Mixing
(Updates on this title are welcome..) Function-valued functions in differential equations
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I'm trying to prove the problem below:
Let $Omega = (omega_ij)$ be a skew-symmetric $3×3$ matrix, i.e., $Omega^T=-Omega$.
Let $e_1(s)$, $e_2(s)$ and $e_3(s)$ be smooth vector-valued functions of a parametre $s$ satisfying the differential equations $fracddse_i=sum^3_k=1 omega_ik e_k$, where $i=1,2,3$.
Suppose that for some parametre value $s_0$, the vectors $e_1(s_0)$, $e_2(s_0)$ and $e_3(s_0)$ are orthonormal.
Show that the vectors $e_1(s)$, $e_2(s)$ and $e_3(s)$ are orthonormal for all $s$.
=
But what I have been doing is just expanding out the differential equations $fracddse_i$ and translating the condition of "orthonormal" for $e_1(s_0)$, $e_2(s_0)$ and $e_3(s_0)$.
I have also defined a function $f_ik(s)=e_i(s) cdot e_k(s)$, where $i,k=1,2,3$, and found its derivative by the product rule: $fracddsf_ik(s)=fracddse_i(s) cdot e_k(s) + e_i(s) cdot fracddse_k(s)$.
Any ideas on how to carrying on this proof? Thanks!
differential-equations differential-geometry orthonormal
asked Aug 6 at 22:32
Evelyn Venne
334
add a comment |Â
up vote
0
down vote
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I'm trying to prove the problem below:
Let $Omega = (omega_ij)$ be a skew-symmetric $3×3$ matrix, i.e., $Omega^T=-Omega$.
Let $e_1(s)$, $e_2(s)$ and $e_3(s)$ be smooth vector-valued functions of a parametre $s$ satisfying the differential equations $fracddse_i=sum^3_k=1 omega_ik e_k$, where $i=1,2,3$.
Suppose that for some parametre value $s_0$, the vectors $e_1(s_0)$, $e_2(s_0)$ and $e_3(s_0)$ are orthonormal.
Show that the vectors $e_1(s)$, $e_2(s)$ and $e_3(s)$ are orthonormal for all $s$.
=
But what I have been doing is just expanding out the differential equations $fracddse_i$ and translating the condition of "orthonormal" for $e_1(s_0)$, $e_2(s_0)$ and $e_3(s_0)$.
I have also defined a function $f_ik(s)=e_i(s) cdot e_k(s)$, where $i,k=1,2,3$, and found its derivative by the product rule: $fracddsf_ik(s)=fracddse_i(s) cdot e_k(s) + e_i(s) cdot fracddse_k(s)$.
Any ideas on how to carrying on this proof? Thanks!
differential-equations differential-geometry orthonormal
asked Aug 6 at 22:32
Evelyn Venne
334
If you want to learn more about similar equations, a fantastic resource is the book by Murray, Li and Sastry on "A Mathematical Introduction to Robotic Manipulation" cds.caltech.edu/~murray/books/MLS/pdf/mls94-complete.pdf
– WalterJ
Aug 6 at 23:35
1
Thanks heaps would look into it :)
– Evelyn Venne
Aug 7 at 2:19
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm trying to prove the problem below:
Let $Omega = (omega_ij)$ be a skew-symmetric $3×3$ matrix, i.e., $Omega^T=-Omega$.
Let $e_1(s)$, $e_2(s)$ and $e_3(s)$ be smooth vector-valued functions of a parametre $s$ satisfying the differential equations $fracddse_i=sum^3_k=1 omega_ik e_k$, where $i=1,2,3$.
Suppose that for some parametre value $s_0$, the vectors $e_1(s_0)$, $e_2(s_0)$ and $e_3(s_0)$ are orthonormal.
Show that the vectors $e_1(s)$, $e_2(s)$ and $e_3(s)$ are orthonormal for all $s$.
=
But what I have been doing is just expanding out the differential equations $fracddse_i$ and translating the condition of "orthonormal" for $e_1(s_0)$, $e_2(s_0)$ and $e_3(s_0)$.
I have also defined a function $f_ik(s)=e_i(s) cdot e_k(s)$, where $i,k=1,2,3$, and found its derivative by the product rule: $fracddsf_ik(s)=fracddse_i(s) cdot e_k(s) + e_i(s) cdot fracddse_k(s)$.
Any ideas on how to carrying on this proof? Thanks!
differential-equations differential-geometry orthonormal
asked Aug 6 at 22:32
Evelyn Venne
334
I'm trying to prove the problem below:
Let $Omega = (omega_ij)$ be a skew-symmetric $3×3$ matrix, i.e., $Omega^T=-Omega$.
Let $e_1(s)$, $e_2(s)$ and $e_3(s)$ be smooth vector-valued functions of a parametre $s$ satisfying the differential equations $fracddse_i=sum^3_k=1 omega_ik e_k$, where $i=1,2,3$.
Suppose that for some parametre value $s_0$, the vectors $e_1(s_0)$, $e_2(s_0)$ and $e_3(s_0)$ are orthonormal.
Show that the vectors $e_1(s)$, $e_2(s)$ and $e_3(s)$ are orthonormal for all $s$.
=
But what I have been doing is just expanding out the differential equations $fracddse_i$ and translating the condition of "orthonormal" for $e_1(s_0)$, $e_2(s_0)$ and $e_3(s_0)$.
I have also defined a function $f_ik(s)=e_i(s) cdot e_k(s)$, where $i,k=1,2,3$, and found its derivative by the product rule: $fracddsf_ik(s)=fracddse_i(s) cdot e_k(s) + e_i(s) cdot fracddse_k(s)$.
Any ideas on how to carrying on this proof? Thanks!
differential-equations differential-geometry orthonormal
asked Aug 6 at 22:32
Evelyn Venne
334
edited Aug 6 at 22:47
asked Aug 6 at 22:32
Evelyn Venne
334
asked Aug 6 at 22:32
Evelyn Venne
334
asked Aug 6 at 22:32
Evelyn Venne
334
334
If you want to learn more about similar equations, a fantastic resource is the book by Murray, Li and Sastry on "A Mathematical Introduction to Robotic Manipulation" cds.caltech.edu/~murray/books/MLS/pdf/mls94-complete.pdf
– WalterJ
Aug 6 at 23:35
1
Thanks heaps would look into it :)
– Evelyn Venne
Aug 7 at 2:19
add a comment |Â
If you want to learn more about similar equations, a fantastic resource is the book by Murray, Li and Sastry on "A Mathematical Introduction to Robotic Manipulation" cds.caltech.edu/~murray/books/MLS/pdf/mls94-complete.pdf
– WalterJ
Aug 6 at 23:35
1
Thanks heaps would look into it :)
– Evelyn Venne
Aug 7 at 2:19
If you want to learn more about similar equations, a fantastic resource is the book by Murray, Li and Sastry on "A Mathematical Introduction to Robotic Manipulation" cds.caltech.edu/~murray/books/MLS/pdf/mls94-complete.pdf
– WalterJ
Aug 6 at 23:35
If you want to learn more about similar equations, a fantastic resource is the book by Murray, Li and Sastry on "A Mathematical Introduction to Robotic Manipulation" cds.caltech.edu/~murray/books/MLS/pdf/mls94-complete.pdf
– WalterJ
Aug 6 at 23:35
1
1
Thanks heaps would look into it :)
– Evelyn Venne
Aug 7 at 2:19
Thanks heaps would look into it :)
– Evelyn Venne
Aug 7 at 2:19
add a comment |Â
1 Answer
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oldest
votes
up vote
0
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You can write the set of differential equations as
beginequation
fracpartialpartial sbeginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix = Omega beginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix
endequation
with the solution
beginalign
beginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix &= e^Omega sbeginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix\
&= e^-Omega^top sbeginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix.
endalign
It turns out that $e^Omega s$ is a rotation matrix (formally you can look into the special orthogonal group), which means that for any $s$ the frame $beginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix$ is rotated, which preserves relative orientation, e.g. pick up a box, you can rotate it all day long, the sides remain fixed with respect to each other.
answered Aug 6 at 23:25
WalterJ
795611
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
You can write the set of differential equations as
beginequation
fracpartialpartial sbeginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix = Omega beginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix
endequation
with the solution
beginalign
beginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix &= e^Omega sbeginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix\
&= e^-Omega^top sbeginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix.
endalign
It turns out that $e^Omega s$ is a rotation matrix (formally you can look into the special orthogonal group), which means that for any $s$ the frame $beginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix$ is rotated, which preserves relative orientation, e.g. pick up a box, you can rotate it all day long, the sides remain fixed with respect to each other.
answered Aug 6 at 23:25
WalterJ
795611
add a comment |Â
up vote
0
down vote
accepted
You can write the set of differential equations as
beginequation
fracpartialpartial sbeginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix = Omega beginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix
endequation
with the solution
beginalign
beginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix &= e^Omega sbeginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix\
&= e^-Omega^top sbeginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix.
endalign
It turns out that $e^Omega s$ is a rotation matrix (formally you can look into the special orthogonal group), which means that for any $s$ the frame $beginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix$ is rotated, which preserves relative orientation, e.g. pick up a box, you can rotate it all day long, the sides remain fixed with respect to each other.
answered Aug 6 at 23:25
WalterJ
795611
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
You can write the set of differential equations as
beginequation
fracpartialpartial sbeginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix = Omega beginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix
endequation
with the solution
beginalign
beginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix &= e^Omega sbeginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix\
&= e^-Omega^top sbeginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix.
endalign
It turns out that $e^Omega s$ is a rotation matrix (formally you can look into the special orthogonal group), which means that for any $s$ the frame $beginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix$ is rotated, which preserves relative orientation, e.g. pick up a box, you can rotate it all day long, the sides remain fixed with respect to each other.
answered Aug 6 at 23:25
WalterJ
795611
You can write the set of differential equations as
beginequation
fracpartialpartial sbeginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix = Omega beginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix
endequation
with the solution
beginalign
beginpmatrix e_1(s) & e_2(s) & e_3(s) endpmatrix &= e^Omega sbeginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix\
&= e^-Omega^top sbeginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix.
endalign
It turns out that $e^Omega s$ is a rotation matrix (formally you can look into the special orthogonal group), which means that for any $s$ the frame $beginpmatrix e_1(s_0) & e_2(s_0) & e_3(s_0) endpmatrix$ is rotated, which preserves relative orientation, e.g. pick up a box, you can rotate it all day long, the sides remain fixed with respect to each other.
answered Aug 6 at 23:25
WalterJ
795611
answered Aug 6 at 23:25
WalterJ
795611
answered Aug 6 at 23:25
WalterJ
795611
answered Aug 6 at 23:25
WalterJ
795611
795611
add a comment |Â
add a comment |Â
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If you want to learn more about similar equations, a fantastic resource is the book by Murray, Li and Sastry on "A Mathematical Introduction to Robotic Manipulation" cds.caltech.edu/~murray/books/MLS/pdf/mls94-complete.pdf
– WalterJ
Aug 6 at 23:35
1
Thanks heaps would look into it :)
– Evelyn Venne
Aug 7 at 2:19