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Hamilton's equations have another advantage over Lagrange's equations: if a system has a symmetry, so that some coordinate does not occur in the Hamiltonian (i.e. a cyclic coordinate), the corresponding momentum coordinate is conserved along each trajectory, and that coordinate can be reduced to a constant in the other equations of the set.
Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q 1, q 2, ..., q N) between two specified states q 1 = q(t 1) and q 2 = q(t 2) at two specified times t 1 and t 2 is a stationary point (a point where the variation is zero) of the action functional [] = ((), ˙ (),) where (, ˙,) is the Lagrangian function for the system.
2.4 Euler–Lagrange equations and Hamilton's principle. 2.5 Lagrange multipliers and constraints. ... With these definitions, Lagrange's equations of the first kind ...
Starting with Hamilton's principle, the local differential Euler–Lagrange equation can be derived for systems of fixed energy. The action S {\displaystyle S} in Hamilton's principle is the Legendre transformation of the action in Maupertuis' principle.
The general results presented above for Hamilton's principle can be applied to optics using the Lagrangian defined in Fermat's principle.The Euler-Lagrange equations with parameter σ =x 3 and N=2 applied to Fermat's principle result in ˙ = with k = 1, 2 and where L is the optical Lagrangian and ˙ = /.
The formulation of Lagrange's equations for this system yields six equations in the four Cartesian coordinates x i, y i (i = 1, 2) and the two Lagrange multipliers λ i (i = 1, 2) that arise from the two constraint equations.
The Euler–Lagrange equation was developed in connection with their studies of the tautochrone problem. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in ...
Covariant Hamilton equations are equivalent to the Euler–Lagrange equations in the case of hyperregular Lagrangians. Covariant Hamiltonian field theory is developed in the Hamilton–De Donder, [4] polysymplectic, [5] multisymplectic [6] and k-symplectic [7] variants.