d

56dc514e-4886-49ce-bef7-36cf44e85c39 70 4 5 false false 0 decimal

&[DATE] &[TIME] - &[FILENAME]

d

Derivative utility X\032E\ t\032E\ n\032E\ p\032E\ DIFF R\0324\ 0 : R\0324\ 0 : k\0324\ 0 : 13 mat x\0324\ X\032E\ str\0324\ diff(x\0324\ num2str, t\032E\ num2str, n\032E\ num2str) concat : R\0324\ k\0324\ k\0324\ 1 + : el str\0324\ str2num p\032E\ * : 21 line for k\0324\ 1 ≡ R\0324\ 1 el k\0324\ 0 : r\0324\ 1 X\032E\ rows range c\0324\ 1 X\032E\ cols range A\0324\ r\0324\ c\0324\ el R\0324\ k\0324\ k\0324\ 1 + : el : for for A\0324\ 31 line if 31 line :

d.Aux2 try to catch the differential variable and it's order. t\032E\ d.Aux2 U\0324\ t\032E\ Unknowns : R\0324\ 0 : 12 mat k\0324\ 1 : k\0324\ U\0324\ length ≤ k\0324\ k\0324\ 1 + : str\0324\ U\0324\ k\0324\ el num2str : Unknowns(dSymbol str\0324\) concat str2num num2str 0 ≡ 0 l\0324\ 05 range t\032E\ U\0324\ k\0324\ el l\0324\ 1 + 1 DIFF num2str 0 ≡ break continue if for p\0324\ dSymbol str\0324\ concat dSymbol dSymbol concat dSymbol strrep str2num : n\0324\ str\0324\ dSymbol findstr : (1 - ≡ (R\0324\ k\0324\ 1 el R\0324\ k\0324\ 2 el R\0324\ k\0324\ 3 el 13 mat U\0324\ k\0324\ el l\0324\ p\0324\ 13 mat : m\0324\ str\0324\ 1 n\0324\ rows substr dSymbol findstr : (1 - ≡ R\0324\ k\0324\ 1 el R\0324\ k\0324\ 2 el R\0324\ k\0324\ 3 el 13 mat U\0324\ k\0324\ el l\0324\ n\0324\ rows + 1 - p\0324\ 13 mat : R\0324\ k\0324\ 1 el R\0324\ k\0324\ 2 el R\0324\ k\0324\ 3 el 13 mat U\0324\ k\0324\ el l\0324\ m\0324\ rows + 1 - p\0324\ 13 mat : if if 31 line if 21 line for R\0324\ 31 line :

∂(x,t) returns x'*∂t x\032E\ t\032E\ ∂ R\0324\ t\032E\ d.Aux2 : D\0324\ x\032E\ : 12 mat R\0324\ num2str 0 ≡ D\0324\ 0 : k\0324\ 1 R\0324\ rows range D\0324\ D\0324\ R\0324\ k\0324\ 1 el R\0324\ k\0324\ 2 el R\0324\ k\0324\ 3 el (R\0324\ k\0324\ 2 el^DIFF : 11 line for if D\0324\ 31 line :

d(x) returns the total differential of x x\032E\ d U\0324\ x\032E\ Unknowns : s\0324\ 0 : 12 mat k\0324\ 1 : k\0324\ U\0324\ length ≤ k\0324\ k\0324\ 1 + : s\0324\ s\0324\ x\032E\ U\0324\ k\0324\ el ∂ + : for s\0324\ 31 line :

equrep

\032E\a TrueQ \032E\a num2str 1 ≡ 11 line :

\032E\a FalseQ \032E\a num2str 0 ≡ 11 line :

\032E\a b\032E\ EqualQ \032E\a num2str b\032E\ num2str ≡ 11 line :

E\032E\ \032E\a b\032E\ EquRep equrep(E\032E\ num2str, \032E\a num2str, b\032E\ num2str) concat str2num 11 line :

E\032E\ \032E\a b\032E\ q\032E\ EquRep equrep(E\032E\ num2str, \032E\a num2str,if(TrueQ(q\032E\ num2str), b\032E\ num2str, a# num2str)) concat str2num 11 line :

EQ EQ EQ(a/b,z) EQ.B(a,b,z,a/b,z/a,z*b) 1 quotient 14 mat stack :

a#

b#

q#

E\032E\ Rules\032E\ EquRep ans\0324\ E\032E\ : n\0324\ 1 Rules\032E\ rows range ans\0324\ ans\0324\ Rules\032E\ n\0324\ 1 el Rules\032E\ n\0324\ 2 el EquRep : for ans\0324\ 31 line :

E\032E\ Rules\032E\ EquRepN ans\0324\ E\032E\ : N\0324\ 1 EquRepN range ans\0324\ ans\0324\ ans\0324\ Rules\032E\ EquRep : EqualQ break continue if 11 line for ans\0324\ 31 line :

E\032E\ equal2EQ ans\0324\ E\032E\ num2str : ans\0324\ \2261\ findstr 1 - ≡ ans\0324\ ans\0324\ \2261\0 concat : n\0324\ ans\0324\ \2261\ findstr 1 el : 12 mat n\0324\ ans\0324\ \2261\ findstr 1 el : if ans\0324\ 1 n\0324\ 1 - substr str2num ans\0324\ n\0324\ 1 + ans\0324\ strlen n\0324\ - substr str2num 12 mat 31 line :

E\032E\ x\032E\ depends E\032E\ x\032E\ x\032E\_DEPENDS.X# EquRep num2str x\032E\_DEPENDS.X# num2str findstr 1 - ≠ 11 line :

\032E\a b\032E\ x\032E\ depends \032E\a x\032E\ depends b\032E\ x\032E\ depends & 11 line :

E\032E\ x\032E\ DIFF diff(E\032E\ x\032E\_DIFF# EquRep num2str,_DIFF#) concat str2num_DIFF# x\032E\ EquRep 11 line :

E\032E\ x\032E\ n\032E\ DIFF diff(E\032E\ x\032E\_DIFF# EquRep num2str,_DIFF#, n\032E\ num2str) concat str2num_DIFF# x# EquRep 11 line :

EquRepN 20 :

isol

EQ EQ(-a,b) EQ(a,0-b) 1 minus 14 mat :

EQ EQ EQ(a^b,z) EQ.B(a,b,z,a^b,ln(z)/ln(a),'sign*z^(1/b)) 1 power 14 mat stack :

Not usual but better EQ.B(a,b,z,a^b,ln(z)/ln(a),exp(ln(z)/b))

EQ EQ EQ(a/b,z) EQ.B(a,b,z,a/b,z/a,z*b) 1 quotient 14 mat stack :

EQ EQ EQ(a*b,z) EQ.B(a,b,z,a*b,z/a,z/b) 1 times 14 mat stack :

EQ EQ EQ(a+b,z) EQ.B(a,b,z,a+b,z-a,z-b) 1 plus 14 mat stack :

EQ EQ EQ(a,z) EQ.L(a,z) 1 lineal 14 mat stack :

EQ EQ EQ(x*(b+a*x),z) EQ.P2(a,b,x,z) 1 Poly2 14 mat stack :

EQA sin cos tan sec csc cot sinh cosh tanh sech csch coth exp_EQ#.10 erf asin acos atan asec acsc acot asinh acosh atanh asech acsch acoth ln log10 erfinv 215 mat transpose :

EQA rows 15

_k# 1 EQA rows range EQ EQ EQ(EQA_k# 1 el (a),b) concat EQ.A(a,b, EQA_k# 1 el (a), EQA_k# 2 el (b) concat 1 EQA_k# 1 el 14 mat stack : for

_k# 1 EQA rows range EQ EQ EQ(EQA_k# 2 el (a),b) concat EQ.A(a,b, EQA_k# 2 el (a), EQA_k# 1 el (b) concat 1 EQA_k# 2 el 14 mat stack : for

E\032E\ x\032E\ isolate a\0324\_EQ#.10 10 a\0324\^11 line : a\0324\ b\0324\ f\0324\ a\0324\ EQ.A a\0324\ x\032E\ depends a\0324\ a\0324\ EQ f\0324\ b\0324\ EQ if 11 line : a\0324\ b\0324\ z\0324\ f\0324\ a\0324\ j\0324\ EQ.B a\0324\ x\032E\ depends b\0324\ x\032E\ depends \0324\k f\0324\ x\032E\ 0 EquRep : f\0324\ \0324\k - z\0324\ \0324\k - EQ 21 line a\0324\ j\0324\ EQ if b\0324\ x\032E\ depends b\0324\ a\0324\ EQ z\0324\ f\0324\ EQ if if 11 line : a\0324\ z\0324\ EQ.L a\0324\ x\032E\ 2 DIFF FalseQ x\032E\ z\0324\ a\0324\ x\032E\ 0 EquRep - a\0324\ x\032E\ DIFF / EQ a\0324\ z\0324\ EQ if 11 line : a\0324\ b\0324\ \0324\X z\0324\ EQ.P2 \0324\X x\032E\ EqualQ a\0324\ b\0324\ x\032E\ depends ¬ (& a\0324\ \0324\X b\0324\ 2 a\0324\ * / + (2^* z\0324\ b\0324\ 2^4 a\0324\ * / + EQ \0324\X b\0324\ a\0324\ \0324\X * + (* z\0324\ EQ if 11 line : ans\0324\ E\032E\ equal2EQ : A\0324\ ans\0324\ 1 el numden : B\0324\ ans\0324\ 2 el numden : 13 mat ans\0324\ A\0324\ 1 el B\0324\ 2 el * A\0324\ 2 el B\0324\ 1 el * - 0 EQ EQ EquRepN : a\0324\ b\0324\ EQ a\0324\ b\0324\ ≡ (: ans\0324\ 91 line :

E\032E\ x\032E\ isol ans\0324\ E\032E\ x\032E\ isolate equal2EQ : ans\0324\ 1 el x\032E\ EqualQ ans\0324\ 2 el x\032E\ depends ¬ & ans\0324\ 2 el ans\0324\ 1 el ans\0324\ 2 el - if 21 line :

sign 1 :

Differentials samples

Differentials

dSymbol d :

Enter an expression, and say that have some constants

z x y 2^* y a x * / + :

da 0 :

dz z d : y 1 x 2^y * a * - (* dx * - 12 x 2^* y * a * + (dy * x * + a x 2^* /

Total differential

Partial derivatives

dz dx ∂ dx / y 1 x 2^y * a * - (* a x 2^* / -

dz dy ∂ dy / 12 x 2^* y * a * + x a * /

z x ∂ dx / y 1 x 2^y * a * - (* a x 2^* / -

z y ∂ dy / 12 x 2^* y * a * + a x * /

Some examples

dSymbol \2202\. :

r θ cos * r θ sin * 21 mat d θ cos ∂.r * r θ sin * ∂.θ * - θ sin ∂.r * r θ cos * ∂.θ * + 21 mat

dSymbol \2202\ :

∂a 0 :

a e b x *^* d e b x *^a * x ∂b * b ∂x * + (*

a e x sin^* d e x sin^x cos * a * ∂x *

x 2^x y * ln * d x 12 x y * ln * + (∂x * y * x ∂y * + (* y /

∂a Clear 1

d1 a b * d : b ∂a * a ∂b * +

d2 d1 d : ∂a b ∂b + ∂b + (* a ∂b * +

d3 d2 d : ∂a b 3 ∂b * + ∂b + (* a 2 ∂a * + (∂b * +

∂a 0 :

d1 a b * d : a ∂b *

d2 d1 d : a ∂b *

d3 d2 d : a ∂b *

Chain rule

c a x g f * d : a x g f x diff * ∂x *

x f x ln :

x g x cos :

c a x tan * ∂x * -

a x cos ln * d x sin a * ∂x * x cos / -

Total differential

t x x' f d t x x' f t diff ∂t * t x x' f x diff ∂x * + t x x' f x' diff ∂x' * +

Alvaro

Lagrangians

Lagrangian Dynamics

dSymbol \2202\ :

a g θ θ' θ'' Clear 1

Simple pendulum

Constants

∂a 0 :

∂m 0 :

∂g 0 :

Gen Coords

∂θ θ' ∂t * :

∂θ' θ'' ∂t * :

Position

x a θ sin * :

y a θ cos * - :

ẋ x d ∂t / :

ẏ y d ∂t / :

Lagrangian

L 12 / m * ẋ 2^ẏ 2^+ (* m g * y * - :

Equations of motion

eq 1 \2202\t / L θ' ∂ ∂θ' / d * L θ ∂ \2202\θ / - θ'' isol 11 sys : g θ sin * a / - 11 sys

k m x ẋ ẍ Clear 1

Harmonic Oscillator

Constants

∂k 0 :

∂m 0 :

Gen Coords

∂x ẋ ∂t * :

∂ẋ ẍ ∂t * :

Position

x

Lagrangian

L 12 / m * ẋ 2^* 12 / k * x 2^* - :

Equations of motion

eq 1 \2202\t / L ẋ ∂ \2202\ẋ / d * L x ∂ \2202\x / - ẍ isol 11 sys : ∂t ẍ * ẍ m * x k * + (* m ẋ * ∂ẍ * + 11 sys

a m.1 m.2 g θ θ' θ'' x.1 ẋ.1 ẍ.1 Clear 1

Pendulum with Horizontal support

Constants

∂a 0 :

∂m.1 0 :

∂m.2 0 :

∂g 0 :

Gen Coords

∂θ θ' ∂t * :

∂θ' θ'' ∂t * :

∂x.1 ẋ.1 ∂t * :

∂ẋ.1 ẍ.1 ∂t * :

Position

x.2 x.1 a θ sin * + :

y.2 a θ cos * - :

ẋ.2 x.2 d ∂t / :

ẏ.2 y.2 d ∂t / :

L 12 / m.1 * ẋ.1 2^* 12 / m.2 * ẋ.2 2^ẏ.2 2^+ (* + m.2 g * y.2 * - :

Lagrangian

Equations of motion

eq 1 \2202\t / L ẋ.1 ∂ ∂ẋ.1 / d * L x.1 ∂ ∂x.1 / - ẍ.1 isol 1 \2202\t / L θ' ∂ ∂θ' / d * L θ ∂ ∂θ / - θ'' isol 21 sys : a m.2 * θ sin θ' 2^* - θ cos θ'' * + (* m.1 m.2 + / θ sin g * θ cos ẍ.1 * + a / - 21 sys

α m g φ φ' φ'' r ṙ r̈ Clear 1

Particle in a Cone

Constants

∂α 0 :

∂m 0 :

∂g 0 :

Gen Coords

∂φ φ' ∂t * :

∂φ' φ'' ∂t * :

∂r ṙ ∂t * :

∂ṙ r'' ∂t * :

Position

x r φ cos * :

y r φ sin * :

z r α cot * :

ẋ x d ∂t / :

ẏ y d ∂t / :

ż z d ∂t / :

Lagrangian

L 12 / m * ẋ 2^ẏ 2^+ ż 2^+ (* m g * z * - :

Equations of motion

eq 1 \2202\t / L φ' ∂ ∂φ' / d * L φ ∂ ∂φ / - φ'' isol 1 \2202\t / L ṙ ∂ ∂ṙ / d * L r ∂ ∂r / - r'' isol 21 sys :

α 45 deg * :

eq φ sin φ cos ṙ * r φ sin * φ' * - (* - φ cos φ sin ṙ * r φ cos * φ' * + (* + r φ' * + (ṙ * r 2^/ - g α cot * - φ' φ sin φ cos ṙ * r φ sin * φ' * - (* - φ cos φ sin ṙ * r φ cos * φ' * + (* + (* + φ cos 2^φ sin 2^+ α cot 2^+ / 21 sys

For small angles

eq a sin 0 a cos 122 mat EquRep 2 φ' * ṙ * r / - g α cot * - φ' 2^r * + 1 α cot 2^+ / 21 sys

a.1 a.2 m.1 m.2 g θ.1 θ'.1 θ''.1 θ.2 θ''.2 Clear 1

Double Pendulum

Constants

∂m.1 0 :

∂m.2 0 :

∂g 0 :

∂a.1 0 :

∂a.2 0 :

Gen Coords

∂θ.1 θ'.1 ∂t * :

∂θ'.1 θ''.1 ∂t * :

∂θ.2 θ'.2 ∂t * :

∂θ'.2 θ''.2 ∂t * :

Position

x.1 a.1 θ.1 sin * :

y.1 a.1 θ.1 cos * - :

x.2 a.1 θ.1 sin * a.2 θ.2 sin * + :

y.1 a.1 θ.1 cos * - a.2 θ.2 cos * - :

ẋ.1 x.1 d ∂t / :

ẏ.1 y.1 d ∂t / :

ẋ.2 x.2 d ∂t / :

ẏ.2 y.2 d ∂t / :

Lagrangian

L 12 / m.1 * ẋ.1 2^ẏ.1 2^+ (* 12 / m.2 * ẋ.2 2^y'.2 2^+ (* + m.1 g * y.1 * - m.2 g * y.2 * - :

Equations of motion

eq 1 \2202\t / L θ'.1 ∂ ∂θ'.1 / d * L θ.1 ∂ ∂θ.1 / - θ''.1 isol 1 \2202\t / L θ'.2 ∂ ∂θ'.2 / d * L θ.2 ∂ ∂θ.2 / - θ''.2 isol 21 sys :

Too long for display. For m.1 = 0, it's independent for the other mass too

eq m.1 012 mat EquRep a.2 θ'.2 2^θ.2 sin * - θ.2 cos θ''.2 * + (* θ'.1 2^θ.1 sin * a.1 * - θ.1 cos a.1 * / - θ'.2 2^θ.2 sin * a.2 * - a.1 θ'.1 2^θ.1 sin * - θ.1 cos θ''.1 * + (* + θ.2 cos a.2 * / - 21 sys

For small angles

eq a sin 0 a cos 122 mat EquRep a.2 m.2 * θ''.2 * a.1 m.1 m.2 + (* / - a.1 θ''.1 * a.2 / - 21 sys

m a R α g x ẋ ẍ Clear 1

Hoop rolling without sliding down an inclined plane

Constants

∂m 0 :

∂a 0 :

∂g 0 :

∂R 0 :

∂α 0 :

Gen Coords

∂x ẋ ∂t * :

∂ẋ ẍ ∂t * :

Restrictions

θ' ẋ R / :

Non sliding condition.

Position

x

Lagrangian

L 12 / m * ẋ 2^* 12 / m * R 2^* θ' 2^* + m g * a x - (* α sin * - :

Equations of motion

eq 1 \2202\t / L ẋ ∂ ∂ẋ / d * L x ∂ ∂x / - ẍ isol 11 sys : g α sin * 2 / - 11 sys

∂a ∂α ∂m ∂R ∂g θ θ' θ'' Clear 1

Pendulum whose support rotates with constant angular speed

Constants

∂m 0 :

∂a 0 :

∂g 0 :

∂R 0 :

∂ω 0 :

Gen Coords

∂θ θ' ∂t * :

∂θ' θ'' ∂t * :

Position

x R ω t * cos * a θ sin * + :

y R ω t * sin * a θ cos * - :

ẋ x d ∂t / :

ẏ y d ∂t / :

Lagrangian

L 12 / m * ẋ 2^ẏ 2^+ (* m g * y * - :

Equations of motion

eq 1 \2202\t / L θ' ∂ ∂θ' / d * L θ ∂ ∂θ / - θ'' isol 11 sys :

eq ω 2^R * ω t * cos θ cos * ω t * sin θ sin * + (* - g θ sin * + a / - 11 sys

For zero angular speed, we recover the simple pendulum

eq ω 0 EquRep g θ sin * a / - 11 sys

m a k g θ θ' θ'' Clear 1

Pendulum whose support rotates with constant angular speed

Constants

∂m 0 :

∂a 0 :

∂g 0 :

Gen Coords

∂θ θ' ∂t * :

∂θ' θ'' ∂t * :

Position

x r θ cos * :

y r θ sin * - :

ẋ x d ∂t / :

ẏ y d ∂t / :

Lagrangian

L 12 / m * ẋ 2^ẏ 2^+ (* m g * y * - 12 / k * r a - (2^* - :

Equations of motion

eq 1 \2202\t / L θ' ∂ ∂θ' / d * L θ ∂ ∂θ / - θ'' isol 11 sys :

eq θ sin θ cos ṙ * r θ sin * θ' * - (* - θ cos θ sin ṙ * r θ cos * θ' * + (* + θ' r * + (ṙ * - r g * θ cos * + r 2^/ 11 sys

For small angles

eq a sin 0 a cos 122 mat EquRep g 2 θ' * ṙ * - r / 11 sys

Alvaro