what are the hamilton equations of motion for homogeneous lagrangians?by pellman Tags: equations, hamilton, homogeneous, lagrangians, motion 

#1
Feb2113, 07:55 PM

P: 565

For a Lagrangian [itex]L(x^k,\dot{x}^k)[/itex] which is homogeneous in the [itex]\dot{x}^k[/itex] in the first degree, the usual Hamiltonian vanishes identically. Instead an alternative conjugate momenta is defined as
[itex]y_j=L\frac{\partial L}{\partial \dot{x}^j}[/itex] which can then be inverted to give the velocities as a function of the position and momenta [itex]\dot{x}^i=\phi^{i}(x^k,y_k)[/itex] The Hamiltonian is then equal to the Lagrangian with the velocities replaced with this function [itex]H(x^k,y_k)=L(x^k,\phi^{k}(x^l,y_l))[/itex] We then find that [itex]\dot{x}^i=H\frac{\partial H}{\partial y_i}[/itex] which is one half of the Hamilton equations of motion. But what about [itex]\dot{y}_i[/itex]? I am following Hanno Rund The HamiltonJacobi equation in the Calculus of Variations. But Rund moves on from this point to the HJ equation, leaving me wondering about this question. 



#2
Feb2213, 05:21 AM

P: 565

I found the answer to this



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