Routhian

In analytical mechanics, a branch of theoretical physics, Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions. R ( q 1 , … , q n , ζ 1 , … , ζ s , p 1 , … , p n , ζ ˙ 1 , … , ζ ˙ s , t ) = ∑ i = 1 n p i q ˙ i ( p i ) − L ( q 1 , … , q n , ζ 1 , … , ζ s , q ˙ 1 ( p 1 ) , … , q ˙ n ( p n ) , ζ ˙ 1 , … , ζ ˙ s , t ) , {displaystyle R(q_{1},ldots ,q_{n},zeta _{1},ldots ,zeta _{s},p_{1},ldots ,p_{n},{dot {zeta }}_{1},ldots ,{dot {zeta }}_{s},t)=sum _{i=1}^{n}p_{i}{dot {q}}_{i}(p_{i})-L(q_{1},ldots ,q_{n},zeta _{1},ldots ,zeta _{s},{dot {q}}_{1}(p_{1}),ldots ,{dot {q}}_{n}(p_{n}),{dot {zeta }}_{1},ldots ,{dot {zeta }}_{s},t),,} q ˙ i = ∂ R ∂ p i , p ˙ i = − ∂ R ∂ q i , {displaystyle {dot {q}}_{i}={frac {partial R}{partial p_{i}}},,quad {dot {p}}_{i}=-{frac {partial R}{partial q_{i}}},,} In analytical mechanics, a branch of theoretical physics, Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions. The Routhian, like the Hamiltonian, can be obtained from a Legendre transform of the Lagrangian, and has a similar mathematical form to the Hamiltonian, but is not exactly the same. The difference between the Lagrangian, Hamiltonian, and Routhian functions are their variables. For a given set of generalized coordinates representing the degrees of freedom in the system, the Lagrangian is a function of the coordinates and velocities, while the Hamiltonian is a function of the coordinates and momenta. The Routhian differs from these functions in that some coordinates are chosen to have corresponding generalized velocities, the rest to have corresponding generalized momenta. This choice is arbitrary, and can be done to simplify the problem. It also has the consequence that the Routhian equations are exactly the Hamiltonian equations for some coordinates and corresponding momenta, and the Lagrangian equations for the rest of the coordinates and their velocities. In each case the Lagrangian and Hamiltonian functions are replaced by a single function, the Routhian. The full set thus has the advantages of both sets of equations, with the convenience of splitting one set of coordinates to the Hamilton equations, and the rest to the Lagrangian equations. Often the Routhian approach may offer no advantage, but one notable case where this is useful is when a system has cyclic coordinates (also called 'ignorable coordinates'), by definition those coordinates which do not appear in the original Lagrangian. The Lagrangian equations are powerful results, used frequently in theory and practice, since the equations of motion in the coordinates are easy to set up. However, if cyclic coordinates occur there will still be equations to solve for all the coordinates, including the cyclic coordinates despite their absence in the Lagrangian. The Hamiltonian equations are useful theoretical results, but less useful in practice because coordinates and momenta are related together in the solutions - after solving the equations the coordinates and momenta must be eliminated from each other. Nevertheless, the Hamiltonian equations are perfectly suited to cyclic coordinates because the equations in the cyclic coordinates trivially vanish, leaving only the equations in the non cyclic coordinates.