In classical mechanics, Routh's procedure or Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh, who formulated it about 1860. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions. Although Routhian mechanics is equivalent to Lagrangian mechanics and Hamiltonian mechanics, and introduces no new physics, it offers an alternative way to solve mechanical problems.
Definitions 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. In the case of Lagrangian mechanics, the generalized coordinates q1, q2, ... and the corresponding velocities dq1/dt, dq2/dt, ..., and possibly time t, enter the Lagrangian,
L ( q 1 , q 2 , … , q ˙ 1 , q ˙ 2 , … , t ) , q ˙ i = d q i d t , {\displaystyle L(q_{1},q_{2},\ldots ,{\dot {q}}_{1},{\dot {q}}_{2},\ldots ,t)\,,\quad {\dot {q}}_{i}={\frac {dq_{i}}{dt}}\,,}
where the overdots denote time derivatives. In Hamiltonian mechanics, the generalized coordinates q1, q2, ... and the corresponding generalized momenta p1, p2, ..., and possibly time, enter the Hamiltonian,
H ( q 1 , q 2 , … , p 1 , p 2 , … , t ) = ∑ i q ˙ i p i − L ( q 1 , q 2 , … , q ˙ 1 ( p 1 ) , q ˙ 2 ( p 2 ) , … , t ) , p i = ∂ L ∂ q ˙ i , {\displaystyle H(q_{1},q_{2},\ldots ,p_{1},p_{2},\ldots ,t)=\sum _{i}{\dot {q}}_{i}p_{i}-L(q_{1},q_{2},\ldots ,{\dot {q}}_{1}(p_{1}),{\dot {q}}_{2}(p_{2}),\ldots ,t)\,,\quad p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}\,,}
where the second equation is the definition of the generalized momentum pi corresponding to the coordinate qi (partial derivatives are denoted using ∂). The velocities dqi/dt are expressed as functions of their corresponding momenta by inverting their defining relation. In this context, pi is said to be the momentum "canonically conjugate" to qi. The Routhian is intermediate between L and H; some coordinates q1, q2, ..., qn are chosen to have corresponding generalized momenta p1, p2, ..., pn, the rest of the coordinates ζ1, ζ2, ..., ζs to have generalized velocities dζ1/dt, dζ2/dt, ..., dζs/dt, and time may appear explicitly;
where again the generalized velocity dqi/dt is to be expressed as a function of generalized momentum pi via its defining relation. The choice of which n coordinates are to have corresponding momenta, out of the n + s coordinates, is arbitrary. The above is used by Landau and Lifshitz, and Goldstein. Some authors may define the Routhian to be the negative of the above definition. Given the length of the general definition, a more compact notation is to use boldface for tuples (or vectors) of the variables, thus q = (q1, q2, ..., qn), ζ = (ζ1, ζ2, ..., ζs), p = (p1, p2, ..., pn), and d ζ/dt = (dζ1/dt, dζ2/dt, ..., dζs/dt), so that
R ( q , ζ , p , ζ ˙ , t ) = p ⋅ q ˙ − L ( q , ζ , q ˙ , ζ ˙ , t ) , {\displaystyle R(\mathbf {q} ,{\boldsymbol {\zeta }},\mathbf {p} ,{\dot {\boldsymbol {\zeta }}},t)=\mathbf {p} \cdot {\dot {\mathbf {q} }}-L(\mathbf {q} ,{\boldsymbol {\zeta }},{\dot {\mathbf {q} }},{\dot {\boldsymbol {\zeta }}},t)\,,}
where · is the dot product defined on the tuples, for the specific example appearing here:
p ⋅ q ˙ = ∑ i = 1 n p i q ˙ i . {\displaystyle \mathbf {p} \cdot {\dot {\mathbf {q} }}=\sum _{i=1}^{n}p_{i}{\dot {q}}_{i}\,.}
Equations of motion For reference, the Euler-Lagrange equations for s degrees of freedom are a set of s coupled second order ordinary differential equations in the coordinates
d d t ∂ L ∂ q ˙ j = ∂ L ∂ q j , {\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}={\frac {\partial L}{\partial q_{j}}}\,,}
where j = 1, 2, ..., s, and the Hamiltonian equations for n degrees of freedom are a set of 2n coupled first order ordinary differential equations in the coordinates and momenta
q ˙ i = ∂ H ∂ p i , p ˙ i = − ∂ H ∂ q i . {\displaystyle {\dot {q}}_{i}={\frac {\partial H}{\partial p_{i}}}\,,\quad {\dot {p}}_{i}=-{\frac {\partial H}{\partial q_{i}}}\,.}
Below, the Routhian equations of motion are obtained in two ways, in the process other useful derivatives are found that can be used elsewhere.
Two degrees of freedom Consider the case of a system with two degrees of freedom, q and ζ, with generalized velocities dq/dt and dζ/dt, and the Lagrangian is time-dependent. (The generalization to any number of degrees of freedom follows exactly the same procedure as with two). The Lagrangian of the system will have the form
L ( q , ζ , q ˙ , ζ ˙ , t ) {\displaystyle L(q,\zeta ,{\dot {q}},{\dot {\zeta }},t)}
The differential of L is
d L = ∂ L ∂ q d q + ∂ L ∂ ζ d ζ + ∂ L ∂ q ˙ d q ˙ + ∂ L ∂ ζ ˙ d ζ ˙ + ∂ L ∂ t d t . {\displaystyle dL={\frac {\partial L}{\partial q}}dq+{\frac {\partial L}{\partial \zeta }}d\zeta +{\frac {\partial L}{\partial {\dot {q}}}}d{\dot {q}}+{\frac {\partial L}{\partial {\dot {\zeta }}}}d{\dot {\zeta }}+{\frac {\partial L}{\partial t}}dt\,.}
Now change variables, from the set (q, ζ, dq/dt, dζ/dt) to (q, ζ, p, dζ/dt), simply switching the velocity dq/dt to the momentum p. This change of variables in the differentials is the Legendre transformation. The differential of the new function to replace L will be a sum of differentials in dq, dζ, dp, d(dζ/dt), and dt. Using the definition of generalized momentum and Lagrange's equation for the coordinate q:
p = ∂ L ∂ q ˙ , p ˙ = d d t ∂ L ∂ q ˙ = ∂ L ∂ q {\displaystyle p={\frac {\partial L}{\partial {\dot {q}}}}\,,\quad {\dot {p}}={\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}}}={\frac {\partial L}{\partial q}}}
we have
d L = p ˙ d q + ∂ L ∂ ζ d ζ + p d q ˙ + ∂ L ∂ ζ ˙ d ζ ˙ + ∂ L ∂ t d t {\displaystyle dL={\dot {p}}dq+{\frac {\partial L}{\partial \zeta }}d\zeta +pd{\dot {q}}+{\frac {\partial L}{\partial {\dot {\zeta }}}}d{\dot {\zeta }}+{\frac {\partial L}{\partial t}}dt}
and to replace pd(dq/dt) by (dq/dt)dp, recall the product rule for differentials, and substitute
p d q ˙ = d ( q ˙ p ) − q ˙ d p {\displaystyle pd{\dot {q}}=d({\dot {q}}p)-{\dot {q}}dp}
to obtain the differential of a new function in terms of the new set of variables:
d ( L − p q ˙ ) = p ˙ d q + ∂ L ∂ ζ d ζ − q ˙ d p + ∂ L ∂ ζ ˙ d ζ ˙ + ∂ L ∂ t d t . {\displaystyle d(L-p{\dot {q}})={\dot {p}}dq+{\frac {\partial L}{\partial \zeta }}d\zeta -{\dot {q}}dp+{\frac {\partial L}{\partial {\dot {\zeta }}}}d{\dot {\zeta }}+{\frac {\partial L}{\partial t}}dt\,.}
Introducing the Routhian
R ( q , ζ , p , ζ ˙ , t ) = p q ˙ ( p ) − L {\displaystyle R(q,\zeta ,p,{\dot {\zeta }},t)=p{\dot {q}}(p)-L}
where again the velocity dq/dt is a function of the momentum p, we have
d R = − p ˙ d q − ∂ L ∂ ζ d ζ + q ˙ d p − ∂ L ∂ ζ ˙ d ζ ˙ − ∂ L ∂ t d t , {\displaystyle dR=-{\dot {p}}dq-{\frac {\partial L}{\partial \zeta }}d\zeta +{\dot {q}}dp-{\frac {\partial L}{\partial {\dot {\zeta }}}}d{\dot {\zeta }}-{\frac {\partial L}{\partial t}}dt\,,}
but from the above definition, the differential of the Routhian is
d R = ∂ R ∂ q d q + ∂ R ∂ ζ d ζ + ∂ R ∂ p d p + ∂ R ∂ ζ ˙ d ζ ˙ + ∂ R ∂ t d t . {\displaystyle dR={\frac {\partial R}{\partial q}}dq+{\frac {\partial R}{\partial \zeta }}d\zeta +{\frac {\partial R}{\partial p}}dp+{\frac {\partial R}{\partial {\dot {\zeta }}}}d{\dot {\zeta }}+{\frac {\partial R}{\partial t}}dt\,.}
Comparing the coefficients of the differentials dq, dζ, dp, d(dζ/dt), and dt, the results are Hamilton's equations for the coordinate q,
q ˙ = ∂ R ∂ p , p ˙ = − ∂ R ∂ q , {\displaystyle {\dot {q}}={\frac {\partial R}{\partial p}}\,,\quad {\dot {p}}=-{\frac {\partial R}{\partial q}}\,,}
and Lagrange's equation for the coordinate ζ
d d t ∂ R ∂ ζ ˙ = ∂ R ∂ ζ {\displaystyle {\frac {d}{dt}}{\frac {\partial R}{\partial {\dot {\zeta }}}}={\frac {\partial R}{\partial \zeta }}}
which follow from
∂ L ∂ ζ = − ∂ R ∂ ζ , ∂ L ∂ ζ ˙ = − ∂ R ∂ ζ ˙ , {\displaystyle {\frac {\partial L}{\partial \zeta }}=-{\frac {\partial R}{\partial \zeta }}\,,\quad {\frac {\partial L}{\partial {\dot {\zeta }}}}=-{\frac {\partial R}{\partial {\dot {\zeta }}}}\,,}
and taking the total time derivative of the second equation and equating to the first. Notice the Routhian replaces the Hamiltonian and Lagrangian functions in all the equations of motion. The remaining equation states the partial time derivatives of L and R are negatives
∂ L ∂ t = − ∂ R ∂ t . {\displaystyle {\frac {\partial L}{\partial t}}=-{\frac {\partial R}{\partial t}}\,.}
Any number of degrees of freedom For n + s coordinates as defined above, with Routhian
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 {\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}
the equations of motion can be derived by a Legendre transformation of this Routhian as in the previous section, but another way is to simply take the partial derivatives of R with respect to the coordinates qi and ζj, momenta pi, and velocities dζj/dt, where i = 1, 2, ..., n, and j = 1, 2, ..., s. The derivatives are
∂ R ∂ q i = − ∂ L ∂ q i = − d d t ∂ L ∂ q ˙ i = − p ˙ i {\displaystyle {\frac {\partial R}{\partial q_{i}}}=-{\frac {\partial L}{\partial q_{i}}}=-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}=-{\dot {p}}_{i}}
∂ R ∂ p i = q ˙ i {\displaystyle {\frac {\partial R}{\partial p_{i}}}={\dot {q}}_{i}}
∂ R ∂ ζ j = − ∂ L ∂ ζ j , {\displaystyle {\frac {\partial R}{\partial \zeta _{j}}}=-{\frac {\partial L}{\partial \zeta _{j}}}\,,}
∂ R ∂ ζ ˙ j = − ∂ L ∂ ζ ˙ j , {\displaystyle {\frac {\partial R}{\partial {\dot {\zeta }}_{j}}}=-{\frac {\partial L}{\partial {\dot {\zeta }}_{j}}}\,,}
∂ R ∂ t = − ∂ L ∂ t . {\displaystyle {\frac {\partial R}{\partial t}}=-{\frac {\partial L}{\partial t}}\,.}
The first two are identically the Hamiltonian equations. Equating the total time derivative of the fourth set of equations with the third (for each value of j) gives the Lagrangian equations. The fifth is just the same relation between time partial derivatives as before. To summarize
The total number of equations is 2n + s, there are 2n Hamiltonian equations plus s Lagrange equations.
Energy Since the Lagrangian has the same units as energy, the units of the Routhian are also energy. In SI units this is the Joule. Taking the total time derivative of the Lagrangian leads to the general result
∂ L ∂ t = d d t ( ∑ i = 1 n q ˙ i ∂ L ∂ q ˙ i + ∑ j = 1 s ζ ˙ j ∂ L ∂ ζ ˙ j − L ) . {\displaystyle {\frac {\partial L}{\partial t}}={\frac {d}{dt}}\left(\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}+\sum _{j=1}^{s}{\dot {\zeta }}_{j}{\frac {\partial L}{\partial {\dot {\zeta }}_{j}}}-L\right)\,.}
If the Lagrangian is independent of time, the partial time derivative of the Lagrangian is zero, ∂L/∂t = 0, so the quantity under the total time derivative in brackets must be a constant, it is the total energy of the system
E = ∑ i = 1 n q ˙ i ∂ L ∂ q ˙ i + ∑ j = 1 s ζ ˙ j ∂ L ∂ ζ ˙ j − L . {\displaystyle E=\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}+\sum _{j=1}^{s}{\dot {\zeta }}_{j}{\frac {\partial L}{\partial {\dot {\zeta }}_{j}}}-L\,.}
(If there are external fields interacting with the constituents of the system, they can vary throughout space but not time). This expression requires the partial derivatives of L with respect to all the velocities dqi/dt and dζj/dt. Under the same condition of R being time independent, the energy in terms of the Routhian is a little simpler, substituting the definition of R and the partial derivatives of R with respect to the velocities dζj/dt,
E = R − ∑ j = 1 s ζ ˙ j ∂ R ∂ ζ ˙ j . {\displaystyle E=R-\sum _{j=1}^{s}{\dot {\zeta }}_{j}{\frac {\partial R}{\partial {\dot {\zeta }}_{j}}}\,.}
Notice only the partial derivatives of R with respect to the velocities dζj/dt are needed. In the case that s = 0 and the Routhian is explicitly time-independent, then E = R, that is, the Routhian equals the energy of the system. The same expression for R in when s = 0 is also the Hamiltonian, so in all E = R = H. If the Routhian has explicit time dependence, the total energy of the system is not constant. The general result is
∂ R ∂ t = d d t ( R − ∑ j = 1 s ζ ˙ j ∂ R ∂ ζ ˙ j ) , {\displaystyle {\frac {\partial R}{\partial t}}={\dfrac {d}{dt}}\left(R-\sum _{j=1}^{s}{\dot {\zeta }}_{j}{\frac {\partial R}{\partial {\dot {\zeta }}_{j}}}\righ