What does it mean to say a 'double' Legendre transform?
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Ref. 1 mentons that you can achieve the momentum space Lagrangian by doing a so called double Legendre transform. It goes on to write:
K(p,p˙,t) = L(q,q˙,t)−pq˙−qp˙,(5.63)(5.63)K(p,p˙,t) = L(q,q˙,t)−pq˙−qp˙,
where K(p,p˙,t)K(p,p˙,t) is the momentum space Lagrangian. I don't see how this is a Legendre transform as there seems to be a sign error in the way the Legendre transform has been done. In general when we say a Legendre transform we mean H(p,q,t)=pq˙−L(q,q˙,t)H(p,q,t)=pq˙−L(q,q˙,t), so according to that for our momentum space Lagrangian it should be K(p,p˙,t)=pq˙+qp˙−L(q,q˙,t)K(p,p˙,t)=pq˙+qp˙−L(q,q˙,t) but that doesn't seem to be the case. Can someone point me in the right direction because the sign matters when we try to get different forms of the generating functions.
K(p,p˙,t) = L(q,q˙,t)−pq˙−qp˙,(5.63)(5.63)K(p,p˙,t) = L(q,q˙,t)−pq˙−qp˙,
where K(p,p˙,t)K(p,p˙,t) is the momentum space Lagrangian. I don't see how this is a Legendre transform as there seems to be a sign error in the way the Legendre transform has been done. In general when we say a Legendre transform we mean H(p,q,t)=pq˙−L(q,q˙,t)H(p,q,t)=pq˙−L(q,q˙,t), so according to that for our momentum space Lagrangian it should be K(p,p˙,t)=pq˙+qp˙−L(q,q˙,t)K(p,p˙,t)=pq˙+qp˙−L(q,q˙,t) but that doesn't seem to be the case. Can someone point me in the right direction because the sign matters when we try to get different forms of the generating functions.
Answered by
0
Ref. 1 mentons that you can achieve the momentum space Lagrangian by doing a so called double Legendre transform. It goes on to write:
K(p,p˙,t) = L(q,q˙,t)−pq˙−qp˙,(5.63)(5.63)K(p,p˙,t) = L(q,q˙,t)−pq˙−qp˙,
where K(p,p˙,t)K(p,p˙,t) is the momentum space Lagrangian. I don't see how this is a Legendre transform as there seems to be a sign error in the way the Legendre transform has been done. In general when we say a Legendre transform we mean H(p,q,t)=pq˙−L(q,q˙,t)H(p,q,t)=pq˙−L(q,q˙,t), so according to that for our momentum space Lagrangian it should be K(p,p˙,t)=pq˙+qp˙−L(q,q˙,t)K(p,p˙,t)=pq˙+qp˙−L(q,q˙,t) but that doesn't seem to be the case. Can someone point me in the right direction because the sign matters when we try to get different forms of the generating functions.
References:
Hand & Finch, Analytical mechanics, Ch 5, pg 190, eq. (5.63).
K(p,p˙,t) = L(q,q˙,t)−pq˙−qp˙,(5.63)(5.63)K(p,p˙,t) = L(q,q˙,t)−pq˙−qp˙,
where K(p,p˙,t)K(p,p˙,t) is the momentum space Lagrangian. I don't see how this is a Legendre transform as there seems to be a sign error in the way the Legendre transform has been done. In general when we say a Legendre transform we mean H(p,q,t)=pq˙−L(q,q˙,t)H(p,q,t)=pq˙−L(q,q˙,t), so according to that for our momentum space Lagrangian it should be K(p,p˙,t)=pq˙+qp˙−L(q,q˙,t)K(p,p˙,t)=pq˙+qp˙−L(q,q˙,t) but that doesn't seem to be the case. Can someone point me in the right direction because the sign matters when we try to get different forms of the generating functions.
References:
Hand & Finch, Analytical mechanics, Ch 5, pg 190, eq. (5.63).
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