’s
principle
to
include
Lagrange’s
equations,
their
equations
of
variations,
and
the
end
points
contributions.-
3.6
Generalization
of
Hamilton’s
principle
to
include
a
symplectic
structure.-
3.7
Generalization
of
Hamilton’s
principle
for
the
unified
treatment
of
the
Inverse
Problem
in
configuration
and
phase
space.-
3.8
Self-adjointness
of
first-order
Lagrange’s
equations.-
3.9
The
fundamental
analytic
theorem
for
first-order
equations
of
motion
in
configuration
space.-
3.10
A
unified
treatment
of
the
conditions
of
self-adjointness
for
first-,
second-,
and
higher-order
ordinary
differential
equations.-
3.11
Engels’
methods
for
the
construction
of
a
Lagrangian.-
3.12
Mertens’approach
to
complex
Lagrangians.-
3.13
Bateman’s
approach
to
the
Inverse
Problem.-
3.14
Douglas’approach
to
the
Inverse
Problem.-
3.15
Rapoport’s
approach
to
the
Inverse
Problem.-
3.16
Vainberg’s
approach
to
the
Inverse
Problem.-
3.17
Tonti’s
approach
to
the
Inverse
Problem.-
3.18
Analytic,
algebraic
and
geometrical
significance
of
the
conditions
of
variational
self-adjointness.-
Examples.-
Problems.-
Appendix:
Newtonian
Systems.-
A.
1
Newton’s
equations
of
motion.-
A.2
Constraints.-
A.3
Generalized
coordinates.-
A.4
Conservative
systems.-
A.5
Dissipative
systems.-
A.6
Dynamical
systems.-
A.7
The
fundamental
form
of
the
equations
of
motion
in
configuration
space.-
A.l
Galilean
relativity.-
A.2
Ignorable
coordinates
and
conservation
laws.-
A.3
Impulsive
motion.-
A.4
Arrow
of
time
and
entropy.-
A.5
Gauss
principle
of
least
constraint.-
A.6
The
Gibbs-Appel
equations.-
A.7
Virial
theorem.-
A.8
Liouville’s
theorem
for
conservative
systems.-
A.9
Generalizations
of
Liouville’s
theorem
to
dynamical
systems.-
A.
10
The
method
of
Lagrange
undetermined
multipliers.-
A.
11
Geometric
approach
to
Newtonian
systems.-
A.
12
Tensor
calculus
for
linear
coordinate
transformations.-
A.
13
Tensor
calculus
for
nonlinear
coordinate
transformations.-
A.
14
Dynamical
systems
in
curvilinear
coordinates.-
Examples.-
Problems.-
References.