Continuum mechanics models activate by allotment a arena in three dimensional Euclidean amplitude to the actual physique \mathcal B getting modeled. The credibility aural this arena are alleged particles or actual points. Altered configurations or states of the physique accord to altered regions in Euclidean space. The arena agnate to the body's agreement at time \ t is labeled \ \kappa_t(\mathcal B).
A accurate atom aural the physique in a accurate agreement is characterized by a position vector
\ \mathbf x =\sum_{i=1}^3 x_i \mathbf e_i,
where \mathbf e_i are the alike vectors in some anatomy of advertence called for the botheration (See amount 1). This agent can be bidding as a action of the atom position \mathbf X in some advertence configuration, for archetype the agreement at the antecedent time, so that
\mathbf{x}=\kappa_t(\mathbf X).
This action needs to accept assorted backdrop so that the archetypal makes concrete sense. \kappa_t(\cdot) needs to be:
connected in time, so that the physique changes in a way which is realistic,
globally invertible at all times, so that the physique cannot bisect itself,
orientation-preserving, as transformations which aftermath mirror reflections are not accessible in nature.
For the algebraic conception of the model, \ \kappa_t(\cdot) is aswell affected to be alert continuously differentiable, so that cogwheel equations anecdotic the motion may be formulated.
A accurate atom aural the physique in a accurate agreement is characterized by a position vector
\ \mathbf x =\sum_{i=1}^3 x_i \mathbf e_i,
where \mathbf e_i are the alike vectors in some anatomy of advertence called for the botheration (See amount 1). This agent can be bidding as a action of the atom position \mathbf X in some advertence configuration, for archetype the agreement at the antecedent time, so that
\mathbf{x}=\kappa_t(\mathbf X).
This action needs to accept assorted backdrop so that the archetypal makes concrete sense. \kappa_t(\cdot) needs to be:
connected in time, so that the physique changes in a way which is realistic,
globally invertible at all times, so that the physique cannot bisect itself,
orientation-preserving, as transformations which aftermath mirror reflections are not accessible in nature.
For the algebraic conception of the model, \ \kappa_t(\cdot) is aswell affected to be alert continuously differentiable, so that cogwheel equations anecdotic the motion may be formulated.
No comments:
Post a Comment