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Principle of least action and Euler-Lagrange equation
Clash Royale CLAN TAG#URR8PPP
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Define the action
$S[g]=displaystylefrac12int^1_0 Tr(I(g^-1dot g)~g^-1dot g)~dt.$
$I:SO(N)to SO(N)$ denotes the endomorphism $omega to I(omega)$ with $I(omega)_ij=omega_ij/F_ij$
$g:[0,1]to SO(N)$
How to use the principle of least action or Euler-Lagrange equation to derive something like this (A differential equation with Lie bracket):
$dot A=[A,B],~~dot g =gB,$ where $B_ij=F_ijA_ij$ and $F$ is a symmetric matrix with strictly positive entries. (Actually, this is not the true result.)
functional-analysis mathematical-physics calculus-of-variations classical-mechanics
asked Jul 24 at 20:39
learner
29916
add a comment |Â
up vote
1
down vote
favorite
Define the action
$S[g]=displaystylefrac12int^1_0 Tr(I(g^-1dot g)~g^-1dot g)~dt.$
$I:SO(N)to SO(N)$ denotes the endomorphism $omega to I(omega)$ with $I(omega)_ij=omega_ij/F_ij$
$g:[0,1]to SO(N)$
How to use the principle of least action or Euler-Lagrange equation to derive something like this (A differential equation with Lie bracket):
$dot A=[A,B],~~dot g =gB,$ where $B_ij=F_ijA_ij$ and $F$ is a symmetric matrix with strictly positive entries. (Actually, this is not the true result.)
functional-analysis mathematical-physics calculus-of-variations classical-mechanics
asked Jul 24 at 20:39
learner
29916
Where does this originate from?
– mvw
Jul 24 at 20:48
@mvw It is from a paper that my professor assign me to read
– learner
Jul 24 at 21:07
Crossposted from physics.stackexchange.com/q/419302/2451
– Qmechanic
Jul 25 at 8:25
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Define the action
$S[g]=displaystylefrac12int^1_0 Tr(I(g^-1dot g)~g^-1dot g)~dt.$
$I:SO(N)to SO(N)$ denotes the endomorphism $omega to I(omega)$ with $I(omega)_ij=omega_ij/F_ij$
$g:[0,1]to SO(N)$
How to use the principle of least action or Euler-Lagrange equation to derive something like this (A differential equation with Lie bracket):
$dot A=[A,B],~~dot g =gB,$ where $B_ij=F_ijA_ij$ and $F$ is a symmetric matrix with strictly positive entries. (Actually, this is not the true result.)
functional-analysis mathematical-physics calculus-of-variations classical-mechanics
asked Jul 24 at 20:39
learner
29916
Define the action
$S[g]=displaystylefrac12int^1_0 Tr(I(g^-1dot g)~g^-1dot g)~dt.$
$I:SO(N)to SO(N)$ denotes the endomorphism $omega to I(omega)$ with $I(omega)_ij=omega_ij/F_ij$
$g:[0,1]to SO(N)$
How to use the principle of least action or Euler-Lagrange equation to derive something like this (A differential equation with Lie bracket):
$dot A=[A,B],~~dot g =gB,$ where $B_ij=F_ijA_ij$ and $F$ is a symmetric matrix with strictly positive entries. (Actually, this is not the true result.)
functional-analysis mathematical-physics calculus-of-variations classical-mechanics
asked Jul 24 at 20:39
learner
29916
asked Jul 24 at 20:39
learner
29916
asked Jul 24 at 20:39
learner
29916
asked Jul 24 at 20:39
learner
29916
29916
Where does this originate from?
– mvw
Jul 24 at 20:48
@mvw It is from a paper that my professor assign me to read
– learner
Jul 24 at 21:07
Crossposted from physics.stackexchange.com/q/419302/2451
– Qmechanic
Jul 25 at 8:25
add a comment |Â
Where does this originate from?
– mvw
Jul 24 at 20:48
@mvw It is from a paper that my professor assign me to read
– learner
Jul 24 at 21:07
Crossposted from physics.stackexchange.com/q/419302/2451
– Qmechanic
Jul 25 at 8:25
Where does this originate from?
– mvw
Jul 24 at 20:48
Where does this originate from?
– mvw
Jul 24 at 20:48
@mvw It is from a paper that my professor assign me to read
– learner
Jul 24 at 21:07
@mvw It is from a paper that my professor assign me to read
– learner
Jul 24 at 21:07
Crossposted from physics.stackexchange.com/q/419302/2451
– Qmechanic
Jul 25 at 8:25
Crossposted from physics.stackexchange.com/q/419302/2451
– Qmechanic
Jul 25 at 8:25
add a comment |Â
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Where does this originate from?
– mvw
Jul 24 at 20:48
@mvw It is from a paper that my professor assign me to read
– learner
Jul 24 at 21:07
Crossposted from physics.stackexchange.com/q/419302/2451
– Qmechanic
Jul 25 at 8:25