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Abstract

We prove that in elastic anisotropic inhomogeneous media, rays and wavefronts are orthogonal to each other with respect to the metric induced by the phase-velocity function. The standard orthogonality of rays and wavefronts in elastic isotropic inhomogeneous media is a special case of this formulation.
Generalized orthogonality between rays and
wavefronts in anisotropic inhomogeneous
media
Ioan BucataruMichael A. Slawinski
Abstract
We prove that in elastic anisotropic inhomogeneous media, rays
and wavefronts are orthogonal to each other with respect to the metric
induced by the phase-velocity function. The standard orthogonality
of rays and wavefronts in elastic isotropic inhomogeneous media is a
special case of this formulation.
1 Introduction
It has been shown in several papers, e.g., Antonelli et al (2003), Bona and
Slawinski (2002) and Antonelli et al (2002) that Finsler geometry provides
a fruitful platform for the study of seismic ray theory in anisotropic inho-
mogeneous media. In this paper, we show that the Finsler metric, discussed
in the above papers, provides a natural context for the study of rays and
wavefronts in such media.
In general, in anisotropic media, rays and wavefronts are not orthogonal
to each other in the sense of Euclidean geometry. However, in this paper we
prove that rays and wavefronts are orthogonal to each other in the sense of
the geometry imposed by the properties of the medium, which are stated in
Dept. of Earth Sciences, Memorial University, St. John’s, Newfoundland A1B 3X7,
Canada. E-mail: nbucataru@esd.mun.ca, and Faculty of Mathematics, “Al.I.Cuza” Uni-
versity, Ia¸si, Romania. E-mail: bucataru@uaic.ro
Dept. of Earth Sciences, Memorial University, St. John’s, Newfoundland A1B 3X7,
Canada. E-mail: mslawins@mun.ca
1
the context of a given phase-velocity function. Since this imposed geometry is
different from the Euclidean geometry we need to introduce its fundamental
concepts in the context of differential geometry.
We begin this paper by introducing the differential-geometry concepts
that are pertinent to ray theory in anisotropic inhomogeneous media. Therein,
we state definitions and equations that we need to investigate rays and wave-
fronts. Subsequently, we use our ray-theory Hamiltonian to obtain a general-
ized metric. Then, we show the generalized orthogonality between rays and
wavefronts in anisotropic inhomogeneous media. We conclude with a dis-
cussion of several related ray-theory properties. Orthogonality of geodesics
(rays) and geodesic balls (wavefronts) in the context of Finsler geometry is
known as Gauss Lemma, see Bao et al (2000). The proof we give here is dif-
ferent and uses Hamilton equations of rays. Moreover we prefer to work on
the cotangent space of a manifold rather then on the tangent space because
the eikonal equations are defined there and the Hamiltonian metric may not
be positive definite everywhere. This implies that the Legendre transforma-
tion may not be well defined somewhere and therefore we cannot use Euler
Lagrange equations to give the usual proof for the orthogonality.
2 Geometrical background
2.1 Geometric space
To obtain the results of this paper, in this section we define several necessary
entities and the spaces to which they belong.
The geometry of an elastic medium is the geometry of a triple (M, ρ, c),
where Mis a three-dimensional space, ρis the mass-density function, and
cis a fourth-rank tensor with particular properties. We note that — in a
mathematical context — Mis a manifold.
To study rays and wavefronts in the context of differential geometry, we
identify the physical space, M, of the elastic medium with an open subset of
the Euclidean three-dimensional space, where the coordinates are given by
functions x1,x2,x3. At every point xof M, we consider the differential of
functions ψat this point, namely,
dψ|x=
3
X
i=1
∂ψ
∂xidxi. (1)
2
The set of the differentials at xof all functions on Mis a three-dimensional
vector space. We denote this space by T
xM, while we denote the collection
of T
xMat all points xby TM. We note that TMis often called the
momentum space in classical mechanics, the phase space in Hamiltonian
mechanics and corresponds to the cotangent space in differential geometry.
Our geometry must be associated with TMsince the key entities that are
discussed below — such as the phase slowness, the Hamiltonian, Hamilton’s
equations and the Hamiltonian metric — are defined in this space.
Each element pof T
xMcan be expressed as p=p1dx1+p2dx2+p3dx3,
where — in view of expression (1) — we have
pi:= ∂ψ
∂xi,i∈ {1,2,3}. (2)
Then, the coordinates of TMare denoted by (xi, pi), where i= 1,2,3.
Symbol := emphasizes that expression (2) is a definition not an equation.
Since the entities that we are about to define are, in general, not differ-
entiable at p= 0, we remove (x,0) from TM. We denote the remaining
space by g
TM. Furthermore, as these entities are homogeneous in p, if we
do not remove (x,0), these entities would be reduced to special cases that
would limit the generality of our subsequent formulation.
To give a seismological motivation, we note that expression (2) plays an
important role in ray theory. If we consider a particular function ψ(x), we
can write a moving wavefront as ψ(x) = t, where tdenotes time. At a given
instant, wavefronts are level sets of this particular function. Hence, in view
of expression (2) — in this case — pis normal to the wavefront. Since —
in this case — units of the piare the units of slowness, we may refer to p
as the phase-slowness vector. We note that in this context, the removal of
p= 0 has an immediate physical meaning since we remove the case that
would correspond to phase slowness that is equal to zero.
At this point, we also wish to emphasize the distinction between the
formulation of various entities in TMand their meaning along particular
curves in this space that correspond to rays and wavefronts in ray theory.
Rigorous treatment of ray theory cannot be performed without global defini-
tions. In other words, while we are interested only in the properties of rays
and wavefronts, to study these properties, we require global definitions for
our Hamiltonian and its related entities.
Let us now return to triple (M, ρ, c). At each point xof M, the elasticity
tensor is a fourth-rank tensor whose components in our coordinate system
3
are cijkl (x), where i, j, k, l ∈ {1,2,3}. Using standard methods for solving
equations of motion in anisotropic inhomogeneous media, which employ an
asymptotic trial solution (e.g., B´ona and Slawinski, 2003), we obtain
£Γij (x,p)δij ¤Aj(x) = 0,i, j ∈ {1,2,3}, (3)
where A(x) is the wavefront amplitude and Γ(x,p) is the matrix whose
entries are
Γij (x,p) =
3
X
k=1
3
X
l=1
cikjl (x)
ρ(x)pkpl,i, j ∈ {1,2,3}, (4)
with ρ(x) being mass density. Matrix Γij (x,p) is called Christoffel’s matrix.
Due to the properties of the elasticity tensor, cijk l (x), Christoffel’s matrix
is symmetric and positive-definite. Consequently, the three eigenvalues of
this matrix are real and positive. We denote them by Gα(x,p), where α
{1,2,3}. It can be shown that the eigenvalues, Gα, are homogeneous of degree
2 in p(e.g., ˇ
Cerven´y, 2001, p. 22). In other words, Gα(x, λp) = λ2Gα(x,p),
for any real number λ. Also, functions Gαare differentiable on g
TM. We
note that — in view of the homogeneity of Gα— including p= 0 in the
domain of differentiability would have limited Gαto a quadratic function
in p. Consequently, our subsequent formulation would have been limited to
elliptical anisotropy.
We wish to consider a given eigenvalue, Gα. Let us refer to it as Gand,
for convenience, let
H(x,p) := 1
2G(x,p), (5)
to which we shall refer as the ray-theory Hamiltonian. In view of the prop-
erties of G, we see that His also homogeneous of degree 2 in p. Since His
homogeneous of degree 2 in p, by factoring p2=p·p, we rewrite Hamiltonian
(5) as
H(x,p) = 1
2p2v2(x,p) , (6)
where, as we immediately see,
v2(x,p) := 2H(x,p)
p2. (7)
4
Function vis homogeneous of degree 0 in p. In other words, v(xi, λpi) =
v(xi, pi). This property means that the value of vdepends on direction of p
but not on the magnitude of p.
Expression (6) is the form of our Hamiltonian to be used in the remainder
of this paper. We note that our Hamiltonian is a product of two functions,
namely, p2and v2(x,p). The former function accounts for the homogeneity
of H, while the latter one, which is homogeneous of degree 0 in p, contains
all the information about the medium. For instance, if vdoes not depend on
p, the medium is isotropic.
Having defined the phase-slowness vector, p, function v, and the ray-
theory Hamiltonian, H, which are all associated with TM, we are now
ready to formulate the equations that will allow us to discuss wavefronts and
rays. At this point, we will focus on level sets on TMthat correspond to
the wavefronts.
2.2 Wavefront and ray equations
To discuss the wavefronts, we invoke equation (3) and study the eigenvalues
that are associated with this equation. For nontrivial solvability of equation
(3), we require
det £Γij (x,p)δij ¤= 0, (8)
which is an eigenvalue equation. Hence — in the context of equation (3) —
each of the three corresponding eigenvalues of matrix (4) gives us an equation,
which can be written as
G(x,p) = 1. (9)
This is an eikonal equation allowing us to describe the wavefronts. Following
expressions (5) and (6), we can restate eikonal equation (9) as
p2=1
v2(x,p). (10)
In view of equation (10) and since wavefronts are loci of constant phase, we
refer to v, defined in expression (7), as the phase-velocity function. Equations
(9) and (10) are valid for anisotropic inhomogeneous media since, considering
wavefronts and their normals, we see that xrefers to the dependence on
position and prefers to the dependence on direction.
5
We wish to explicitly state the eikonal equation as a differential equation.
In view of expression (2), we can write equation (10) as
3
X
i=1 µ∂ψ
∂xi2
=1
v2µxi,∂ψ
∂xi. (11)
This is a first-order nonlinear partial differential equation for function ψ. To
state equation (11) in terms of our Hamiltonian, H, in view of expression
(6), we can write
2H(x,p)2Hµxi,∂ψ
∂xi= 1, (12)
which is akin to equation (9) and is a standard form of the Hamilton-Jacobi
equation. The solution of equation (12) is function ψ, whose level sets are
the wavefronts.
Solving the eikonal equation by the method of characteristics (e.g., Courant
and Hilbert, 1989), we obtain Hamilton’s ray equations, namely,
dxi
dt=∂H
∂pi
dpi
dt=∂H
∂xi
,i∈ {1,2,3}. (13)
Rays are curves whose components, [x1(t), x2(t), x3(t)], are solutions of
system (13).
Having formulated the equations describing wavefronts and rays, we are
now ready to study their geometrical relation, which is a function of a metric
that characterizes a given geometry. Thus, we begin by formulating pertinent
metrics.
2.3 Hamiltonian metric
In general, a given elastic medium exhibits two metric structures that are of
interest in our study. They are the Euclidean metric δij , where δij is Kro-
necker’s delta, and metric gij(x,p), to which we refer to as the Hamiltonian
metric. The Euclidean metric is intrinsically associated with the physical
6
space, while the Hamiltonian metric is induced by the phase-velocity func-
tion that corresponds to a given type of waves that propagate in the medium.
Since, in general, there are three distinct phase-velocity functions, which cor-
respond to the three types of waves that propagate in an anisotropic inhomo-
geneous medium, at every point of the medium there are three Hamiltonian
metrics.
Now, we wish to obtain the Hamiltonian metric that corresponds to the
properties of a given medium.
Consider the ray-theory Hamiltonian given by expression (6). Using this
Hamiltonian, we write a convenient metric (e.g., Rund, 1959) whose compo-
nents are given by
gij (x,p) := 2H
∂pipj
(x,p) , i, j ∈ {1,2,3}. (14)
Inserting expression (6) into expression (14), we can obtain an explicit ex-
pression for the components of this metric, namely,
gij =v2δij + 2vµpi∂v
∂pj
+pj∂v
∂pi+p2µv2v
∂pipj
+∂v
∂pi
∂v
∂pj,i, j ∈ {1,2,3}.
(15)
Thus, our Hamiltonian metric naturally emerges from the properties of a
given medium, which are contained in the phase-velocity function, v.
We require that Hbe regular, which means that the matrix with entries
given in expression (14) be nondegenerate on g
TM. In other words, det [gij ]6=
0. We note that, physically, det [gij ] = 0 corresponds to the inflection points
of a phase-slowness surface. In this paper, however, we do not study these
singular points. In view of expression (6), the assumption of differentiability
of His equivalent to the assumption of differentiability of v.
Examining expression (14), we observe the following properties of the
Hamiltonian metric (e.g., Miron, et al., 2001). Since His homogeneous of
degree 2 in the pi, it follows that the components of the Hamiltonian metric
are homogeneous of degree 0 in the pi. Furthermore, since det [gij ]6= 0, we
also have the inverse of the Hamiltonian metric, which we denote by gij (x,p);
in other words,
3
X
j=1
gij (x,p)gjk (x,p) = δk
i,i, k ∈ {1,2,3}. (16)
7
Examining expression (15), we recognize that we can also view this expres-
sion as the relation between the Hamiltonian metric, gij, and the Euclidean
metric, δij . We note that for an isotropic medium, where vis independent
of p, equation (15) reduces to gij =v2δij . This means that in an isotropic
medium the two metrics differ by a multiplicative scalar factor; in other
words, they are conformal to one another. This also justifies our choice of
metric (14).
Having formulated the Hamiltonian metric, we are now ready to complete
our study of a geometrical relation between rays and wavefronts, namely,
their orthogonality.
3 Orthogonality between rays and wavefronts
3.1 Euclidean and Hamiltonian gradients
To discuss orthogonality, we use the fact that a gradient of a function with
respect to a given metric is a vector that is orthogonal to the level sets of
this function with respect to this metric. For function ψon M, we can define
its gradient as being either the vector whose components are
(ψ)i=
3
X
j=1
δij ∂ψ
∂xj,i∈ {1,2,3}, (17)
with respect to the Euclidean metric, or as the vector whose components are
(gψ)i=
3
X
j=1
gij ∂ψ
∂xj,i∈ {1,2,3}, (18)
with respect to the Hamiltonian metric (e.g., Anastasiei, 1998). In other
words, expression (17) defines the Euclidean gradient, while expression (18)
defines the Hamiltonian gradient. Consequently, ψis orthogonal to the
level sets of ψwith respect to the Euclidean metric, while gψis orthogonal
to the level sets of ψwith respect to the Hamiltonian metric. For conciseness
of terminology, when dealing with the latter case, we will refer to it as the
Hamilton-orthogonality. In the next section, we show that gψis Hamilton-
orthogonal to the level sets of ψ. Before showing this orthogonality, we wish
to show that gradients ψand gψare distinct from one another.
8
In expression (15), we obtained the relation between the Euclidean and
Hamiltonian metrics. Using this relation we can also derive an analytical
expression relating the components of the two corresponding gradients, as
follows. In view of expressions (15), (17) and (18), we can write
(gψ)i=v2(ψ)i+vp2v
∂pi
,i∈ {1,2,3}. (19)
Thus, in general, the two gradients are distinct from one another.
3.2 Hamilton-orthogonality of gψand wavefronts
Herein, we rigorously show that vector gψis Hamilton-orthogonal to the
wavefronts. This is equivalent to saying that gψis Hamilton-orthogonal to
any curve that belongs to a level set of ψ.
Consider such a curve described by our coordinate functions as
ψ(x(t)) = C, (20)
where Cdenotes a constant. Taking the derivative of both sides of equation
(20) with respect to t, we obtain
3
X
i=1
∂ψ
∂xi
dxi
dt= 0. (21)
To state expression (21) in terms of the Hamiltonian gradient, using ex-
pression (16), we can rewrite expression (21) as
3
X
i=1
3
X
j=1
3
X
k=1
gji gjk ψ
∂xk
dxi
dt= 0. (22)
Recognizing that P3
k=1 gjk ψ/∂xkis expression (18), we can write expression
(22) as
3
X
i=1
3
X
j=1
gji (gψ)jdxi
dt= 0. (23)
Equation (23) states that the scalar product — with respect to the Hamil-
tonian metric — of the Hamiltonian gradient and the vector tangent to the
wavefront vanishes. Thus, the Hamiltonian gradient is Hamilton-orthogonal
to the wavefront.
9
3.3 Hamilton-orthogonality of rays and wavefronts
To show the Hamilton-orthogonality of rays and wavefronts, it now suffices
to show that the vector tangent to the ray coincides with the Hamiltonian-
gradient vector gψ.
Consider a ray described in our coordinates as a curve given by x(t).
The vector tangent to this ray can be written as dx/dt. In view of the first
equation of system (13), we can write the components of dx/dtas
dxi
dt=∂H
∂pi
,i∈ {1,2,3}. (24)
We want to show that H/∂piare the components of the Hamiltonian gra-
dient of ψ.
Since His homogeneous of degree 2 in the pi, by Euler’s homogeneous-
function theorem, we can write
3
X
i=1
∂H
∂pi
pi= 2H. (25)
Furthermore, since His homogeneous of degree 2 in the pi, it follows that
∂H/∂piis homogeneous of degree 1 in the pi. Following Euler’s homogeneous-
function theorem, we obtain
3
X
j=1
∂pjµH
∂pipj=H
∂pi
,i∈ {1,2,3}. (26)
Multiplying both sides of equation (26) by piand summing over i, we get
3
X
i=1
3
X
j=1
2H
∂pipj
pipj=
3
X
i=1
∂H
∂pi
pi. (27)
Following expressions (14) and (25), we can rewrite equation (27) as
2H(x,p) =
3
X
i=1
3
X
j=1
gij (x,p)pipj. (28)
Taking partial derivatives of equation (28) with respect to pi, we obtain
2∂H
∂pi
= 2
3
X
j=1
gij pj+
3
X
j=1
3
X
k=1
∂gkj
∂pi
pkpj,i∈ {1,2,3}. (29)
10
Now, we will show that the double summation is identically zero. Recall-
ing expression (14), we can write
∂gkj
∂pi
=1
2
3H
∂pipkpj
=∂gij
∂pk
,i, j, k ∈ {1,2,3}, (30)
where the second equality results from the equality of the mixed partial
derivatives. Using expression (30), we can rewrite the double summation as
3
X
j=1
3
X
k=1
∂gkj
∂pi
pkpj=
3
X
j=1 Ã3
X
k=1
∂gij
∂pk
pk!pj,i∈ {1,2,3}. (31)
Since gij is homogeneous of degree 0 in the pi, by Euler’s homogeneous-
function theorem, the term in parentheses vanishes. Consequently, the double
summation is identically zero and expression (29) is reduced to
∂H
∂pi
=
3
X
j=1
gij pj,i∈ {1,2,3}. (32)
Using expression (32), we can now write equation (24) as
dxi
dt=
3
X
j=1
gij pj,i∈ {1,2,3}. (33)
In view of expressions (2) and (18), we can rewrite equation (33) as
dxi
dt= (gψ)i,i∈ {1,2,3}. (34)
Equation (34) states that the vector tangent to the ray coincides with the
Hamiltonian gradient of ψ, whose level sets are the wavefronts. Hence, the
proof of the statement that rays and wavefronts are Hamilton-orthogonal to
each other is complete.
4 Discussion and conclusions
This paper proves the Hamilton-orthogonality between rays and wavefronts
in anisotropic inhomogeneous media. In other words, the Euclidean orthog-
onality associated with isotropic media is extended to the anisotropic ones.
11
This can be explained in the following way. The Hamiltonian metric, which
is derived from the angle-dependent phase velocity, contains this angular de-
pendence. This means that the anisotropy has been accounted for by the
metric itself.
In addition, this study allows us to illustrate the following ray-theory
properties.
We recall that the Hamiltonian metric is expressed in terms of the phase-
velocity function, as shown in expression (15). It is also possible to obtain
the phase-velocity function from the Hamiltonian metric alone. To see this,
consider expression (28). Dividing both sides of equation (28) by p2and in
view of expression (6), we obtain
v2(x,p) =
3
X
i=1
3
X
j=1
gij (x,p)pi
p
pj
p=
3
X
i=1
3
X
j=1
gij (x,n)ninj. (35)
In expression (35), p=p·pand ni=pi/p.
We note that the fact that rays and wavefronts are Hamilton-orthogonal
to each other means that each elementary wavefront generated by a point
source is a unit ball of the geometry induced by the phase-velocity function.
In other words — in the context of the Hamiltonian metric — each elementary
wavefront can be viewed as a sphere with unit radius. To see this, we can
use expression (28) to rewrite Hamilton-Jacobi equation (12) as
3
X
i=1
3
X
j=1
gij µxk,∂ψ
∂xkψ
∂xi
∂ψ
∂xj= 1. (36)
Equation (36) can be expressed in terms of the Hamiltonian gradient, as fol-
lows. The squared magnitude of the Hamiltonian gradient gψwith respect
to the Hamiltonian metric is
(gψ)2=
3
X
i=1
3
X
j=1
3
X
k=1
3
X
l=1
gklgki ψ
∂xiglj ψ
∂xj=
3
X
i=1
3
X
j=1
gij ∂ψ
∂xi
∂ψ
∂xj. (37)
In view of expression (37), we can concisely write equation (36) as
(gψ)2= 1, (38)
which is a unit ball. Notably, this is the indicatrix of the Finsler geometry
(e.g., Bao et al., 2000), and we see that the geodesics are orthogonal to
indicatrices.
12
The results we have shown include, in particular, the case of isotropy. If
a medium is isotropic, then the phase-velocity function depends on position
only, namely, v=v(x). In such a case, expression (19) can be written as
gψ=v2ψ. (39)
Also, in such a case, expression (15) reduces to
gij =v2δij ,i, j ∈ {1,2,3}, (40)
which means that, for isotropy, the Hamiltonian and Euclidean metrics are
conformal to one another. From equations (39) and (40), we see that, in
isotropic media, the Hamilton-orthogonality reduces to the standard Eu-
clidean orthogonality.
Acknowledgements
We would like to acknowledge the valuable editorial work of Cathy Beveridge,
the fruitful collaboration with Drs. Andrej B´ona and Michael Rochester.
This study has been done in the context of The Geomechanics Project.
Hence, the authors wish to acknowledge the support of EnCana Energy,
Husky Energy and Talisman Energy.
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13
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14
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... 2. φ(t) and p(t) are real, 3. the imaginary part of H(t) = (H ij (t)) ij is positive definite. ...
... The next two propositions give some geometric information about F t . Proposition 2.1 is proved in [3] for real phase functions. ...
... Let us recall Gauss' lemma in Riemannian geometry [3,19]. It states that if B is a geodesic ball around a point p, then geodesics from p will intersect the boundary of B orthogonally. ...
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We study the complex Riccati tensor equation D ú cG + GCG − R = 0 on a geodesic c on a Riemannian 3-manifold. This non-linear equation appears in the study of Gaussian beams. Gaussian beams are asymptotic solutions to hyperbolic equations that at each time instant are concentrated around one point in space. When time moves forward, Gaussian beams move along geodesics, and the Riccati equation determines the Hessian of the phase function for the Gaussian beam. The imaginary part of a solution G describes how a Gaussian beam decays in different directions of space. The main result of the present work is that the real part of G is the shape operator of the phase front for the Gaussian beam. This result generalizes a known result for the Riccati equation in R3. The idea of the proof is to express the Riccati equation in Fermi coordinates adapted to the underlying geodesic. In Euclidean ge- ometry we also study when the phase front is contained in the area of influence, or light cone.
... Cone structures appear in different parts of Mathematics and they are the basis of Causality in standard Relativity as well as in recent extensions such as Finsler spacetimes (see [25,29,32] and references therein). As pointed out by some authors [5,12,15,17,35], the viewpoint of spacetimes can be used in non-relativistic settings to describe the propagation of certain physical phenomena that propagate through a medium at finite speed, e.g., wildfires or sound waves, and the framework can be extended to other phenomena such as seismic waves [3,7,33], water waves, etc. Indeed, this applies in some situations related to the classical Fermat's principle such as Zermelo's navigation problem, which seeks the fastest trajectory between two prescribed points for a moving object with respect to a medium, which may also move with respect to the observer (see the recent detailed study in [9,25]). ...
... In our setup, this means that all J ± (p) are closed (thus, equal to the closure I ± (p)).6 In our setup, all J + (p) ∩ J − (q) compact, see[6,34,25] for background.7 More precisely, it is easy to check that if a causal curveγ(t) = (t, γ(t)), t ∈ I ⊂ R, cannot be continuously extended to the endpoints of I, then I = R. ...
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A general framework for the description of classic wave propagation is introduced. This relies on a cone structure $C$ determined by an intrinsic space $\Sigma$ of velocities of propagation (point, direction and time-dependent) and an observers' vector field $\partial_t$ whose integral curves provide both a Zermelo problem for the wave and an auxiliary Lorentz-Finsler metric $G$ compatible with $C$. The PDE for the wavefront is reduced to the ODE for the $t$-parametrized cone geodesics of $C$. Particular cases include time-independence ($\partial_t$ is Killing for $G$), infinitesimally ellipsoidal propagation ($G$ can be replaced by a Lorentz metric) or the case of a medium which moves with respect to $\partial_t$ faster than the wave (the strong wind case of a sound wave), where a conic time-dependent Finsler metric emerges. The specific case of wildfire propagation is revisited.
... From this approximation arises the convenient fiction of seismic rays, which are both the characteristics of the HJE and solutions of Hamilton's equations. Although there are disadvantages to its use in treating seismic wave propagation, e.g., dynamic interactions are not modeled and spectral information is lost, the Hamiltonian approach has proven insightful, particularly when dealing with anisotropic media Antonelli et al. (2003); Slawinski (2002, 2003); Bucataru and Slawinski (2005);Červený (2002); Klimeš (2002); Yajima and Nagahama (2009); Yajima et al. 135 (2011). ...
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The rate of erosion of a geomorphic surface depends on its local gradient and on the material fluxes over it. Since both quantities are functions of the shape of the catchment surface, this dependence constitutes a mathematical straitjacket, in the sense that – subject to simplifying assumptions about the erosion process, and absent variations in external forcing and erodibility – the rate of change of surface geometry is solely a function of surface geometry. Here we demonstrate how to use this geometric self-constraint to convert an erosion model into its equivalent Hamiltonian, and explore the implications of having a Hamiltonian description of the erosion process. To achieve this conversion, we recognize that the rate of erosion defines the velocity of surface motion in its orthogonal direction, and we express this rate in its reciprocal form as the surface-normal slowness. By rewriting surface tilt in terms of normal slowness components, and by deploying a substitution developed in geometric mechanics, we extract what is known as the fundamental metric function of the model phase space; its square is the Hamiltonian. Such a Hamiltonian provides several new ways of solving for the evolution of an erosion surface: here we use it to derive Hamilton's ray tracing equations, which describe both the velocity of a surface point and the rate of change of the surface-normal slowness at that point. In this context, erosion involves two distinct directions: (i) the surface-normal direction, which points subvertically downwards, and (ii) the erosion ray direction, which points upstream at a generally small angle to horizontal with a sign controlled by the scaling of erosion with slope. If the model erosion rate scales faster than linearly with gradient, the rays point obliquely upwards; but if erosion scales sublinearly with gradient, the rays point obliquely downwards. Analysis of the Hamiltonian shows that these rays carry boundary-condition information upstream, and that they are geodesics, meaning that erosion takes the path of least erosion time. This constitutes a definition of the variational principle governing landscape evolution. In contrast with previous studies of network self-organization, neither energy nor energy dissipation is invoked in this variational principle, only geometry.
... Moreover, the geometric structures can be found within a framework of a classical Newtonian mechanics. For example, a propagation of an elastic wave through anisotropic media is geometrically described in the Finsler space because of a direction-dependence of a crystal [2][3][4][5][6]. Similarly, a fluids flow followed by the Darcy's law through inhomogeneous porous media gives the Finslerian metric function because a hydraulic conductivity depends on a position by an inhomogeneity [7][8][9]. ...
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A theory of non-Riemannian geometry (Riemann-Cartan geometry) can be applied to a free rotation of a rigid body system. The Euler equations of angular velocities are transformed into equations of the Euler angle. This transformation is geometrically non-holonomic, and the Riemann-Cartan structure is associated with the system of the Euler angles. Then, geometric objects such as torsion and curvature tensors are related to a singularity of the Euler angle. When a pitch angle becomes singular ±π/2, components of the torsion tensor diverge for any shape of the rigid body while components of the curvature tensor do not diverge in case of a symmetric rigid body. Therefore, the torsion tensor is related to the singularity of dynamics of the rigid body rather than the curvature tensor. This means that the divergence of the torsion tensor is interpreted as the occurrence of the gimbal lock. Moreover, attitudes of the rigid body for the singular pitch angles ±π/2 are distinguished by the condition that a path-dependence vector of the Euler angles diverges or converges.
... This formula has been used in the above cited papers in order to obtain the Gaussian curvature for the Remark. Throughout this paper we shall alternatively call I x the wavefront at x, following the paper [9]. ...
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The present paper deals with the Gaussian curvature K of a particular indicatrix in a general Finsler manifold. A formula for K similar to the one obtained by Shin-ichi Nishimura and Masao Hashiguchi [Rep. Fac. Sci., Kagoshima Univ., Math. Phys. Chem. 24, 33–41 (1991; Zbl 0770.53016)] and Masao Hashiguchi [Rep. Fac. Sci., Kagoshima Univ., Math. Phys. Chem. 25, 21–27 (1992; Zbl 0795.53022)], is derived for (α,β)-metrics and then specialized to Randers, Kropina, Matsumoto and Riemann–type metrics. For the Funk two-dimensional case the class of Randers surfaces is detailed, and the Randers-Funk metric on the unit disk is investigated. Alternative expressions for K are provided, in terms of Finslerian metric and angular metric, and in terms of Berwald frame for n=2. It is shown that K is h-covariant constant with respect to the Cartan and Chern-Rund connections. The last section describes the pseudo-Finsler locally-Minkowski Berwald-Moor case.
Article
The rate of erosion of a landscape depends largely on local gradient and material fluxes. Since both quantities are functions of the shape of the catchment surface, this dependence constitutes a mathematical straitjacket, in the sense that – subject to simplifying assumptions about the erosion process, and absent variations in external forcing and erodibility – the rate of change of surface geometry is solely a function of surface geometry. Here we demonstrate how to use this geometric self-constraint to convert a gradient-dependent erosion model into its equivalent Hamiltonian, and explore the implications of having a Hamiltonian description of the erosion process. To achieve this conversion, we recognize that the rate of erosion defines the velocity of surface motion in its orthogonal direction, and we express this rate in its reciprocal form as the surface-normal slowness. By rewriting surface tilt in terms of normal slowness components and deploying a substitution developed in geometric mechanics, we extract what is known as the fundamental metric function of the model phase space; its square is the Hamiltonian. Such a Hamiltonian provides several new ways to solve for the evolution of an erosion surface: here we use it to derive Hamilton's ray-tracing equations, which describe both the velocity of a surface point and the rate of change of the surface-normal slowness at that point. In this context, gradient-dependent erosion involves two distinct directions: (i) the surface-normal direction, which points subvertically downwards, and (ii) the erosion ray direction, which points upstream at a generally small angle to horizontal with a sign controlled by the scaling of erosion with slope. If the model erosion rate scales faster than linearly with gradient, the rays point obliquely upwards, but if erosion scales sublinearly with gradient, the rays point obliquely downwards. This dependence of erosional anisotropy on gradient scaling explains why, as previous studies have shown, model knickpoints behave in two distinct ways depending on the gradient exponent. Analysis of the Hamiltonian shows that the erosion rays carry boundary-condition information upstream, and that they are geodesics, meaning that surface evolution takes the path of least erosion time. Correspondingly, the time it takes for external changes to propagate into and change a landscape is set by the velocity of these rays. The Hamiltonian also reveals that gradient-dependent erosion surfaces have a critical tilt, given by a simple function of the gradient scaling exponent, at which ray-propagation behaviour changes. Channel profiles generated from the non-dimensionalized Hamiltonian have a shape entirely determined by the scaling exponents and by a dimensionless erosion rate expressed as the surface tilt at the downstream boundary.
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One of the finest and most powerful assets of Finsler geometry is its ability to model, describe, and analyse in precise geometric terms an abundance of physical phenomena that are genuinely asymmetric, see e.g. Antonelli et al. (1993, 2003), Yajima and Nagahama (2009), Bao et al. (2004), Cvetič and Gibbons (2012), Gibbons et al. (2007), Astola and Florack (2011), Caponio et al. (2011), Yajima and Nagahama (2015). In this paper we show how wildfires can be naturally included into this family. Specifically we show how the celebrated and much applied Richards' equations for the large scale elliptic wildfire spreads have a rather simple Finsler-geometric formulation. The general Finsler framework can be explicitly 'integrated' to provide detailed-and curvature sensitive-geodesic solutions to the wildfire spread problem. The methods presented here stem directly from first principles of 2-dimensional Finsler geometry, and they can be readily extracted from the seminal monographs Shen (2001) and Bao et al. (2000), but we will take special care to introduce and exemplify the necessary framework for the implementation of the geometric machinery into this new application - not least in order to facilitate and support the dialog between geometers and the wildfire modelling community. The 'integration' part alluded to above is obtained via the geodesics of the ensuing Finsler metric which represents the local fire templates. The 'paradigm' part of the present proposal is thus concerned with the corresponding shift of attention from the actual fire-lines to consider instead the geodesic spray-the 'fire-particles'-which together, side by side, mould the fire-lines at each instant of time and thence eventually constitute the local and global structure of the wildfire spread.
Article
Fluids flow followed by Darcy’s law through inhomogeneous porous media is studied by the theory of Finsler geometry. According to Fermat’s variational principle, the nonlinear paths of fluids flow called Darcy’s flow are described by geodesics in a Finsler space. For inhomogeneous media, the direction dependence of Darcy’s flow gives a Finsler metric called Kropina metric. Then, the influence of direction dependence on the Darcy’s flow is shown by the differences between Riemannian geodesics and Finslerian geodesics. In this case, the deviation curvature tensor implies that the trajectory of Darcy’s flow is Jacobi unstable for the deviation of geodesics. Moreover, similar to Darcy’s flow, the seismic ray path in anisotropic media can be defined in Finsler space, and the metric of seismic ray path is given by the th root metric. It is shown that the relationship between the variational problems of Darcy’s flow and seismic ray path in Finsler space.
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The potential energy surface of a molecule can be decomposed into equipotential hypersurfaces of the level sets. It is a foliation. The main result is that the contours are the wave fronts of a certain hyperbolic partial differential equation, a wave equation. It is connected with the gradient lines, as well as with a corresponding eikonal equation. The energy seen as an additional coordinate plays the central role in this treatment. A solution of the wave equation can be a sharp front in the form of a delta distribution. We discuss a general Huygens’ principle: there is no wake of the wave solution in every dimension.
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We prove that, in general, for anisotropic nonuniform continua, seismic rays are geodesics in Finsler geometry. In particular, for separable velocity functions, the geometry is Wagnerian. We provide concrete examples with theoretical discussions and introduce the seismic Finsler metric.
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We prove that rays in linearly elastic anisotropic nonuniform media obey Fermat's principle of stationary traveltime. First, we formulate the concept of rays, which emerges from the Hamilton equations. Then, we show that these rays are solutions of the variational problem stated by Fermat's principle. This proof is valid for all rays except the ones associated with infection points on the phase-slowness surface.
Article
The title of this book is no surprise for people working in the field of Analytical Mechanics. However, the geometric concepts of Lagrange space and Hamilton space are completely new. The geometry of Lagrange spaces, introduced and studied in [76],[96], was ext- sively examined in the last two decades by geometers and physicists from Canada, Germany, Hungary, Italy, Japan, Romania, Russia and U.S.A. Many international conferences were devoted to debate this subject, proceedings and monographs were published [10], [18], [112], [113],... A large area of applicability of this geometry is suggested by the connections to Biology, Mechanics, and Physics and also by its general setting as a generalization of Finsler and Riemannian geometries. The concept of Hamilton space, introduced in [105], [101] was intensively studied in [63], [66], [97],... and it has been successful, as a geometric theory of the Ham- tonian function the fundamental entity in Mechanics and Physics. The classical Legendre’s duality makes possible a natural connection between Lagrange and - miltonspaces. It reveals new concepts and geometrical objects of Hamilton spaces that are dual to those which are similar in Lagrange spaces. Following this duality Cartan spaces introduced and studied in [98], [99],..., are, roughly speaking, the Legendre duals of certain Finsler spaces [98], [66], [67]. The above arguments make this monograph a continuation of [106], [113], emphasizing the Hamilton geometry.
Chapter
Finsler and Cartan geometries are shown to provide seismological science’s Seismic Ray Theory, with a streamlined mathematical formalism which is then applied to signals in materials with azimuthal symmetry whose (qP)-wave solution of the eiconal equation, yields a regular Hamiltonian. The Legendre transformation theory plus use of software, MAPLE, provides a means to obtain ray velocities as functions of ray angle with exact formulas. The use of “polar constructions” as is traditional in this field can thus be circumvented, once tables are constructed using explicit elasticity constants.
Article
Since the first volume of this work came out in Germany in 1924, this book, together with its second volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's second and final revision of 1953. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. All rights reserved.
Book
I: Calculus of Variations. Minkowskian Spaces.- 1. Problems in the calculus of variations in parametric form.- 2. The tangent space. The indicatrix.- 3. The metric tensor and the osculating indicatrix.- 4. The dual tangent space. The figuratrix.- 5. The Hamiltonian function.- 6. The trigonometric functions and orthogonality.- 7. Definitions of angle.- 8. Area and Volume.- II: Geodesics: Covariant Differentiation.- 1. The differential equations satisfied by the geodesics.- 2. The explicit expression for the second derivatives in the differential equations of the geodesies.- 3. The differential of a vector.- 4. Partial differentiation of vectors.- 5. Elementary properties of ?-differentiation.- III: The "Euclidean Connection" of E. Cartan.- 1. The fundamental postulates of Cartan.- 2. Properties of the covariant derivative.- 3. The general geometry of paths: the connection of Berwald.- 4. Further connections arising from the general geometry of paths.- 5. The osculating Riemannian space.- 6. Normal coordinates.- IV: The Theory of Curvature.- 1. The commutation formulae.- 1 . Commutation formulae resulting from ?-differentiation.- 2 . The three curvature tensors of Cartan.- 3 . Alternative derivation of the curvature tensors by means of exterior forms.- 2. Identities satisfied by the curvature tensors.- 3. The Bianchi identities.- 4. Geodesic deviation Ill.- 5. The first and second variations of the length integral.- 6. The curvature tensors arising from Berwald's connection.- 7. Spaces of constant curvature.- 8. The projective curvature tensors.- 1 . The generalised Weyl tensor.- 2 . The projective connection.- 3 . Projectively flat spaces spaces with rectilinear geodesies.- V: The Theory of Subspaces.- 1. The theory of curves.- 2. The projection factors.- 3. The induced connection parameters. .- 4. Fundamental aspects of the theory of subspaces based on the euclidean connection.- 1 . The normal curvature and associated tensors.- 2 . The D-symbolism.- 3 . The generalised equations of Gauss, Codazzi and Kuhne.- 5. The Lie derivative and its application to the theory of subspaces.- 6. Surfaces imbedded in an F3.- 7. Fundamental aspects of the theory of subspaces from the point of view of the locally Minkowskian metric.- 1 . Normal curvature.- 2 . The two second fundamental forms.- 3 . Principal directions.- 4 . The equations of Gauss and Codazzi.- 5 . Subspaces of arbitrary dimension.- 8. The differential geometry of the indicatrix and the geometrical significance of the tensor Sijhk.- 9. Comparison between the induced and the intrinsic connection parameters.- VI: Miscellaneous Topics.- 1. Groups of motions.- 2. Conformai geometry.- 3. The equivalence problem.- 4. The theory of non-linear connections.- 5. The local imbedding theories.- 6. Two-dimensional Finsler spaces.- 1 . Formal Aspects.- 2 . Certain projective changes applied to F2. Spaces with rectilinear geodesics.- 3 . Two-dimensional Finsler spaces whose principal scalar is a function of position only. Landsberg spaces.- Appendix: Bibliographical references to related topics.- Symbols.
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This book focuses on the elementary but essential items among these results.
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Preliminary remarks about linear differential equationsSystems of a finite number of degrees of freedomThe vibrating stringThe vibrating rodThe vibrating membraneThe vibrating plateGeneral remarks on the eigenfunction methodVibration of three-dimensional continua. Separation of variablesEigenfunctions and the boundary value problem of potential theoryProblems of the Sturm-Liouville type. Singular boundary pointsThe asymptotic behavior of the solutions of Sturm-Liouville equationsEigenvalue problems with a continuous spectrumPerturbation theoryGreen's function (influence function) and reduction of differential equations to integral equationsExamples of Green's functionSupplement to Chapter V