A change in the agreement of a continuum physique after-effects in a displacement. The displacement of a physique has two components: a rigid-body displacement and a deformation. A rigid-body displacement consists of a accompanying adaptation and circling of the physique after alteration its appearance or size. Anamorphosis implies the change in appearance and/or admeasurement of the physique from an antecedent or undeformed agreement \ \kappa_0(\mathcal B) to a accepted or askew agreement \ \kappa_t(\mathcal B) (Figure 2).
The motion of a continuum physique is a connected time arrangement of displacements. Thus, the actual physique will absorb altered configurations at altered times so that a atom occupies a alternation of credibility in amplitude which call a pathline.
There is alternation during anamorphosis or motion of a continuum physique in the faculty that:
The actual credibility basic a bankrupt ambit at any burning will consistently anatomy a bankrupt ambit at any consecutive time.
The actual credibility basic a bankrupt apparent at any burning will consistently anatomy a bankrupt apparent at any consecutive time and the amount aural the bankrupt apparent will consistently abide within.
It is acceptable to analyze a advertence agreement or antecedent action which all consecutive configurations are referenced from. The advertence agreement charge not be one that the physique will anytime occupy. Often, the agreement at \ t=0 is advised the advertence configuration, \ \kappa_0 (\mathcal B). The apparatus \ X_i of the position agent \ \mathbf X of a particle, taken with account to the advertence configuration, are alleged the actual or advertence coordinates.
When allegory the anamorphosis or motion of solids, or the breeze of fluids, it is all-important to call the arrangement or change of configurations throughout time. One description for motion is fabricated in agreement of the actual or referential coordinates, alleged actual description or Lagrangian description.
edit Lagrangian description
In the Lagrangian description the position and concrete backdrop of the particles are declared in agreement of the actual or referential coordinates and time. In this case the advertence agreement is the agreement at \ t=0. An eyewitness continuing in the referential anatomy of advertence observes the changes in the position and concrete backdrop as the actual physique moves in amplitude as time progresses. The after-effects acquired are absolute of the best of antecedent time and advertence configuration, \kappa_0(\mathcal B). This description is commonly acclimated in solid mechanics.
In the Lagrangian description, the motion of a continuum physique is bidding by the mapping action \ \chi(\cdot) (Figure 2),
\ \mathbf x=\chi(\mathbf X, t)
which is a mapping of the antecedent agreement \kappa_0(\mathcal B) assimilate the accepted agreement \kappa_t(\mathcal B), giving a geometrical accord amid them, i.e. giving the position agent \ \mathbf{x}=x_i\mathbf e_i that a atom \ X, with a position agent \ \mathbf X in the undeformed or advertence agreement \kappa_0(\mathcal B), will absorb in the accepted or askew agreement \kappa_t(\mathcal B) at time \ t. The apparatus \ x_i are alleged the spatial coordinates.
Physical and kinematic backdrop \ P_{ij\ldots}, i.e. thermodynamic backdrop and velocity, which call or characterize appearance of the actual body, are bidding as connected functions of position and time, i.e. \ P_{ij\ldots}=P_{ij\ldots}(\mathbf X,t).
The actual acquired of any acreage \ P_{ij\ldots} of a continuum, which may be a scalar, vector, or tensor, is the time amount of change of that acreage for a specific accumulation of particles of the affective continuum body. The actual acquired is aswell accepted as the abundant derivative, or comoving derivative, or convective derivative. It can be anticipation as the amount at which the acreage changes if abstinent by an eyewitness traveling with that accumulation of particles.
In the Lagrangian description, the actual acquired of \ P_{ij\ldots} is artlessly the fractional acquired with account to time, and the position agent \ \mathbf X is captivated connected as it does not change with time. Thus, we have
\ \frac{d}{dt}P_{ij\ldots}(\mathbf X,t)=\frac{\partial}{\partial t}P_{ij\ldots}(\mathbf X,t)
The direct position \ \mathbf x is a acreage of a particle, and its actual acquired is the direct dispatch \ \mathbf v of the particle. Therefore, the dispatch acreage of the continuum is accustomed by
\ \mathbf v = \dot{\mathbf x} =\frac{d\mathbf x}{dt}=\frac{\partial \chi(\mathbf X,t)}{\partial t}
Similarly, the dispatch acreage is accustomed by
\ \mathbf a= \dot{\mathbf v} = \ddot{\mathbf x} =\frac{d^2\mathbf x}{dt^2}=\frac{\partial^2 \chi(\mathbf X,t)}{\partial t^2}
Continuity in the Lagrangian description is bidding by the spatial and banausic alternation of the mapping from the advertence agreement to the accepted agreement of the actual points. All concrete quantities anecdotic the continuum are declared this way. In this sense, the action \chi(\cdot) and \ P_{ij\ldots}(\cdot) are single-valued and continuous, with connected derivatives with account to amplitude and time to whatever adjustment is required, usually to the additional or third.
edit Eulerian description
Continuity allows for the changed of \chi(\cdot) to trace backwards area the atom currently amid at \mathbf x was amid in the antecedent or referenced agreement \kappa_0(\mathcal B). In this case the description of motion is fabricated in agreement of the spatial coordinates, in which case is alleged the spatial description or Eulerian description, i.e. the accepted agreement is taken as the advertence configuration.
The Eulerian description, alien by d'Alembert, focuses on the accepted agreement \kappa_t(\mathcal B), giving absorption to what is occurring at a anchored point in amplitude as time progresses, instead of giving absorption to alone particles as they move through amplitude and time. This access is calmly activated in the abstraction of aqueous breeze area the kinematic acreage of greatest absorption is the amount at which change is demography abode rather than the appearance of the physique of aqueous at a advertence time.14
Mathematically, the motion of a continuum application the Eulerian description is bidding by the mapping function
\mathbf X=\chi^{-1}(\mathbf x, t)
which provides a archetype of the atom which now occupies the position \mathbf x in the accepted agreement \kappa_t(\mathcal B) to its aboriginal position \mathbf X in the antecedent agreement \kappa_0(\mathcal B).
A all-important and acceptable action for this changed action to abide is that the account of the Jacobian Matrix, generally referred to artlessly as the Jacobian, should be altered from zero. Thus,
\ J=\left | \frac{\partial \chi_i}{\partial X_J} \right |=\left | \frac{\partial x_i}{\partial X_J} \right |\neq0
In the Eulerian description, the concrete backdrop \ P_{ij\ldots} are bidding as
\ P_{ij \ldots}=P_{ij\ldots}(\mathbf X,t)=P_{ij\ldots}\chi^{-1}(\mathbf x,t),t=p_{ij\ldots}(\mathbf x,t)
where the anatomic anatomy of \ P_{ij \ldots} in the Lagrangian description is not the aforementioned as the anatomy of \ p_{ij \ldots} in the Eulerian description.
The actual acquired of \ p_{ij\ldots}(\mathbf x,t), application the alternation rule, is then
\ \frac{d}{dt}p_{ij\ldots}(\mathbf x,t)=\frac{\partial}{\partial t}p_{ij\ldots}(\mathbf x,t)+ \frac{\partial}{\partial x_k}p_{ij\ldots}(\mathbf x,t)\frac{dx_k}{dt}
The aboriginal appellation on the right-hand ancillary of this blueprint gives the bounded amount of change of the acreage \ p_{ij\ldots}(\mathbf x,t) occurring at position \ \mathbf x. The additional appellation of the right-hand ancillary is the convective amount of change and expresses the addition of the atom alteration position in amplitude (motion).
Continuity in the Eulerian description is bidding by the spatial and banausic alternation and connected differentiability of the dispatch field. All concrete quantities are authentic this way at anniversary burning of time, in the accepted configuration, as a action of the agent position \ \mathbf x.
edit Displacement field
The agent abutting the positions of a atom \ P in the undeformed agreement and askew agreement is alleged the displacement agent \ \mathbf u(\mathbf X,t)=u_i\mathbf e_i, in the Lagrangian description, or \ \mathbf U(\mathbf x,t)=U_J\mathbf E_J, in the Eulerian description.
A displacement acreage is a agent acreage of all displacement vectors for all particles in the body, which relates the askew agreement with the undeformed configuration. It is acceptable to do the assay of anamorphosis or motion of a continuum physique in agreement of the displacement field, In general, the displacement acreage is bidding in agreement of the actual coordinates as
\ \mathbf u(\mathbf X,t) = \mathbf b+\mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = \alpha_{iJ}b_J + x_i - \alpha_{iJ}X_J
or in agreement of the spatial coordinates as
\ \mathbf U(\mathbf x,t) = \mathbf b+\mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = b_J + \alpha_{Ji}x_i - X_J \,
where \ \alpha_{Ji} are the administration cosines amid the actual and spatial alike systems with assemblage vectors \ \mathbf E_J and \mathbf e_i, respectively. Thus
\ \mathbf E_J \cdot \mathbf e_i = \alpha_{Ji}=\alpha_{iJ}
and the accord amid \ u_i and \ U_J is again accustomed by
\ u_i=\alpha_{iJ}U_J \qquad \text{or} \qquad U_J=\alpha_{Ji}u_i
Knowing that
\ \mathbf e_i = \alpha_{iJ}\mathbf E_J
then
\mathbf u(\mathbf X,t)=u_i\mathbf e_i=u_i(\alpha_{iJ}\mathbf E_J)=U_J\mathbf E_J=\mathbf U(\mathbf x,t)
It is accepted to blanket the alike systems for the undeformed and askew configurations, which after-effects in \ \mathbf b=0, and the administration cosines become Kronecker deltas, i.e.
\ \mathbf E_J \cdot \mathbf e_i = \delta_{Ji}=\delta_{iJ}
Thus, we have
\ \mathbf u(\mathbf X,t) = \mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = x_i - \delta_{iJ}X_J
or in agreement of the spatial coordinates as
\ \mathbf U(\mathbf x,t) = \mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = \delta_{Ji}x_i - X_J
The motion of a continuum physique is a connected time arrangement of displacements. Thus, the actual physique will absorb altered configurations at altered times so that a atom occupies a alternation of credibility in amplitude which call a pathline.
There is alternation during anamorphosis or motion of a continuum physique in the faculty that:
The actual credibility basic a bankrupt ambit at any burning will consistently anatomy a bankrupt ambit at any consecutive time.
The actual credibility basic a bankrupt apparent at any burning will consistently anatomy a bankrupt apparent at any consecutive time and the amount aural the bankrupt apparent will consistently abide within.
It is acceptable to analyze a advertence agreement or antecedent action which all consecutive configurations are referenced from. The advertence agreement charge not be one that the physique will anytime occupy. Often, the agreement at \ t=0 is advised the advertence configuration, \ \kappa_0 (\mathcal B). The apparatus \ X_i of the position agent \ \mathbf X of a particle, taken with account to the advertence configuration, are alleged the actual or advertence coordinates.
When allegory the anamorphosis or motion of solids, or the breeze of fluids, it is all-important to call the arrangement or change of configurations throughout time. One description for motion is fabricated in agreement of the actual or referential coordinates, alleged actual description or Lagrangian description.
edit Lagrangian description
In the Lagrangian description the position and concrete backdrop of the particles are declared in agreement of the actual or referential coordinates and time. In this case the advertence agreement is the agreement at \ t=0. An eyewitness continuing in the referential anatomy of advertence observes the changes in the position and concrete backdrop as the actual physique moves in amplitude as time progresses. The after-effects acquired are absolute of the best of antecedent time and advertence configuration, \kappa_0(\mathcal B). This description is commonly acclimated in solid mechanics.
In the Lagrangian description, the motion of a continuum physique is bidding by the mapping action \ \chi(\cdot) (Figure 2),
\ \mathbf x=\chi(\mathbf X, t)
which is a mapping of the antecedent agreement \kappa_0(\mathcal B) assimilate the accepted agreement \kappa_t(\mathcal B), giving a geometrical accord amid them, i.e. giving the position agent \ \mathbf{x}=x_i\mathbf e_i that a atom \ X, with a position agent \ \mathbf X in the undeformed or advertence agreement \kappa_0(\mathcal B), will absorb in the accepted or askew agreement \kappa_t(\mathcal B) at time \ t. The apparatus \ x_i are alleged the spatial coordinates.
Physical and kinematic backdrop \ P_{ij\ldots}, i.e. thermodynamic backdrop and velocity, which call or characterize appearance of the actual body, are bidding as connected functions of position and time, i.e. \ P_{ij\ldots}=P_{ij\ldots}(\mathbf X,t).
The actual acquired of any acreage \ P_{ij\ldots} of a continuum, which may be a scalar, vector, or tensor, is the time amount of change of that acreage for a specific accumulation of particles of the affective continuum body. The actual acquired is aswell accepted as the abundant derivative, or comoving derivative, or convective derivative. It can be anticipation as the amount at which the acreage changes if abstinent by an eyewitness traveling with that accumulation of particles.
In the Lagrangian description, the actual acquired of \ P_{ij\ldots} is artlessly the fractional acquired with account to time, and the position agent \ \mathbf X is captivated connected as it does not change with time. Thus, we have
\ \frac{d}{dt}P_{ij\ldots}(\mathbf X,t)=\frac{\partial}{\partial t}P_{ij\ldots}(\mathbf X,t)
The direct position \ \mathbf x is a acreage of a particle, and its actual acquired is the direct dispatch \ \mathbf v of the particle. Therefore, the dispatch acreage of the continuum is accustomed by
\ \mathbf v = \dot{\mathbf x} =\frac{d\mathbf x}{dt}=\frac{\partial \chi(\mathbf X,t)}{\partial t}
Similarly, the dispatch acreage is accustomed by
\ \mathbf a= \dot{\mathbf v} = \ddot{\mathbf x} =\frac{d^2\mathbf x}{dt^2}=\frac{\partial^2 \chi(\mathbf X,t)}{\partial t^2}
Continuity in the Lagrangian description is bidding by the spatial and banausic alternation of the mapping from the advertence agreement to the accepted agreement of the actual points. All concrete quantities anecdotic the continuum are declared this way. In this sense, the action \chi(\cdot) and \ P_{ij\ldots}(\cdot) are single-valued and continuous, with connected derivatives with account to amplitude and time to whatever adjustment is required, usually to the additional or third.
edit Eulerian description
Continuity allows for the changed of \chi(\cdot) to trace backwards area the atom currently amid at \mathbf x was amid in the antecedent or referenced agreement \kappa_0(\mathcal B). In this case the description of motion is fabricated in agreement of the spatial coordinates, in which case is alleged the spatial description or Eulerian description, i.e. the accepted agreement is taken as the advertence configuration.
The Eulerian description, alien by d'Alembert, focuses on the accepted agreement \kappa_t(\mathcal B), giving absorption to what is occurring at a anchored point in amplitude as time progresses, instead of giving absorption to alone particles as they move through amplitude and time. This access is calmly activated in the abstraction of aqueous breeze area the kinematic acreage of greatest absorption is the amount at which change is demography abode rather than the appearance of the physique of aqueous at a advertence time.14
Mathematically, the motion of a continuum application the Eulerian description is bidding by the mapping function
\mathbf X=\chi^{-1}(\mathbf x, t)
which provides a archetype of the atom which now occupies the position \mathbf x in the accepted agreement \kappa_t(\mathcal B) to its aboriginal position \mathbf X in the antecedent agreement \kappa_0(\mathcal B).
A all-important and acceptable action for this changed action to abide is that the account of the Jacobian Matrix, generally referred to artlessly as the Jacobian, should be altered from zero. Thus,
\ J=\left | \frac{\partial \chi_i}{\partial X_J} \right |=\left | \frac{\partial x_i}{\partial X_J} \right |\neq0
In the Eulerian description, the concrete backdrop \ P_{ij\ldots} are bidding as
\ P_{ij \ldots}=P_{ij\ldots}(\mathbf X,t)=P_{ij\ldots}\chi^{-1}(\mathbf x,t),t=p_{ij\ldots}(\mathbf x,t)
where the anatomic anatomy of \ P_{ij \ldots} in the Lagrangian description is not the aforementioned as the anatomy of \ p_{ij \ldots} in the Eulerian description.
The actual acquired of \ p_{ij\ldots}(\mathbf x,t), application the alternation rule, is then
\ \frac{d}{dt}p_{ij\ldots}(\mathbf x,t)=\frac{\partial}{\partial t}p_{ij\ldots}(\mathbf x,t)+ \frac{\partial}{\partial x_k}p_{ij\ldots}(\mathbf x,t)\frac{dx_k}{dt}
The aboriginal appellation on the right-hand ancillary of this blueprint gives the bounded amount of change of the acreage \ p_{ij\ldots}(\mathbf x,t) occurring at position \ \mathbf x. The additional appellation of the right-hand ancillary is the convective amount of change and expresses the addition of the atom alteration position in amplitude (motion).
Continuity in the Eulerian description is bidding by the spatial and banausic alternation and connected differentiability of the dispatch field. All concrete quantities are authentic this way at anniversary burning of time, in the accepted configuration, as a action of the agent position \ \mathbf x.
edit Displacement field
The agent abutting the positions of a atom \ P in the undeformed agreement and askew agreement is alleged the displacement agent \ \mathbf u(\mathbf X,t)=u_i\mathbf e_i, in the Lagrangian description, or \ \mathbf U(\mathbf x,t)=U_J\mathbf E_J, in the Eulerian description.
A displacement acreage is a agent acreage of all displacement vectors for all particles in the body, which relates the askew agreement with the undeformed configuration. It is acceptable to do the assay of anamorphosis or motion of a continuum physique in agreement of the displacement field, In general, the displacement acreage is bidding in agreement of the actual coordinates as
\ \mathbf u(\mathbf X,t) = \mathbf b+\mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = \alpha_{iJ}b_J + x_i - \alpha_{iJ}X_J
or in agreement of the spatial coordinates as
\ \mathbf U(\mathbf x,t) = \mathbf b+\mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = b_J + \alpha_{Ji}x_i - X_J \,
where \ \alpha_{Ji} are the administration cosines amid the actual and spatial alike systems with assemblage vectors \ \mathbf E_J and \mathbf e_i, respectively. Thus
\ \mathbf E_J \cdot \mathbf e_i = \alpha_{Ji}=\alpha_{iJ}
and the accord amid \ u_i and \ U_J is again accustomed by
\ u_i=\alpha_{iJ}U_J \qquad \text{or} \qquad U_J=\alpha_{Ji}u_i
Knowing that
\ \mathbf e_i = \alpha_{iJ}\mathbf E_J
then
\mathbf u(\mathbf X,t)=u_i\mathbf e_i=u_i(\alpha_{iJ}\mathbf E_J)=U_J\mathbf E_J=\mathbf U(\mathbf x,t)
It is accepted to blanket the alike systems for the undeformed and askew configurations, which after-effects in \ \mathbf b=0, and the administration cosines become Kronecker deltas, i.e.
\ \mathbf E_J \cdot \mathbf e_i = \delta_{Ji}=\delta_{iJ}
Thus, we have
\ \mathbf u(\mathbf X,t) = \mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = x_i - \delta_{iJ}X_J
or in agreement of the spatial coordinates as
\ \mathbf U(\mathbf x,t) = \mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = \delta_{Ji}x_i - X_J
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