Lagrangian Formulation of Elastic Wave Equation on Riemannian Manifolds

RWE-C3-EAFIT Lagrangian Formulation of Elastic Wave Equation on Riemannian Manifolds Hector Roman Quiceno E. Advisors Ph.D Jairo Alberto Villegas G. Ph.D Diego Alberto Gutierrez I. Centro de Ciencias de la Computacion C3-ITM Universidad EAFIT June 01, 2017

Outline RWE-C3-EAFIT

Outline 1. Lagrangian Formulation of Mechanics

RWE-C3-EAFIT Outline 1. Lagrangian Formulation of Mechanics 2. Lagrangian Formulation on Riemannian Manifolds

RWE-C3-EAFIT Outline 1. Lagrangian Formulation of Mechanics 2. Lagrangian Formulation on Riemannian Manifolds 3. Lagrangian Formulation for an Elastic Space of Configurations

RWE-C3-EAFIT Outline 1. Lagrangian Formulation of Mechanics 2. Lagrangian Formulation on Riemannian Manifolds 3. Lagrangian Formulation for an Elastic Space of Configurations 4. Further Works

Motivation RWE-C3-EAFIT

RWE-C3-EAFIT Motivation It is a reformulation of Newton s mechanics

RWE-C3-EAFIT Motivation It is a reformulation of Newton s mechanics Based on least action principles (Calculus of variations)

RWE-C3-EAFIT Motivation It is a reformulation of Newton s mechanics Based on least action principles (Calculus of variations) Allows to work with different fields, such as the electromagnetic field in one simple formulation

Motivation It is a reformulation of Newton s mechanics Based on least action principles (Calculus of variations) Allows to work with different fields, such as the electromagnetic field in one simple formulation It only considers the forces that give rise to motions

Motivation It is a reformulation of Newton s mechanics Based on least action principles (Calculus of variations) Allows to work with different fields, such as the electromagnetic field in one simple formulation It only considers the forces that give rise to motions Give rise to nondynamical symmetries because of the way in which we formulate the action

Lagrangian formulation of mechanics The action Examples

Lagrangian formulation of mechanics The action Examples

RWE-C3-EAFIT Lagrangian formulation of mechanics The action L(x, ẋ, t) : C R and the action is We need to solve the problem t2 S = Ldt. t 1 δs = 0. Examples

Lagrangian formulation of mechanics The action Examples

Lagrangian formulation of mechanics The action Examples Consider the Lagrangian L(x, y, v) = mv 2 mgy 2

RWE-C3-EAFIT Lagrangian formulation of mechanics The action Examples Consider the Lagrangian which gives the equations L(x, y, v) = mv 2 mgy 2 ẍ = 0 ÿ = g.

Lagrangian formulation of mechanics The action Examples Consider the Lagrangian L(x, y, v) = mv 2 mgy 2 Consider L(r, ṙ) = 1 [ṙ 2 m 2 + r 2 θ2 + φ ] 2 sin 2 (θ) V (r) (1)

Lagrangian formulation of mechanics The action Examples Consider the Lagrangian Consider we get the system of equations L(x, y, v) = mv 2 mgy 2 L(r, ṙ) = 1 [ṙ 2 m 2 + r 2 θ2 + φ ] 2 sin 2 (θ) V (r) (1) r = r θ 2 + r sin 2 θ φ 2 1 m θ = 2 r ṙ θ + sin θ cos θ φ 2 φ = 2 r ṙ φ 2 cot θ θ φ dv dr in particular, for θ = π 2 r = r φ 2 1 m θ = 0 φ = 2 r ṙ φ dv dr

Lagrangian formulation of mechanics RWE-C3-EAFIT

RWE-C3-EAFIT Lagrangian formulation of mechanics Consider a scalar field φ, and the Lagrangian L(φ, φ) = 1 2 g µν ( µφ) ( νφ) 1 2 m2 φ 2 V (φ)

RWE-C3-EAFIT Lagrangian formulation of mechanics Consider a scalar field φ, and the Lagrangian L(φ, φ) = 1 2 g µν ( µφ) ( νφ) 1 2 m2 φ 2 V (φ) The Euler-Lagrange equations for this field are we get ( ) L φ r L q µ φ r = 0,µ

Lagrangian formulation of mechanics Consider a scalar field φ, and the Lagrangian L(φ, φ) = 1 2 g µν ( µφ) ( νφ) 1 2 m2 φ 2 V (φ) where ( + m 2 ) φ = V φ, 2 t 2 2

Lagrangian Formulation on Riemannian Manifolds

Lagrangian Formulation on Riemannian Manifolds Let (M, g), (N, G) be Riemannian manifolds, φ F(M) and ψ : M N. The Lagrangian is a function which depends on ψ and ψ ;i. L : TN R,

Lagrangian Formulation on Riemannian Manifolds Let (M, g), (N, G) be Riemannian manifolds, φ F(M) and ψ : M N. The Lagrangian is a function L : TN R, which depends on ψ and ψ ;i. We define the variations of ψ as a one-parameter family of functions ψ(s, x), where s ( ε, ε)

Lagrangian Formulation on Riemannian Manifolds Let (M, g), (N, G) be Riemannian manifolds, φ F(M) and ψ : M N. The Lagrangian is a function L : TN R, which depends on ψ and ψ ;i. We define the variations of ψ as a one-parameter family of functions ψ(s, x), where s ( ε, ε) The integral I = Ldv g D M is called stationary under a variation of ψ if di ds s=0 = 0

RWE-C3-EAFIT Lagrangian Formulation on Riemannian Manifolds Let (M, g), (N, G) be Riemannian manifolds, φ F(M) and ψ : M N. The Lagrangian is a function L : TN R, which depends on ψ and ψ ;i. We define the variations of ψ as a one-parameter family of functions ψ(s, x), where s ( ε, ε) The integral I = Ldv g D M is called stationary under a variation of ψ if di ds s=0 = 0 If the integral is stationary, then it satisfies the Euler-Lagrange equations on the manifold ( ) m L (ψ i k=1 ;k ) = L ψ i ;k

Lagrangian Formulation for an Elastic Space of Configurations

Lagrangian Formulation for an Elastic Space of Configurations Let us consider a body manifold B with an atlas (ψ i, U i ) where B R n and ψ i (U i ) R n ; regard this manifold as the undeformed state of any elastic medium. Let S be ambient manifold with an atlas (φ(u i ), θ i ) and φ : B S be a configuration of B into S. We can consider the phase space (φ t(x ), T x φ t(u i )) for X U i and x = φ t(x )

Lagrangian Formulation for an Elastic Space of Configurations Let us consider a body manifold B with an atlas (ψ i, U i ) where B R n and ψ i (U i ) R n ; regard this manifold as the undeformed state of any elastic medium. Let S be ambient manifold with an atlas (φ(u i ), θ i ) and φ : B S be a configuration of B into S. We can consider the phase space (φ t(x ), T x φ t(u i )) for X U i and x = φ t(x ) After a motion, say φ(x, t) = x + u(x, t), where u is the small displacement vector field on S, we can see that the strain tensor is given by ε ij (x) dx i dx j = 1 2 { φ ds(x) 2 ds(x) 2}, and after calculations on this expression we get ε ij = 1 2 (Lug) ij, where (L ug) ij = 1 ( gil 2 j u l + g lj i u l ), are the components of the Lie derivative of the metric with respect to the displacement vector field.

Lagrangian Formulation for an Elastic Space of Configurations RWE-C3-EAFIT

RWE-C3-EAFIT Lagrangian Formulation for an Elastic Space of Configurations Consider the existence of the following functions and vector fields e(x, t), internal energy functional b(x, t), external force vector field, t(x, t, ñ), traction for which exists a two-tensor σ such that t(x, y, ñ) = σ(x, t) ñ, where n is the normal outward to the manifold at every point. Since changes of the metric on the manifold S affect the accelerations of the particles, the internal energy must depend parametrically on the metric g, and if we have balance of energy d ρ (e + 12 ) dt < v, v > dv = ρ < b, v > dv + < t, ṽ > da, φ(u) φ(u) φ(u) it can be proved that σ ij = 2ρ e g.

RWE-C3-EAFIT Lagrangian Formulation for an Elastic Space of Configurations Consider the Lagrangian L : T S R given by L(x, v) = 1 < v, v > e(x, t, g), 2 and the Euler-Lagrange equations ( ) d L dt v µ = L x µ.

Lagrangian Formulation for an Elastic Space of Configurations RWE-C3-EAFIT

Lagrangian Formulation for an Elastic Space of Configurations then we have the system of equations ( e g µi a i = x µ + σ ij 2ρ g ) ij x µ. (2) If we consider small perturbations, on which Hook s law is valid, we can take the energy functional to be e(x, t, g) =< Cε, ε > which clearly depends on the metric; and σ = Cε then we rewrite equation the above equation as ( < Cε, ε > ρg µi a i = x µ + σ 2ρ g ) ij x µ, (3) after some manipulations and the use of Leibnitz s rule we have [ ρg µi a i = C ijkl x µ ε kl, ε ij + g ij 1 2 2 εkl x µ, g ij ]. (4) Since we are assuming the time invariance of the medium, the causality of the wave motion is going to be taken into account. Let ω be the time-fourier parameter for the field u and denote û(x, ω), εˆ ij the associated fields after Fourier transform on the time variable, then we have the equation ω 2 ρg µi (x)û + C ijkl x µ εˆ kl, εˆ ij + g ij 1 ˆ 2 2 C εkl ijkl x µ, g ij = 0 (5)

Lagrangian Formulation for an Elastic Space of Configurations RWE-C3-EAFIT

Lagrangian Formulation for an Elastic Space of Configurations Since we are interested in a particular direction of propagation, say a geodesic one, in geodesic coordinates suppose j = 3; we get the following system of equations ω 2 ρg µ3 û 3 + x µ ˆσ 33, ˆε 33 + g 33 2 1 2 ω 2 ρg µνû ν + = 0 x µ ˆσ 33, g 33 x µ ˆσ ν3, ˆε ν3 = 0. v = 1, 2

Further Works

Further Works To perform elastic one-way wave equations on a Riemannian manifold in local coordinates: Flux normalization and subprincipal symbol, self-adjoint form.

Further Works To perform elastic one-way wave equations on a Riemannian manifold in local coordinates: Flux normalization and subprincipal symbol, self-adjoint form. Tensor upward/downward continuation with a Riemannian metric. To consider the elastic wave equation for the metric, resulting from the Einstein-Hilbert action on the manifold S.

Some references 1. Yasutomi Y. 2007. Modified Elastic Wave Equations On Riemannian and Kahler Manifolds. Pulb. RIMS, 43,471-504. 2. Bale. R. 2006. Elastic wave equation depth migration of seismic data for Isotropic and Azimuthally Anisotropic media. Ph.D Thesis, University of Calgary. 3. Marsden. J, Hughes. T. 1983. Mathematical Foundations of Elasticity. Dover Publications,Inc. New York. 4. De Hoop. M, De Hoop. A. 1994. Elastic wave up/down decomposition in inhomogeneous and anisotropic media: an operator approach and its approximations. Wave Motion, 20, 57-82.