f103534a-a7ab-480d-8314-132f4a0019b7 42 Valery Ochkov 4 5 false false false 0 0 decimal

x a x.U y.U y a x x.U - a / cosh * a - y.U + :

x a x.U y' x x.U - a / sinh :

x arcsinh x asinh :

For maple outputs

x.1 x.2 a x.U L.c 1 x a x.U y' (2^+ sqrt x x.1 x.2 int maple :

x.1 x.2 a x.U x.cg x 1 x a x.U y' (2^+ sqrt * x x.1 x.2 int x.1 x.2 a x.U L.c / maple :

x.1 x.2 a x.U y.U y.cg x a x.U y.U y 1 x a x.U y' (2^+ sqrt * x x.1 x.2 int x.1 x.2 a x.U L.c / maple :

a x.UL y.UL x.UR y.Ur PE m.c g.e * 0 m * x a x.UL L.c * 0 m * x a x.UL y.UL y.cg * m g.e * x a x.UL y.UL y * m.c g.e * x L a x.UR L.c * x L a x.UR y.UR y.cg * 31 mat sum eval :

L 6 m * :

S 12 m * :

H.L 7 m * :

H.R 5 m * :

m 0.1 kg * :

m.c 0.1 kg m / * :

x 4 m * :

Lagrange's multipliers method function

LM 1 m / S 0 m * x a x.UL L.c x L a x.UR L.c + (- H.L 0 m * a x.UL y.UL y - L a x.UR y.UR y H.R - x a x.UL y.UL y x a x.UR y.UR y - 41 mat * :

Lagrange's "G" function

u i el λ j el L F λ j el G j el * (sum (- ≡

u# LM a x.UL y.UL x.UR y.UR λ.1 λ.2 λ.3 λ.4 19 mat u# transpose : 1 J / a x.UL y.UL x.UR y.Ur PE * λ.1 λ.2 λ.3 λ.4 14 mat transpose LM * - eval 21 line :

Target function: the transposed gradient

r# 19 range :

u# φ a# un# u# * : G# 0 : LMa# a# LM eval : 13 mat G# r# el a# h# r# col + LM LMa# - (h# r# r# el UnitsOf h# r# r# el / * : 21 line :

Numerical solver

uo# 2 m * 2 m * 2 m * 2 m * 2 m * 111119 mat transpose :

guess value

un# uo# UnitsOf vectorize diag :

h# 105 -^un# * :

derivatives forward step

a x.UL y.UL x.UR y.UR λ.1 λ.2 λ.3 λ.4 19 mat un# uo# uo# UnitsOf / vectorize φ al_nleqsolve * transpose :

a 1.8041 m *

x.UL 3.834 m *

y.UL 1.1423 m *

x.UR 2.5377 m *

y.UR 0.5241 m *

a x.UL y.UL x.UR y.Ur PE 40.6931 J *

Sanitiy check:

LM normi 1.784 108 -^*

x.L 0 m * (x x 300 / range :

x.R x L x L x - 300 / + range :

Plot x.L m / x.L a x.UL y.UL y vectorize m / augment x.R m / x.R a x.UR y.UR y vectorize m / augment x.R m / x.R a x.UL y.UL y vectorize m / augment x.L m / x.L a x.UR y.UR y vectorize m / augment 41 sys :

Plot

a 2.5614407 N N m 1 -^* / * :

x.UL 4.4116127 m * :

y.UL 2.163828 m * :

Found in Mathcad

x.UR 2.1353467 m * :

y.UR 1.4877577 m * :

a x.UL y.UL x.UR y.Ur PE 41.275 J *

Sanitiy check:

LM normi 2

LM 2 7.6057 108 -^* 5.7498 108 -^* - 6.7884 108 -^* 41 mat

Not verify this equation:

S 0 m * x a x.UL L.c x L a x.UR L.c + (- 2 m *

Plot x.L m / x.L a x.UL y.UL y vectorize m / augment x.R m / x.R a x.UR y.UR y vectorize m / augment x.R m / x.R a x.UL y.UL y vectorize m / augment x.L m / x.L a x.UR y.UR y vectorize m / augment 41 sys :

Plot