Euclidean transformation

The Euclidean transformation , named after Euclid , is a distance- and thus also angle-preserving transformation of Euclidean space on itself. During the actual Euclidean transformation, the orientation is also retained, i.e. reflections are excluded.

In classical mechanics , the actual Euclidean transformation represents an observer transformation and means a translation and rotation of the observer's reference system . Only the actual transformations are considered, because reflections of material bodies do not occur in mechanics, which only considers physical laws in the macroscopic world determined by gravity . The Euclidean observer transformation can be seen as a rigid body movementImagine a reference system in which the origin and the attached coordinate axes move freely, but the coordinate axes maintain the relative orientation and angle to each other and are not stretched or compressed. The Galileo transformation in Euclidean space is contained as a special case of linear, uniform motion with constant relative speed.

In mechanics, the Euclidean transformation is used to define objective or invariant quantities that are perceived by observers in differently moving reference systems in the same way, see Changing the reference system . Objective variables that describe the state of a material body are of central importance in material theory, because it does not correspond to the experience that a moving observer measures a different material behavior than a stationary one. This law is called material objectivity .

overview

The representation takes place in three dimensions, but can easily be generalized to n dimensions. The Euclidean transformation is a distance-maintaining transformation of a Euclidean space on itself. Since it contains a distance function, the Euclidean space is a metric space and the Euclidean transformation is an isometry . With regard to the sequential execution, the Euclidean transformations form a group .

Depending on which type of Euclidean space is used, there are different formulations:

1. The transformation of the Euclidean point space is a movement (mathematics) , which is also actually called when the orientation is maintained, and improper when this is not the case. ${\displaystyle \mathbb {E} ^{3}}$
2. The Euclidean transformation of the coordinate space with the standard scalar product over the body is a special coordinate transformation in which the transformation matrix is an orthogonal matrix (see description in coordinates ). ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} }$
3. The transformation of the Euclidean vector space (a vector space defined by a scalar product ) is shown here in a physical context. ${\displaystyle \mathbb {V} ^{3}}$ ${\displaystyle \mathbb {R} }$

Euclidean transformation and change of observer

Displacements and vectors

The Euclidean transformation is based on the Euclidean point space of our intuition and assigned a Euclidean vector space , see From Euclidean intuition space to Euclidean vector space . In summary, every Euclidean vector space has an equal-length mapping ${\displaystyle \mathbb {E} ^{3}}$ ${\displaystyle \mathbb {V} ^{3}}$

${\displaystyle {\begin{array}{llll}{\mathcal {V}}:&\mathbb {E} ^{3}\times \mathbb {E} ^{3}&\mapsto &\mathbb {V} ^{3}\\&(R,S)&\mapsto &{\vec {v}}\end{array}}}$

which assigns a vector of the same length to all parallel, same-directional and equally long shifts from points to points , see picture. The importance of this becomes clear in the active interpretation of the change of observer. ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle {\vec {v}}}$

Passive interpretation or coordinate transformation

Illustration of the coordinate transformation

A base of the is designed by selecting four non-coplanar points , of which O is the origin . For the sake of simplicity, the vectors should be perpendicular to one another in pairs and each have a length of one, so that the basis vectors represent an orthonormal basis . Coordinates can now be assigned to each point : ${\displaystyle O,P_{1},P_{2},P_{3}\in \mathbb {E} ^{3}}$ ${\displaystyle {\mathcal {V}}(O,P_{i})={\vec {e}}_{i}}$ ${\displaystyle \mathbb {V} ^{3}}$ ${\displaystyle {\mathcal {V}}(O,P_{i})={\vec {e}}_{i}}$ ${\displaystyle {\vec {e}}_{i}}$ ${\displaystyle X\in \mathbb {E} ^{3}}$ ${\displaystyle {x}_{i}\in \mathbb {R} }$

${\displaystyle {\mathcal {V}}(O,X)=:{\vec {x}}=\sum _{i=1}^{3}x_{i}{\vec {e}}_{i}}$ with the components ${\displaystyle x_{i}:={\vec {e}}_{i}\cdot {\vec {x}}\,.}$

A change of the reference system is now done by selecting four other points , which leads to the new orthonormal basis , see picture. The point has different coordinates in the new reference system  : ${\displaystyle O',P'_{1},P'_{2},P'_{3}\in \mathbb {E} ^{3}}$ ${\displaystyle {\vec {e}}'_{i}}$ ${\displaystyle X}$ ${\displaystyle x'_{i}}$

${\displaystyle {\mathcal {V}}(O',X)=:{\vec {x}}'=\sum _{i=1}^{3}x'_{i}{\vec {e}}'_{i}\,.}$

With

${\displaystyle {\mathcal {V}}(O',O)=:{\vec {s}}=\sum _{i=1}^{3}s_{i}{\vec {e}}'_{i}}$

the vector equation results

${\displaystyle {\begin{aligned}{\mathcal {V}}(O',X)=&{\mathcal {V}}(O',O)+{\mathcal {V}}(O,X)\\\Leftrightarrow \quad \sum _{i=1}^{3}x'_{i}{\vec {e}}'_{i}=&\sum _{i=1}^{3}s_{i}{\vec {e}}'_{i}+\sum _{j=1}^{3}x_{j}{\vec {e}}_{j}=\sum _{i=1}^{3}s_{i}{\vec {e}}'_{i}+\sum _{i,j=1}^{3}x_{j}({\vec {e}}'_{i}\cdot {\vec {e}}_{j}){\vec {e}}'_{i}\,.\end{aligned}}}$

The coordinate transformation describes the relationship between the coordinates

${\displaystyle x'_{i}=s_{i}+\sum _{j=1}^{3}Q_{ij}x_{j}\Leftrightarrow {\begin{pmatrix}x'_{1}\\x'_{2}\\x'_{3}\end{pmatrix}}={\begin{pmatrix}s_{1}\\s_{2}\\s_{3}\end{pmatrix}}+{\begin{pmatrix}Q_{11}&Q_{12}&Q_{13}\\Q_{21}&Q_{22}&Q_{23}\\Q_{31}&Q_{32}&Q_{33}\end{pmatrix}}\cdot {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\,.}$

This is a matrix equation as described in Coordinates or Isometrics . The 3 × 3 matrix Q with the components is an orthogonal matrix and therefore applies ${\displaystyle Q_{ij}={\vec {e}}'_{i}\cdot {\vec {e}}_{j}}$

${\displaystyle Q\cdot Q^{T}=E,}$

where E is the identity matrix. If det ( Q ) = +1 applies to the determinant , the bases and have the same orientation and an actual Euclidean transformation is present. ${\displaystyle {\vec {e}}_{1,2,3}}$ ${\displaystyle {\vec {e}}'_{1,2,3}}$

Active interpretation or coordinate-free mapping

Illustration of the active interpretation of the Euclidean transformation

An observer will usually choose both a different origin and a different assignment of the Euclidean point space to a vector space than another observer. The images of and can at most be twisted because the figure should be true to length and a shift is not significant here, since the same vector is assigned to all parallel, same-directional and equal-length shifts from point to point : ${\displaystyle O'}$ ${\displaystyle {\mathcal {V}}':\mathbb {E} ^{3}\times \mathbb {E} ^{3}\mapsto \mathbb {V} ^{3}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {V}}'}$ ${\displaystyle R}$ ${\displaystyle S}$

${\displaystyle {\mathcal {V}}'(R,S)=\mathbf {Q} \cdot {\mathcal {V}}(R,S)\quad \forall \;R,S\in \mathbb {E} ^{3}}$

where Q is an orthogonal tensor ( Q ^T · Q = 1 with unit tensor 1 ), see picture. Now is

${\displaystyle {\begin{aligned}{\vec {x}}'=&{\mathcal {V}}'(O',X)={\mathcal {V}}'(O',O)+{\mathcal {V}}'(O,X)=\underbrace {{\mathcal {V}}'(O',O)} _{\vec {s}}+\mathbf {Q} \cdot \underbrace {{\mathcal {V}}(O,X)} _{\vec {x}}\\\rightarrow {\vec {x}}'=&{\vec {s}}+\mathbf {Q} \cdot {\vec {x}}\,.\end{aligned}}}$

This vector equation is coordinate-free, i.e. it does not refer to any coordinate or base system. In contrast to the passive interpretation, the ability of tensors (here Q ) is used here to map vectors from one vector space, the image space of , to another, the image space of , whereby in this case the two image spaces are identical. If det ( Q ) = +1 there is again an actual Euclidean transformation. Because every tensor is a linear map, this procedure corresponds to the description with the help of linear maps . ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {V}}'}$

The apparent contradiction is resolved when the various basic systems to the left and right of the equal sign are taken into account: ${\displaystyle \mathbf {Q} \cdot {\vec {x}}={\vec {x}}}$

${\displaystyle \mathbf {Q} \cdot \sum _{i=1}^{3}x_{i}{\vec {e}}'_{i}=\sum _{i=1}^{3}x_{i}{\vec {e}}_{i}\quad \Leftrightarrow \quad \mathbf {Q} \cdot {\vec {e}}'_{i}={\vec {e}}_{i}}$

where Q is the component representation

${\displaystyle \mathbf {Q} =\sum _{i=1}^{3}{\vec {e}}_{i}\otimes {\vec {e}}'_{i}=\sum _{i,j=1}^{3}({\vec {e}}'_{i}\cdot {\vec {e}}_{j}){\vec {e}}_{i}\otimes {\vec {e}}_{j}=\sum _{i,j=1}^{3}Q_{ij}{\vec {e}}_{i}\otimes {\vec {e}}_{j}}$

with Q _ij as in the passive interpretation, which underlines the equivalence of the passive and active interpretations .

Transformation of time

A change of observer also includes a change in the time scale. In the Euclidean transformation only a constant displacement Δ t is provided:

${\displaystyle t'=t+\Delta t}$

What is to be understood in such a way that the observers read different values ​​on their clocks at the same time, but the difference between the values ​​is always the same. So the observers started their stopwatches at different times.

General change of observer

In classical mechanics, a change of observer can be described as follows:

${\displaystyle {\begin{aligned}{\vec {x}}'(t')=&{\vec {s}}(t)+\mathbf {Q} (t)\cdot {\vec {x}}(t)\\t'=&t+\Delta t\\\mathbf {Q} (t)\cdot \mathbf {Q} ^{\top }(t)=&\mathbf {Q} ^{\top }(t)\cdot \mathbf {Q} (t)=\mathbf {1} ,\quad \mathrm {det} (\mathbf {Q} )=+1\,,\end{aligned}}}$

because rotational reflections (with det ( Q ) = -1) do not occur in the macroscopic world dominated by gravity and electromagnetism.

The Galileo transformation is the special case that only includes the so-called Galileo boosts :

${\displaystyle {\ddot {\vec {s}}}(t)={\vec {0}}\Leftrightarrow {\vec {s}}(t)={\vec {s}}_{0}+{\vec {v}}_{0}t}$and Q (t) = Q ₀ = const.

where a constant displacement, a constant velocity and Q ₀ denote a constant twist. These leave the amount and relative angle of accelerations unchanged. ${\displaystyle {\vec {s}}_{0}}$ ${\displaystyle {\vec {v}}_{0}}$

Objective or invariant quantities

Quantities that are perceived unchanged when the frame of reference changes are called objective or invariant. As an introduction, consider the distance between two points: With every Euclidean transformation, the distance between any two points always remains constant and this also applies to a general change of observer. Be and the position vectors of two points P and Q . The square of their distance ${\displaystyle {\vec {p}}={\mathcal {V}}(O,P)}$ ${\displaystyle {\vec {q}}={\mathcal {V}}(O,Q)}$

${\displaystyle {\begin{aligned}d'^{2}=&({\vec {p}}'-{\vec {q}}')\cdot ({\vec {p}}'-{\vec {q}}')=({\vec {s}}+\mathbf {Q} \cdot {\vec {p}}-{\vec {s}}-\mathbf {Q} \cdot {\vec {q}})\cdot ({\vec {s}}+\mathbf {Q} \cdot {\vec {p}}-{\vec {s}}-\mathbf {Q} \cdot {\vec {q}})\\=&[\mathbf {Q} \cdot ({\vec {p}}-{\vec {q}})]\cdot [\mathbf {Q} \cdot ({\vec {p}}-{\vec {q}})]=({\vec {p}}-{\vec {q}})\cdot \mathbf {Q^{\top }\cdot Q} \cdot ({\vec {p}}-{\vec {q}})\\=&({\vec {p}}-{\vec {q}})\cdot ({\vec {p}}-{\vec {q}})={d}^{2}\end{aligned}}}$

thus remains unchanged if Q ^T · Q = 1 and the distance vector according to ${\displaystyle {\vec {p}}-{\vec {q}}}$

${\displaystyle ({\vec {p}}-{\vec {q}})'=\mathbf {Q} \cdot ({\vec {p}}-{\vec {q}})}$

transformed. The latter characterizes objective vectors. The transformation property for objective tensors is derived from the requirement that an objective tensor maps objective vectors onto objective vectors. With objective vectors and should also be objective. From , and results: ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {w}}}$ ${\displaystyle {\vec {w}}\cdot \mathbf {T} \cdot {\vec {v}}}$ ${\displaystyle {\vec {v}}'=\mathbf {Q} \cdot {\vec {v}}}$ ${\displaystyle {\vec {w}}'=\mathbf {Q} \cdot {\vec {w}}}$ ${\displaystyle {\vec {w}}'\cdot \mathbf {T} '\cdot {\vec {v}}'={\vec {w}}\cdot \mathbf {T} \cdot {\vec {v}}}$

${\displaystyle {\vec {w}}\cdot \mathbf {T} \cdot {\vec {v}}=(\mathbf {Q} ^{\top }\cdot {\vec {w}}')\cdot \mathbf {T\cdot Q} ^{\top }\cdot {\vec {v}}'={\vec {w}}'\cdot \mathbf {Q\cdot T\cdot Q} ^{\top }\cdot {\vec {v}}'={\vec {w}}'\cdot \mathbf {T} '\cdot {\vec {v}}'\,.}$

If this is to apply to all objective vectors and , the tensor must be according to ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {w}}}$

${\displaystyle \mathbf {T} '=\mathbf {Q\cdot T\cdot Q} ^{\top }}$

transform.

Transformation properties of objective quantities

A quantity is objective if it transforms itself as follows when there is a change of observer:

Type	condition
Scalar ${\displaystyle \phi }$	${\displaystyle \phi '(t')=\phi (t)}$
vector ${\displaystyle {\vec {v}}}$	${\displaystyle {\vec {v}}'(t')=\mathbf {Q} (t)\cdot {\vec {v}}(t)}$
Tensor ${\displaystyle \mathbf {T} }$	${\displaystyle \mathbf {T} '(t')=\mathbf {Q} (t)\cdot \mathbf {T} (t)\cdot \mathbf {Q} ^{\top }(t)}$

Speeds and acceleration

The speed is because of

${\displaystyle {\vec {v}}'={\frac {\mathrm {d} }{\mathrm {d} t'}}({\vec {s}}+\mathbf {Q} \cdot {\vec {x}})={\frac {\mathrm {d} }{\mathrm {d} t}}({\vec {s}}+\mathbf {Q} \cdot {\vec {x}})={\dot {\vec {s}}}+{\dot {\mathbf {Q} }}\cdot {\vec {x}}+\mathbf {Q} \cdot {\vec {v}}\neq \mathbf {Q} \cdot {\vec {v}}}$

no objective quantity and the same applies to the acceleration:

${\displaystyle {\begin{aligned}{\vec {a}}'=&{\frac {\mathrm {d} {\vec {v}}'}{\mathrm {d} t'}}={\frac {\mathrm {d} }{\mathrm {d} t}}({\dot {\vec {s}}}+{\dot {\mathbf {Q} }}\cdot {\vec {x}}+\mathbf {Q} \cdot {\vec {v}})={\ddot {\vec {s}}}+{\ddot {\mathbf {Q} }}\cdot {\vec {x}}+{\dot {\mathbf {Q} }}\cdot {\vec {v}}+{\dot {\mathbf {Q} }}\cdot {\vec {v}}+\mathbf {Q} \cdot {\vec {a}}\\=&{\ddot {\vec {s}}}+{\ddot {\mathbf {Q} }}\cdot {\vec {x}}+2{\dot {\mathbf {Q} }}\cdot {\vec {v}}+\mathbf {Q} \cdot {\vec {a}}\neq \mathbf {Q} \cdot {\vec {a}}\,.\end{aligned}}}$

Only in the special case of the Galileo transformation is the acceleration due to and objective. However, it can be shown that the absolute speed and the absolute acceleration are objective. ${\displaystyle {\ddot {\vec {s}}}={\vec {0}}}$ ${\displaystyle {\ddot {\mathbf {Q} }}={\dot {\mathbf {Q} }}=\mathbf {0} }$

The time derivative of an objective vector is due to ${\displaystyle {\vec {w}}}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t'}}{\vec {w}}'={\frac {\mathrm {d} }{\mathrm {d} t}}(\mathbf {Q} \cdot {\vec {w}})={\dot {\mathbf {Q} }}\cdot {\vec {w}}+\mathbf {Q} \cdot {\dot {\vec {w}}}\neq \mathbf {Q} \cdot {\dot {\vec {w}}}}$

mostly not objective and the same applies to the time derivative of an objective tensor  : ${\displaystyle \mathbf {T} }$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t'}}\mathbf {T} '={\frac {\mathrm {d} }{\mathrm {d} t}}(\mathbf {Q\cdot T\cdot Q} ^{\top })={\dot {\mathbf {Q} }}\cdot \mathbf {T\cdot Q} ^{\top }+\mathbf {Q} \cdot {\dot {\mathbf {T} }}\cdot \mathbf {Q} ^{\top }+\mathbf {Q\cdot T} \cdot {\dot {\mathbf {Q} }}^{\top }\neq \mathbf {Q} \cdot {\dot {\mathbf {T} }}\cdot \mathbf {Q} ^{\top }\,.}$

For the formulation of rate-dependent material models, objective time derivatives are required for constitutive variables in the spatial approach , because it does not correspond to the experience that an observer in motion measures a different material behavior than a stationary one. Thus, the material models must be formulated with objective time derivatives. The deformation gradient F describes the local deformations at a point in the material and accordingly it also contains all information on deformation rates. It becomes the spatial velocity gradient

${\displaystyle \mathbf {l} :={\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}=\mathbf {d} +\mathbf {w} }$

whose symmetrical part d = ½ ( l + l ^T ) is called the spatial distortion speed tensor and whose asymmetrical part w = ½ ( l - l ^T ) is called the vortex tensor or spin tensor . These tensors are written in lower case here because they are formulated spatially . In materials theory, objective rates of strain tensors and stress tensors are of particular interest . Several rates have been defined, including:

Zaremba- Jaumann derivation: ${\displaystyle {\stackrel {\circ }{\mathbf {T} }}:={\dot {\mathbf {T} }}+\mathbf {T\cdot w} -\mathbf {w\cdot T} }$

Covariant Oldroyd derivation: ${\displaystyle {\stackrel {\triangle }{\mathbf {T} }}:={\dot {\mathbf {T} }}+\mathbf {T\cdot l} +\mathbf {l} ^{\top }\cdot \mathbf {T} ={\stackrel {\circ }{\mathbf {T} }}+\mathbf {T\cdot d} +\mathbf {d\cdot T} }$

Contra variant Oldroyd derivation : ${\displaystyle {\stackrel {\nabla }{\mathbf {T} }}:={\dot {\mathbf {T} }}-\mathbf {l\cdot T} -\mathbf {T} \cdot \mathbf {l} ^{\top }={\stackrel {\circ }{\mathbf {T} }}-\mathbf {T\cdot d} -\mathbf {d\cdot T} }$

Cauchy derivative: ${\displaystyle {\stackrel {\diamond }{\mathbf {T} }}={\dot {\mathbf {T} }}+\operatorname {Sp} (\mathbf {l} )\mathbf {T} -\mathbf {l\cdot T} -\mathbf {T\cdot l} ^{\top }\,.}$

The Zaremba-Jaumann tension speed indicates the change in tension over time in the moving frame of reference. An observer who rotates with the material element observes the change in tension over time. ${\displaystyle {\stackrel {\circ }{\mathbf {T} }}}$ ${\displaystyle \mathbf {T} }$

For an objective vector, these are time derivatives ${\displaystyle {\vec {v}}}$

${\displaystyle {\begin{array}{rclcl}{\stackrel {\circ }{\vec {v}}}&=&{\dot {\vec {v}}}-\mathbf {w} \cdot {\vec {v}}\\{\stackrel {\Delta }{\vec {v}}}&=&{\dot {\vec {v}}}+\mathbf {l} ^{\top }\cdot {\vec {v}}&=&{\stackrel {\circ }{\vec {v}}}+\mathbf {d} \cdot {\vec {v}}\\{\stackrel {\nabla }{\vec {v}}}&=&{\dot {\vec {v}}}-\mathbf {l} \cdot {\vec {v}}&=&{\stackrel {\circ }{\vec {v}}}-\mathbf {d} \cdot {\vec {v}}\end{array}}}$

lens.

Objectivity and algebraic connections

An algebraic combination of objective quantities is again objective. Depending on the type, addition, multiplication, multiplication with a scalar, scalar product , cross product , dyadic product and matrix multiplication can be considered as algebraic operations .

surgery	equation
Scalars
addition	${\displaystyle (a+b)'=a'+b'=a+b}$
multiplication	${\displaystyle (a\cdot b)'=a'\cdot b'=a\cdot b}$
Vectors
addition	${\displaystyle ({\vec {x}}+{\vec {y}})'=\mathbf {Q} \cdot {\vec {x}}+\mathbf {Q} \cdot {\vec {y}}=\mathbf {Q} \cdot ({\vec {x}}+{\vec {y}})}$
Multiplication by a scalar	${\displaystyle (a{\vec {x}})'=a'{\vec {x}}'=a\mathbf {Q} \cdot {\vec {x}}=\mathbf {Q} \cdot (a{\vec {x}})}$
Scalar product	${\displaystyle ({\vec {x}}\cdot {\vec {y}})'={\vec {x}}'\cdot {\vec {y}}'=(\mathbf {Q} \cdot {\vec {x}})\cdot (\mathbf {Q} \cdot {\vec {y}})={\vec {x}}\cdot \mathbf {Q} ^{\top }\cdot \mathbf {Q} \cdot {\vec {y}}={\vec {x}}\cdot {\vec {y}}}$
Cross product	${\displaystyle ({\vec {x}}\times {\vec {y}})'={\vec {x}}'\times {\vec {y}}'=(\mathbf {Q} \cdot {\vec {x}})\times (\mathbf {Q} \cdot {\vec {y}})=\mathbf {Q} \cdot ({\vec {x}}\times {\vec {y}})}$
Dyadic product	${\displaystyle ({\vec {x}}\otimes {\vec {y}})'={\vec {x}}'\otimes {\vec {y}}'=\mathbf {Q} \cdot {\vec {x}}\otimes \mathbf {Q} \cdot {\vec {y}}=\mathbf {Q} \cdot ({\vec {x}}\otimes {\vec {y}})\cdot \mathbf {Q} ^{\top }}$
Tensors
addition	${\displaystyle (\mathbf {S} +\mathbf {T} )'=\mathbf {Q\cdot S\cdot Q} ^{\top }+\mathbf {Q\cdot T\cdot Q} ^{\top }=\mathbf {Q} \cdot (\mathbf {S} +\mathbf {T} )\cdot \mathbf {Q} ^{\top }}$
Multiplication by a scalar	${\displaystyle (a\mathbf {T} )'=a'\mathbf {T} '=a\mathbf {Q\cdot T\cdot Q} ^{\top }=\mathbf {Q} \cdot (a\mathbf {T} )\cdot \mathbf {Q} ^{\top }}$
Scalar product	${\displaystyle (\mathbf {S} :\mathbf {T} )'=(\mathbf {Q\cdot S\cdot Q} ^{\top }):(\mathbf {Q\cdot T\cdot Q} ^{\top })=\mathbf {S} :\mathbf {T} }$
Vector transformation	${\displaystyle (\mathbf {T} \cdot {\vec {x}})'=\mathbf {T} '\cdot {\vec {x}}'=(\mathbf {Q\cdot T\cdot Q} ^{\top })\cdot (\mathbf {Q} \cdot {\vec {x}})=\mathbf {Q} \cdot (\mathbf {T} \cdot {\vec {x}})}$
Matrix multiplication	${\displaystyle (\mathbf {S\cdot T} )'=(\mathbf {Q\cdot S\cdot Q} ^{\top })\cdot (\mathbf {Q\cdot T\cdot Q} ^{\top })=\mathbf {Q} \cdot (\mathbf {S\cdot T} )\cdot \mathbf {Q} ^{\top }}$

List of objective quantities

The following list gives a selection of objective sizes.

Scalars

• Geometry: distance, area (amount), volume (amount). The shape of the volume is objective only for actual Euclidean transformations with det ( Q ) = +1.
• Physics: temperature, mass, internal energy and entropy are objective scalars. From this it can be deduced that the specific quantities related to mass or volume are objective: density, specific internal energy and specific entropy.

Vectors

• Geometry: distance vector, line element ; the surface element is objective only for actual Euclidean transformations with det ( Q ) = +1.
• Kinematics : absolute speed, absolute acceleration, vector invariant
• Physics: force, stress vector, heat flow vector and the temperature gradient

Tensors

The above transformation properties for tensors apply to so-called spatial one-field tensors, the definition and value ranges of which rotate with movement. In addition, in continuum mechanics there are body-related one-field tensors whose definition and value range is materially determined by the reference configuration independent of movement, which is the same for all observers. The image vectors of these configurations are - clearly speaking - pinned to a material point in a legible manner for all observers like a label. Body-related objective tensors transform accordingly

${\displaystyle \mathbf {T'=T} }$.

Furthermore, continuum mechanics knows two-field tensors, which transform vectors and tensors between two configurations. An example of this is the first Piola-Kirchhoff stress tensor , which in convective coordinates has the form

${\displaystyle \mathbf {T} =\sum _{i,j=1}^{3}T^{ij}{\vec {g}}_{i}\otimes {\vec {G}}_{j}}$

owns. This contains basis vectors in the movement-independent material reference configuration, basis vectors in the moving spatial instantaneous configuration and the components of the tensor. If now the basis vectors are the same for all observers, i. H. it applies ${\displaystyle {\vec {G}}_{1,2,3}}$ ${\displaystyle {\vec {g}}_{1,2,3}}$ ${\displaystyle T^{ij}}$ ${\displaystyle {\vec {G}}_{i}}$

${\displaystyle {\vec {G}}_{i}'={\vec {G}}_{i}\quad \forall \;i=1,2,3\,,}$

and the basis vectors are objective, i.e. according to ${\displaystyle {\vec {g}}_{i}}$

${\displaystyle {\vec {g}}_{i}'=\mathbf {Q} \cdot {\vec {g}}_{i}\quad \forall \;i=1,2,3}$

transform, then such a two-field tensor is objective if it is according to ${\displaystyle \mathbf {T} }$

${\displaystyle \mathbf {T} {'}=\sum _{i,j=1}^{3}T^{ij}{'}{\vec {g}}_{i}{'}\otimes {\vec {G}}_{j}{'}=\sum _{i,j=1}^{3}T^{ij}\mathbf {Q} \cdot {\vec {g}}_{i}\otimes {\vec {G}}_{j}=\mathbf {Q\cdot T} \,,}$

thus transformed like an objective vector.

The following tensors are used in mechanics, especially in continuum mechanics. Because mirroring transformations of material bodies are not considered in mechanics, det ( Q ) = +1 is assumed.

Surname	Not objective	Objective (spatial)	Lens (body related)	Objective two-field tensor
Rigid body mechanics
Inertia tensor I , Θ		x
Angular velocity tensor Ω	x
Continuum mechanics
Deformation gradient F = R · U = v · R				F ′ = Q * F
Right stretch tensor U			x
Left stretch tensor v		x
Rotational tensor R				R ′ = Q * R
Right Cauchy-Green tensor C = F T * F = U * U			x
Left Cauchy-Green tensor b = F · F T = v · v		x
Green-Lagrange strain tensor E = ½ ( F T · F - 1 )			x
Euler-Almansi strain tensor e = ½ ( 1 - F T-1 · F −1 )		x
Cauchy's stress tensor σ		x
First Piola-Kirchoff stress tensor P = det ( F ) σ · F T-1				P ′ = Q * P
Second Piola-Kirchoff stress tensor ${\displaystyle {\tilde {\mathbf {T} }}=\mathrm {det} (\mathbf {F} )\mathbf {F} ^{-1}\cdot {\boldsymbol {\sigma }}\cdot \mathbf {F} ^{\top -1}}$			x
Distortion Rates
Spatial velocity gradient ${\displaystyle \mathbf {l} =\mathrm {grad} ({\vec {v}}({\vec {x}},t))={\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}}$	${\displaystyle \mathbf {l} '=\mathbf {Q\cdot l\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }}$
Spatial strain rate tensor d = ½ ( l + l T )		x
Spin tensor w = ½ ( l - l T )	${\displaystyle \mathbf {w} '=\mathbf {Q\cdot w\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }}$
Material strain rate tensor ${\displaystyle {\dot {\mathbf {E} }}}$			x
Material time derivative of e ${\displaystyle {\dot {\mathbf {e} }}}$	longer formula
Oldroyd's derivation of the Euler-Almansi strains ${\displaystyle {\stackrel {\triangle }{\mathbf {e} }}=\mathbf {d} }$		x

Transformations of reference systems in other physical disciplines

Classical mechanics and electrodynamics are invariant under reflections, because the laws in these two disciplines are completely determined by gravity and electromagnetism . In quantum mechanics , however, this symmetry towards reflections is broken by violating the parity of the weak interaction . The other forces of nature, especially gravity and electromagnetism, are parity-preserving interactions.

In the theory of relativity, transformations are used between moving reference systems, which leave the amounts and relative angles of all vectors invariant. The transformation between moving reference systems in four-dimensional space-time, which also leaves the amounts and angles of (four) speeds and pulses invariant, are the Lorentz boosts . The difference to rotations in four-dimensional Euclidean space is that the time coordinate in the four-vector in Minkowski space is assigned a different sign than the space coordinates. As a result, in a rotation matrix for rotations of time and space coordinates, sine and cosine must be replaced by hyperbolic sine and hyperbolic cosine . The Euclidean group that contains this generalization is called the Poincaré group .

example

Evidence of the objectivity of the Zaremba-Jaumann rate of an objective tensor is given. Is to be shown

${\displaystyle ({\stackrel {\circ }{\mathbf {T} }})'=\mathbf {Q} \cdot {\stackrel {\circ }{\mathbf {T} }}\cdot \mathbf {Q} ^{\top }=\mathbf {Q} \cdot ({\dot {\mathbf {T} }}+\mathbf {T\cdot w} -\mathbf {w\cdot T} )\cdot \mathbf {Q} ^{\top }\,.}$

With

${\displaystyle \mathbf {w} '=\mathbf {Q\cdot w\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }}$

and

${\displaystyle (\mathbf {T} '){\dot {}}=(\mathbf {Q\cdot T\cdot Q} ^{\top }){\dot {}}={\dot {\mathbf {Q} }}\cdot \mathbf {T\cdot Q} ^{\top }+\mathbf {Q} \cdot {\dot {\mathbf {T} }}\cdot \mathbf {Q} ^{\top }+\mathbf {Q\cdot T} \cdot {\dot {\mathbf {Q} }}^{\top }}$

you get

${\displaystyle {\begin{aligned}({\stackrel {\circ }{\mathbf {T} }})'=&{\dot {\mathbf {T} }}'+\mathbf {T} '\cdot \mathbf {w} '-\mathbf {w} '\cdot \mathbf {T} '\\=&({\dot {\mathbf {Q} }}\cdot \mathbf {T\cdot Q} ^{\top }+\mathbf {Q} \cdot {\dot {\mathbf {T} }}\cdot \mathbf {Q} ^{\top }+\mathbf {Q\cdot T} \cdot {\dot {\mathbf {Q} }}^{\top })+(\mathbf {Q\cdot T\cdot Q} ^{\top })\cdot (\mathbf {Q\cdot w\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top })\\&-(\mathbf {Q\cdot w\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top })\cdot (\mathbf {Q\cdot T\cdot Q} ^{\top })\\=&{\dot {\mathbf {Q} }}\cdot \mathbf {T\cdot Q} ^{\top }+\mathbf {Q} \cdot {\dot {\mathbf {T} }}\cdot \mathbf {Q} ^{\top }+\mathbf {Q\cdot T} \cdot {\dot {\mathbf {Q} }}^{\top }+\mathbf {Q\cdot T\cdot w\cdot Q} ^{\top }\\&+\mathbf {Q\cdot T\cdot Q} ^{\top }\cdot {\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }-\mathbf {Q\cdot w\cdot T\cdot Q} ^{\top }-{\dot {\mathbf {Q} }}\cdot \mathbf {T\cdot Q} ^{\top }\\=&\mathbf {Q} \cdot ({\dot {\mathbf {T} }}+\mathbf {T\cdot w} -\mathbf {w\cdot T} )\cdot \mathbf {Q} ^{\top }+\mathbf {Q\cdot T} \cdot {\dot {\mathbf {Q} }}^{\top }+\mathbf {Q\cdot T\cdot Q} ^{\top }\cdot {\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }\\=&\mathbf {Q} \cdot {\stackrel {\circ }{\mathbf {T} }}\cdot \mathbf {Q} ^{\top }+\mathbf {Q\cdot T} \cdot {\dot {\mathbf {Q} }}^{\top }+\mathbf {Q\cdot T\cdot Q} ^{\top }\cdot {\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }\,.\end{aligned}}}$

Now you can

${\displaystyle {\dot {\mathbf {1} }}=(\mathbf {Q\cdot Q} ^{\top }){\dot {}}={\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }+\mathbf {Q} \cdot {\dot {\mathbf {Q} }}^{\top }=\mathbf {0} \rightarrow {\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }=-\mathbf {Q} \cdot {\dot {\mathbf {Q} }}^{\top }}$

take advantage of and maintain

${\displaystyle {\begin{aligned}({\stackrel {\circ }{\mathbf {T} }})'=&\mathbf {Q} \cdot {\stackrel {\circ }{\mathbf {T} }}\cdot \mathbf {Q} ^{\top }+\mathbf {Q\cdot T} \cdot {\dot {\mathbf {Q} }}^{\top }-\mathbf {Q\cdot T\cdot Q} ^{\top }\cdot \mathbf {Q} \cdot {\dot {\mathbf {Q} }}^{\top }\\=&\mathbf {Q} \cdot {\stackrel {\circ }{\mathbf {T} }}\cdot \mathbf {Q} ^{\top }+\mathbf {Q\cdot T} \cdot {\dot {\mathbf {Q} }}^{\top }-\mathbf {Q\cdot T} \cdot {\dot {\mathbf {Q} }}^{\top }=\mathbf {Q} \cdot {\stackrel {\circ }{\mathbf {T} }}\cdot \mathbf {Q} ^{\top }\,,\end{aligned}}}$

so the desired result.

See also

• Orthogonal mapping , orthogonal matrix or rotation matrix
• Material model
• Isotropic function
• Main invariant
• Formula collection tensor algebra

literature

• Holm Altenbach: Continuum Mechanics. Introduction to the material-independent and material-dependent equations . 2nd Edition. Springer Vieweg, Berlin a. a. 2012, ISBN 978-3-642-24118-5 .
• P. Haupt: Continuum Mechanics and Theory of Materials . Springer, 2000, ISBN 3-540-66114-X (The 2010 edition has ISBN 978-3-642-07718-0 ).
• Horst Parisch: Solid-state continuum mechanics. From the basic equations to the solution with finite elements . Teubner, Stuttgart a. a. 2003, ISBN 3-519-00434-8 .

Individual evidence

1. ↑ see also the section #Transformations of reference systems in other physical disciplines
2. ↑ ^{a b} Haupt (2000), p. 165
3. ↑ The "active" and "passive" change of observer are not to be confused with the active movement, in which an object moves, and the passive movement, in which the observer moves relative to the object at rest.
4. ↑ The symbols for the objective rates vary from source to source. The ones given here follow P. Haupt, p. 48ff. In H. Altenbach is used for and for . ${\displaystyle \mathbf {T} ^{\nabla }}$ ${\displaystyle {\stackrel {\circ }{\mathbf {T} }}}$ ${\displaystyle \mathbf {T} ^{O}}$ ${\displaystyle {\stackrel {\triangle }{\mathbf {T} }}}$
5. ↑ after James G. Oldroyd (1921 - 1982)
6. ↑ This derivation occurs in the Cauchy elasticity and is also named after C. Truesdell. He himself named the derivative after Cauchy and wrote in 1963 that this rate was named after him for no inventive reason ("came to be named, for no good reason, after [...] me") see C. Truesdell: Remarks on Hypo- Elasticity . In: Journal of Research of the National Bureau of Standards - B. Mathematics and Mathematical Physics , Vol. 67B, No. 3, July-September 1963, p. 141.
7. ↑ H. Altenbach, p. 231.
8. ^ Parisch 2003