Differential equation for a vector : condition to be conservative
$begingroup$
Assume there is a vector $mathbf{N}(t)$ of elements $N_i$ and of dimension $n$ and there are matrices $mathbf{B}_i$ of dimension $ntimes n$, and that :
$begin{equation}left{ begin{split}& frac{dN_i(t)}{dt}=mathbf{N}(t)^Tmathbf{B}_imathbf{N}(t) \ & frac{d(sum_{i=0}^n N_i(t))}{dt}=0 end{split} right. end{equation}$
Are there conditions on $mathbf{B}_i$ and $mathbf{N}(t=0)$ so that the conservation condition over $sum_{i=0}^n N_i$ is realized ?
The second equation leads to $mathbf{N}(t)^T(sum_i mathbf{B}_i)mathbf{N}(t)=0$ with an obvious solution : $sum_i mathbf{B}_i=0$.
But can one say more things apart from this obvious solution ?
matrices ordinary-differential-equations
asked Jan 16 at 15:25
J.AJ.A
1465
$endgroup$
add a comment |
$begingroup$
Assume there is a vector $mathbf{N}(t)$ of elements $N_i$ and of dimension $n$ and there are matrices $mathbf{B}_i$ of dimension $ntimes n$, and that :
$begin{equation}left{ begin{split}& frac{dN_i(t)}{dt}=mathbf{N}(t)^Tmathbf{B}_imathbf{N}(t) \ & frac{d(sum_{i=0}^n N_i(t))}{dt}=0 end{split} right. end{equation}$
Are there conditions on $mathbf{B}_i$ and $mathbf{N}(t=0)$ so that the conservation condition over $sum_{i=0}^n N_i$ is realized ?
The second equation leads to $mathbf{N}(t)^T(sum_i mathbf{B}_i)mathbf{N}(t)=0$ with an obvious solution : $sum_i mathbf{B}_i=0$.
But can one say more things apart from this obvious solution ?
matrices ordinary-differential-equations
asked Jan 16 at 15:25
J.AJ.A
1465
$endgroup$
$begingroup$
The first equation has a dimension mismatch.
$endgroup$
– Ian
Jan 16 at 15:28
$begingroup$
Not unit, dimension. A $n times 1$ vector can't multiply on the left of a $n times n$ matrix.
$endgroup$
– Ian
Jan 16 at 15:29
$begingroup$
Right ! I'm correcting that thx
$endgroup$
– J.A
Jan 16 at 15:31
$begingroup$
This still makes no sense. Do you want $frac{dN_i}{dt}=N^T B_i N$?
$endgroup$
– Ian
Jan 17 at 18:50
add a comment |
$begingroup$
Assume there is a vector $mathbf{N}(t)$ of elements $N_i$ and of dimension $n$ and there are matrices $mathbf{B}_i$ of dimension $ntimes n$, and that :
$begin{equation}left{ begin{split}& frac{dN_i(t)}{dt}=mathbf{N}(t)^Tmathbf{B}_imathbf{N}(t) \ & frac{d(sum_{i=0}^n N_i(t))}{dt}=0 end{split} right. end{equation}$
Are there conditions on $mathbf{B}_i$ and $mathbf{N}(t=0)$ so that the conservation condition over $sum_{i=0}^n N_i$ is realized ?
The second equation leads to $mathbf{N}(t)^T(sum_i mathbf{B}_i)mathbf{N}(t)=0$ with an obvious solution : $sum_i mathbf{B}_i=0$.
But can one say more things apart from this obvious solution ?
matrices ordinary-differential-equations
asked Jan 16 at 15:25
J.AJ.A
1465
$endgroup$
Assume there is a vector $mathbf{N}(t)$ of elements $N_i$ and of dimension $n$ and there are matrices $mathbf{B}_i$ of dimension $ntimes n$, and that :
$begin{equation}left{ begin{split}& frac{dN_i(t)}{dt}=mathbf{N}(t)^Tmathbf{B}_imathbf{N}(t) \ & frac{d(sum_{i=0}^n N_i(t))}{dt}=0 end{split} right. end{equation}$
Are there conditions on $mathbf{B}_i$ and $mathbf{N}(t=0)$ so that the conservation condition over $sum_{i=0}^n N_i$ is realized ?
The second equation leads to $mathbf{N}(t)^T(sum_i mathbf{B}_i)mathbf{N}(t)=0$ with an obvious solution : $sum_i mathbf{B}_i=0$.
But can one say more things apart from this obvious solution ?
matrices ordinary-differential-equations
matrices ordinary-differential-equations
asked Jan 16 at 15:25
J.AJ.A
1465
asked Jan 16 at 15:25
J.AJ.A
1465
edited Jan 18 at 0:09
J.A
asked Jan 16 at 15:25
J.AJ.A
1465
asked Jan 16 at 15:25
J.AJ.A
1465
asked Jan 16 at 15:25
J.AJ.A
1465
1465
$begingroup$
The first equation has a dimension mismatch.
$endgroup$
– Ian
Jan 16 at 15:28
$begingroup$
Not unit, dimension. A $n times 1$ vector can't multiply on the left of a $n times n$ matrix.
$endgroup$
– Ian
Jan 16 at 15:29
$begingroup$
Right ! I'm correcting that thx
$endgroup$
– J.A
Jan 16 at 15:31
$begingroup$
This still makes no sense. Do you want $frac{dN_i}{dt}=N^T B_i N$?
$endgroup$
– Ian
Jan 17 at 18:50
add a comment |
$begingroup$
The first equation has a dimension mismatch.
$endgroup$
– Ian
Jan 16 at 15:28
$begingroup$
Not unit, dimension. A $n times 1$ vector can't multiply on the left of a $n times n$ matrix.
$endgroup$
– Ian
Jan 16 at 15:29
$begingroup$
Right ! I'm correcting that thx
$endgroup$
– J.A
Jan 16 at 15:31
$begingroup$
This still makes no sense. Do you want $frac{dN_i}{dt}=N^T B_i N$?
$endgroup$
– Ian
Jan 17 at 18:50
$begingroup$
The first equation has a dimension mismatch.
$endgroup$
– Ian
Jan 16 at 15:28
$begingroup$
The first equation has a dimension mismatch.
$endgroup$
– Ian
Jan 16 at 15:28
$begingroup$
Not unit, dimension. A $n times 1$ vector can't multiply on the left of a $n times n$ matrix.
$endgroup$
– Ian
Jan 16 at 15:29
$begingroup$
Not unit, dimension. A $n times 1$ vector can't multiply on the left of a $n times n$ matrix.
$endgroup$
– Ian
Jan 16 at 15:29
$begingroup$
Right ! I'm correcting that thx
$endgroup$
– J.A
Jan 16 at 15:31
$begingroup$
Right ! I'm correcting that thx
$endgroup$
– J.A
Jan 16 at 15:31
$begingroup$
This still makes no sense. Do you want $frac{dN_i}{dt}=N^T B_i N$?
$endgroup$
– Ian
Jan 17 at 18:50
$begingroup$
This still makes no sense. Do you want $frac{dN_i}{dt}=N^T B_i N$?
$endgroup$
– Ian
Jan 17 at 18:50
add a comment |
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$begingroup$
The first equation has a dimension mismatch.
$endgroup$
– Ian
Jan 16 at 15:28
$begingroup$
Not unit, dimension. A $n times 1$ vector can't multiply on the left of a $n times n$ matrix.
$endgroup$
– Ian
Jan 16 at 15:29
$begingroup$
Right ! I'm correcting that thx
$endgroup$
– J.A
Jan 16 at 15:31
$begingroup$
This still makes no sense. Do you want $frac{dN_i}{dt}=N^T B_i N$?
$endgroup$
– Ian
Jan 17 at 18:50