Given a Lagrangian function L=L(t,q,v) and a system of n second-order Lagrange equations (I) (Lv(t,q,q′))′=Lq(t,q,q′)=0 (where ′ denotes the derivative with respect to t), it is possible, provided some conditions are satisfied, to introduce the conjugate momentum p as p=Lv(t,q,v) and the Hamiltonian function H(t,q,p) as the Legendre transform of L and rewrite (I) as a system of 2n first-order equations, called Hamilton's equations: (II) q′=Hp(t,q,p), p′=−Hq(t,q,p). The passage from Lagrange's to Hamilton's equations is the object of the paper. Specifically, our aim is to find minimal assumptions on the Lagrangian L guaranteeing that the associated Hamiltonian is a single-valued function on the domain of conjugate momentum and the Hamiltonian system is well defined. We prove it if (a) the function L=L(t,q,v) is defined and of class C1 in a domain I×Ω, where I is an open set of R1+n and Ω is an open the of Rn; (b) the function L(t,q,⋅) satisfies for every fixed (t,q)∈I the following strict convexity condition: L(t,q,v1)−L(t,q,v2)>⟨Lv(t,q,v2),v1−v2⟩ for all v1,v2∈Ω, v1≠v2. With respect to the assumptions customarily postulated in mechanics books, (a) and (b) above are much weaker; moreover, they yield a global (rather than local) result. However, as we remark, if only (a) and (b) are postulated, solutions of the initial value problem for either Lagrange's or Hamilton's equations need not be unique.

On the derivation of Hamilton's equations

PUCCI, Patrizia;
1994

Abstract

Given a Lagrangian function L=L(t,q,v) and a system of n second-order Lagrange equations (I) (Lv(t,q,q′))′=Lq(t,q,q′)=0 (where ′ denotes the derivative with respect to t), it is possible, provided some conditions are satisfied, to introduce the conjugate momentum p as p=Lv(t,q,v) and the Hamiltonian function H(t,q,p) as the Legendre transform of L and rewrite (I) as a system of 2n first-order equations, called Hamilton's equations: (II) q′=Hp(t,q,p), p′=−Hq(t,q,p). The passage from Lagrange's to Hamilton's equations is the object of the paper. Specifically, our aim is to find minimal assumptions on the Lagrangian L guaranteeing that the associated Hamiltonian is a single-valued function on the domain of conjugate momentum and the Hamiltonian system is well defined. We prove it if (a) the function L=L(t,q,v) is defined and of class C1 in a domain I×Ω, where I is an open set of R1+n and Ω is an open the of Rn; (b) the function L(t,q,⋅) satisfies for every fixed (t,q)∈I the following strict convexity condition: L(t,q,v1)−L(t,q,v2)>⟨Lv(t,q,v2),v1−v2⟩ for all v1,v2∈Ω, v1≠v2. With respect to the assumptions customarily postulated in mechanics books, (a) and (b) above are much weaker; moreover, they yield a global (rather than local) result. However, as we remark, if only (a) and (b) are postulated, solutions of the initial value problem for either Lagrange's or Hamilton's equations need not be unique.
1994
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/117197
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