Show that a system of equations to find a curve based on its curvature and torsion has a unique solution.
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Quoted from Pressley's Differential Geometry :
How to show that the three mentioned equations have a unique solution with initial conditions?
Since $k=k(s)$ and $t=t(s)$ are functions not constants so any try for solving them leads to even more complicated system of equations.
differential-equations differential-geometry systems-of-equations
asked Nov 16 at 11:44
72D
50916
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1
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Quoted from Pressley's Differential Geometry :
How to show that the three mentioned equations have a unique solution with initial conditions?
Since $k=k(s)$ and $t=t(s)$ are functions not constants so any try for solving them leads to even more complicated system of equations.
differential-equations differential-geometry systems-of-equations
asked Nov 16 at 11:44
72D
50916
Picard-Lindelöf theorem
– achille hui
Nov 16 at 11:51
@achillehui, there are three functions in not-3-independent-equations) to be solved not one.
– 72D
Nov 16 at 12:08
The $y$ in the theorem can be vector, the wiki book entry for the theorem spell this out explicitly.
– achille hui
Nov 16 at 12:22
@achillehui, I started to learn the one-dim case from an easy textbook now so I hopefully can understand the one in wikibook... Thanks.
– 72D
Nov 16 at 12:52
@achillehui, I understood all thing but one! Let $x(s)=(T_1(s), T_2(s), T_3(s), N_1, ..., B_1, ...)$ and $f(s,x(s))=(k(s)N_1(s), ..., -k(s)T_1+t(s)B_1, ..., -t(s)N_3)$ then $frac{d}{ds}x(s)=f(s,x(s))$ together with existence of $L$ (Lipschitz const) implies that $text{max}_{s in [s_0, s_1]} (k^2(N_1-N'_2)^2+...+(-tN_3+tN'_3)^2) le L^2 text{max}_{s in [s_0, s_1]} (T_1-T'_2)^2+...+(B_3-B'_3)^2)$. But for the smooth functions of $T$, $N$, $B$ in the (differential geo) text (image in OP) how the existence of $L$ is proven/guaranteed? ( ' means y instead of x , not for derivative)
– 72D
Nov 17 at 10:50
|
show 4 more comments
up vote
1
down vote
favorite
1
up vote
1
down vote
favorite
1
1
Quoted from Pressley's Differential Geometry :
How to show that the three mentioned equations have a unique solution with initial conditions?
Since $k=k(s)$ and $t=t(s)$ are functions not constants so any try for solving them leads to even more complicated system of equations.
differential-equations differential-geometry systems-of-equations
asked Nov 16 at 11:44
72D
50916
Quoted from Pressley's Differential Geometry :
How to show that the three mentioned equations have a unique solution with initial conditions?
Since $k=k(s)$ and $t=t(s)$ are functions not constants so any try for solving them leads to even more complicated system of equations.
differential-equations differential-geometry systems-of-equations
differential-equations differential-geometry systems-of-equations
asked Nov 16 at 11:44
72D
50916
asked Nov 16 at 11:44
72D
50916
edited Nov 18 at 9:01
asked Nov 16 at 11:44
72D
50916
asked Nov 16 at 11:44
72D
50916
asked Nov 16 at 11:44
72D
50916
50916
Picard-Lindelöf theorem
– achille hui
Nov 16 at 11:51
@achillehui, there are three functions in not-3-independent-equations) to be solved not one.
– 72D
Nov 16 at 12:08
The $y$ in the theorem can be vector, the wiki book entry for the theorem spell this out explicitly.
– achille hui
Nov 16 at 12:22
@achillehui, I started to learn the one-dim case from an easy textbook now so I hopefully can understand the one in wikibook... Thanks.
– 72D
Nov 16 at 12:52
@achillehui, I understood all thing but one! Let $x(s)=(T_1(s), T_2(s), T_3(s), N_1, ..., B_1, ...)$ and $f(s,x(s))=(k(s)N_1(s), ..., -k(s)T_1+t(s)B_1, ..., -t(s)N_3)$ then $frac{d}{ds}x(s)=f(s,x(s))$ together with existence of $L$ (Lipschitz const) implies that $text{max}_{s in [s_0, s_1]} (k^2(N_1-N'_2)^2+...+(-tN_3+tN'_3)^2) le L^2 text{max}_{s in [s_0, s_1]} (T_1-T'_2)^2+...+(B_3-B'_3)^2)$. But for the smooth functions of $T$, $N$, $B$ in the (differential geo) text (image in OP) how the existence of $L$ is proven/guaranteed? ( ' means y instead of x , not for derivative)
– 72D
Nov 17 at 10:50
|
show 4 more comments
Picard-Lindelöf theorem
– achille hui
Nov 16 at 11:51
@achillehui, there are three functions in not-3-independent-equations) to be solved not one.
– 72D
Nov 16 at 12:08
The $y$ in the theorem can be vector, the wiki book entry for the theorem spell this out explicitly.
– achille hui
Nov 16 at 12:22
@achillehui, I started to learn the one-dim case from an easy textbook now so I hopefully can understand the one in wikibook... Thanks.
– 72D
Nov 16 at 12:52
@achillehui, I understood all thing but one! Let $x(s)=(T_1(s), T_2(s), T_3(s), N_1, ..., B_1, ...)$ and $f(s,x(s))=(k(s)N_1(s), ..., -k(s)T_1+t(s)B_1, ..., -t(s)N_3)$ then $frac{d}{ds}x(s)=f(s,x(s))$ together with existence of $L$ (Lipschitz const) implies that $text{max}_{s in [s_0, s_1]} (k^2(N_1-N'_2)^2+...+(-tN_3+tN'_3)^2) le L^2 text{max}_{s in [s_0, s_1]} (T_1-T'_2)^2+...+(B_3-B'_3)^2)$. But for the smooth functions of $T$, $N$, $B$ in the (differential geo) text (image in OP) how the existence of $L$ is proven/guaranteed? ( ' means y instead of x , not for derivative)
– 72D
Nov 17 at 10:50
Picard-Lindelöf theorem
– achille hui
Nov 16 at 11:51
Picard-Lindelöf theorem
– achille hui
Nov 16 at 11:51
@achillehui, there are three functions in not-3-independent-equations) to be solved not one.
– 72D
Nov 16 at 12:08
@achillehui, there are three functions in not-3-independent-equations) to be solved not one.
– 72D
Nov 16 at 12:08
The $y$ in the theorem can be vector, the wiki book entry for the theorem spell this out explicitly.
– achille hui
Nov 16 at 12:22
The $y$ in the theorem can be vector, the wiki book entry for the theorem spell this out explicitly.
– achille hui
Nov 16 at 12:22
@achillehui, I started to learn the one-dim case from an easy textbook now so I hopefully can understand the one in wikibook... Thanks.
– 72D
Nov 16 at 12:52
@achillehui, I started to learn the one-dim case from an easy textbook now so I hopefully can understand the one in wikibook... Thanks.
– 72D
Nov 16 at 12:52
@achillehui, I understood all thing but one! Let $x(s)=(T_1(s), T_2(s), T_3(s), N_1, ..., B_1, ...)$ and $f(s,x(s))=(k(s)N_1(s), ..., -k(s)T_1+t(s)B_1, ..., -t(s)N_3)$ then $frac{d}{ds}x(s)=f(s,x(s))$ together with existence of $L$ (Lipschitz const) implies that $text{max}_{s in [s_0, s_1]} (k^2(N_1-N'_2)^2+...+(-tN_3+tN'_3)^2) le L^2 text{max}_{s in [s_0, s_1]} (T_1-T'_2)^2+...+(B_3-B'_3)^2)$. But for the smooth functions of $T$, $N$, $B$ in the (differential geo) text (image in OP) how the existence of $L$ is proven/guaranteed? ( ' means y instead of x , not for derivative)
– 72D
Nov 17 at 10:50
@achillehui, I understood all thing but one! Let $x(s)=(T_1(s), T_2(s), T_3(s), N_1, ..., B_1, ...)$ and $f(s,x(s))=(k(s)N_1(s), ..., -k(s)T_1+t(s)B_1, ..., -t(s)N_3)$ then $frac{d}{ds}x(s)=f(s,x(s))$ together with existence of $L$ (Lipschitz const) implies that $text{max}_{s in [s_0, s_1]} (k^2(N_1-N'_2)^2+...+(-tN_3+tN'_3)^2) le L^2 text{max}_{s in [s_0, s_1]} (T_1-T'_2)^2+...+(B_3-B'_3)^2)$. But for the smooth functions of $T$, $N$, $B$ in the (differential geo) text (image in OP) how the existence of $L$ is proven/guaranteed? ( ' means y instead of x , not for derivative)
– 72D
Nov 17 at 10:50
|
show 4 more comments
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Picard-Lindelöf theorem
– achille hui
Nov 16 at 11:51
@achillehui, there are three functions in not-3-independent-equations) to be solved not one.
– 72D
Nov 16 at 12:08
The $y$ in the theorem can be vector, the wiki book entry for the theorem spell this out explicitly.
– achille hui
Nov 16 at 12:22
@achillehui, I started to learn the one-dim case from an easy textbook now so I hopefully can understand the one in wikibook... Thanks.
– 72D
Nov 16 at 12:52
@achillehui, I understood all thing but one! Let $x(s)=(T_1(s), T_2(s), T_3(s), N_1, ..., B_1, ...)$ and $f(s,x(s))=(k(s)N_1(s), ..., -k(s)T_1+t(s)B_1, ..., -t(s)N_3)$ then $frac{d}{ds}x(s)=f(s,x(s))$ together with existence of $L$ (Lipschitz const) implies that $text{max}_{s in [s_0, s_1]} (k^2(N_1-N'_2)^2+...+(-tN_3+tN'_3)^2) le L^2 text{max}_{s in [s_0, s_1]} (T_1-T'_2)^2+...+(B_3-B'_3)^2)$. But for the smooth functions of $T$, $N$, $B$ in the (differential geo) text (image in OP) how the existence of $L$ is proven/guaranteed? ( ' means y instead of x , not for derivative)
– 72D
Nov 17 at 10:50