Back to index/Generated with Clerk from src/emmy/mechanics/routhian.cljc@f807468
(ns emmy.mechanics.routhian
(:refer-clojure :exclude [+ - * / partial])
(:require [emmy.calculus.derivative :refer [D partial]]
[emmy.generic :as g :refer [+ - *]]
[emmy.matrix :as m]
[emmy.mechanics.hamilton :as h]
[emmy.mechanics.lagrange :as l]
[emmy.structure :refer [up]]))

Routhian equations of motion

From the 2nd edition of SICM, p.233.

Assume a Lagrangian of the form L(t; x, y; vx, vy), where x and y may have substructure.

We perform a Legendre transform on vy to get the Routhian.

The equations of motion are Hamilton's equations for the py, y and Lagrange's equations for vx, x.

(defn Lagrangian->Routhian [Lagrangian]
(fn [[t q [vx py]]]
(letfn [(L [vy]
(Lagrangian (up t q (up vx vy))))]
((h/Legendre-transform-procedure L) py))))
#object[emmy.mechanics.routhian$Lagrangian__GT_Routhian 0x4c455ca8 "
emmy.mechanics.routhian$Lagrangian__GT_Routhian@4c455ca8"
]
(defn Routh-equations [Routhian]
(fn [x y py]
(fn [t]
(letfn [(L [[tau q v]]
(Routhian
(up tau (up q (y tau)) (up v (py tau)))))
(H [[tau q p]]
(Routhian
(up tau (up (x tau) q) (up ((D x) tau) p))))]
(up
(((l/Lagrange-equations L) x) t)
(((h/Hamilton-equations H) y py) t))))))
#object[emmy.mechanics.routhian$Routh_equations 0xf44c428 "
emmy.mechanics.routhian$Routh_equations@f44c428"
]
(defn Routhian->acceleration
([R]
(fn [[_ _ [vx] :as s]]
(let [minus-P ((partial 2 0) R)
minus-F (((partial 1 0) R) s)
vy (((partial 2 1) R) s)
pyd ((* -1 ((partial 1 1) R)) s)]
(* (m/s:inverse vx
(((partial 2 0) minus-P) s)
vx)
(- minus-F
(+ (((partial 0) minus-P) s)
(* (((partial 1 0) minus-P) s) vx)
(* (((partial 1 1) minus-P) s) vy)
(* (((partial 2 1) minus-P) s) pyd)))))))
([R dissipation-fn]
(fn [[t q [vx] :as s]]
(let [minus-P ((partial 2 0) R)
minus-F (((partial 1 0) R) s)
vy (((partial 2 1) R) s)
L-state (up t q (up vx vy))
minus-F0 (((partial 2 0) dissipation-fn) L-state)
minus-F1 (((partial 2 1) dissipation-fn) L-state)
pyd (- ((* -1 ((partial 1 1) R)) s)
minus-F1)]
(* (m/s:inverse vx
(((partial 2 0) minus-P) s)
vx)
(+ (- minus-F
(+ (((partial 0) minus-P) s)
(* (((partial 1 0) minus-P) s) vx)
(* (((partial 1 1) minus-P) s) vy)
(* (((partial 2 1) minus-P) s) pyd)))
minus-F0))))))
#object[emmy.mechanics.routhian$Routhian__GT_acceleration 0x36fcff47 "
emmy.mechanics.routhian$Routhian__GT_acceleration@36fcff47"
]
(defn Routhian->state-derivative
([R]
(fn [[_ _ [vx] :as s]]
(let [minus-P ((partial 2 0) R)
minus-F (((partial 1 0) R) s)
vy (((partial 2 1) R) s)
pyd (- (((partial 1 1) R) s))]
(up 1
(up vx vy)
(up
(* (m/s:inverse vx (((partial 2 0) minus-P) s) vx)
(- minus-F
(+ (((partial 0) minus-P) s)
(* (((partial 1 0) minus-P) s) vx)
(* (((partial 1 1) minus-P) s) vy)
(* (((partial 2 1) minus-P) s) pyd))))
pyd)))))
([R dissipation-fn]
(fn [[t q [vx] :as s]]
(let [minus-P ((partial 2 0) R)
minus-F (((partial 1 0) R) s)
vy (((partial 2 1) R) s)
L-state (up t q (up vx vy))
minus-F0 (((partial 2 0) dissipation-fn) L-state)
minus-F1 (((partial 2 1) dissipation-fn) L-state)
pyd (- ((* -1 ((partial 1 1) R)) s) minus-F1)]
(up 1
(up vx vy)
(up
(* (m/s:inverse vx (((partial 2 0) minus-P) s) vx)
(+ (- minus-F
(+ (((partial 0) minus-P) s)
(* (((partial 1 0) minus-P) s) vx)
(* (((partial 1 1) minus-P) s) vy)
(* (((partial 2 1) minus-P) s) pyd)))
minus-F0))
pyd))))))
#object[emmy.mechanics.routhian$Routhian__GT_state_derivative 0x75c41a13 "
emmy.mechanics.routhian$Routhian__GT_state_derivative@75c41a13"
]
(defn Lagrangian-state->Routhian-state [L]
(fn [[t q [vx] :as s]]
(let [py (-> (((partial 2) L) s)
(nth 1))]
(up t q (up vx py)))))
#object[emmy.mechanics.routhian$Lagrangian_state__GT_Routhian_state 0x1382b81f "
emmy.mechanics.routhian$Lagrangian_state__GT_Routhian_state@1382b81f"
]
(defn Routhian-state->Lagrangian-state [R]
(fn [[t q [vx] :as s]]
(let [vy (-> (((partial 2) R) s)
(nth 1))]
(up t q (up vx vy)))))
#object[emmy.mechanics.routhian$Routhian_state__GT_Lagrangian_state 0x3a4e5bc3 "
emmy.mechanics.routhian$Routhian_state__GT_Lagrangian_state@3a4e5bc3"
]