Mathematics

1. ‘LAGRANGE'S EQUATION WITH ONE
APPLICATION’

2. WHO WAS LAGRANGE?
JOSEPH-LOUIS LAGRANGE (25 JANUARY 1736 – 10 APRIL 1813), WAS
MAINLY A FRENCH MATHEMATICIAN.
HE WAS AN ITALIAN ENLIGHTENMENT ERA MATHEMATICIAN &
ASTRONOMER AND MADE SIGNIFICANT CONTRIBUTIONS TO THE FIELDS
OF ANALYSIS, NUMBER THEORY, AND BOTH CLASSICAL AND CELESTIAL
MECHANICS.

3. WHAT IS LAGRANGE EQUATION
Lagrange's equation is a first-order partial differential
equation whose solutions are the functions for which a
given functional is stationary.
Pp +Qq = R
Where p=
𝜕
𝜕𝑥
, 𝑞 =
𝜕
𝜕𝑦
And P,Q,R is the function of (x,y,z)

4. HISTORY OF LAGRANGE’S EQUATION
The Euler–Lagrange equation was developed in the 1750s by Euler and
Lagrange in connection with their studies of the tautochrone problem.
Lagrange solved this problem in 1755 and sent the solution to Euler. Both
further developed Lagrange's method and applied it to mechanics, which led
to the formulation of Lagrangian mechanics. Their correspondence
ultimately led to the calculus of variations, a term coined by Euler himself in
1766.

5. SHORT DESCRIPTION LAGRANGE EQUATION
• The Euler–Lagrange equation is useful for solving optimization problems in
which, given some functional, one seeks the function minimizing (or
maximizing) it. This is analogous to Fermat's theorem in calculus, stating that
at any point where a differentiable function attains a local extremum,
its derivative is zero.
Pp + Qq = R
Where p=
𝜕
𝜕𝑥
, 𝑞 =
𝜕
𝜕𝑦
And P,Q,R is the function of (x,y,z)

6. LAGRANGE’S EQUATION DETAILED
If Pp+Qq=R where ‘Pp, Qq and R’ are the functions of (x,y,z) then φ(u,v)=0.
Given that, u (x,y,z)=a and v (x,y,z) =b the above term ‘φ(u,v)=0’ can be expressed.
More to look, p =
𝜕
𝜕𝑥
, 𝑞 =
𝜕
𝜕𝑦
So, the auxillary equation stands out as-
𝑑𝑥
𝑃
=
𝑑𝑦
𝑄
=
𝑑𝑧
𝑍
And from the above auxillary equation the general solution of Lagrange’s Equation
can be found.

7. SOLUTION OF A LAGRANGE EQUATION
Q: FIND GENERAL SOLUTION OF (mz-ny)p+(nx-lz)q=ly-mx.
SOLUTION: A.E. is ,
𝑑𝑥
𝑚𝑧−𝑛𝑦
=
𝑑𝑦
𝑛𝑥−𝑙𝑧
=
𝑑𝑧
𝑙𝑦−𝑚𝑥
[multiplies x,y,z]
=>
𝑥𝑑𝑥+𝑦𝑑𝑦+𝑧𝑑𝑧
𝑚𝑥𝑧−𝑛𝑥𝑦+𝑛𝑥𝑦−𝑙𝑦𝑧+𝑙𝑦𝑧−𝑚𝑥𝑧
=>
𝑥𝑑𝑥+𝑦𝑑𝑦+𝑧𝑑𝑧
0
Taking fractions,
𝑑𝑥
𝑚𝑧−𝑛𝑦
=
𝑥𝑑𝑥+𝑦𝑑𝑦+𝑧𝑑𝑧
0
=> (mz-ny)(xdx+ydy+zdz)=0
=> xdx+ydy+zdz=0
By integrating=>
𝑥2
2
+
𝑦2
2
+
𝑧2
2
= 𝑎′
𝑥2
+ 𝑦2
+ 𝑧2
= 2𝑎′
= 𝑎
Let, u(x,y,z)=𝑥2 + 𝑦2 + 𝑧2 = 𝑎

8. A.E. IS, =>
𝑑𝑥
𝑚𝑧−𝑛𝑦
=
𝑑𝑦
𝑛𝑥−𝑙𝑧
=
𝑑𝑧
𝑙𝑦−𝑚𝑧
=
𝑙𝑑𝑥+𝑚𝑑𝑦+𝑛𝑑𝑧
𝑚𝑙𝑧−𝑛𝑙𝑦+𝑛𝑥𝑚−𝑙𝑚𝑧+𝑙𝑦𝑛−𝑚𝑥𝑛
[multiplies l,m,n]
=
𝑙𝑑𝑥 + 𝑚𝑑𝑧 + 𝑛𝑑𝑧
0
Taking fraction,
𝑑𝑥
𝑚𝑧−𝑛𝑦
=
𝑙𝑑𝑥+𝑚𝑑𝑦+𝑛𝑑𝑧
0
(mz-ny)(ldx+mdy+ndz)=0
ldx+mdy+ndz=0
By integrating=> lx+my+nz=b
Let, , u(x,y,z)=𝑥2 + 𝑦2 + 𝑧2 = 𝑎
v(x,y,z)=lx+my+nz=b
Therefore, the general solution is,
Φ(u,v)=0
=> Φ(𝑥2 + 𝑦2 + 𝑧2, lx+my+nz)=0

9. A MATHEMATICAL APPLICATION OF LAGRANGE’S EQUATION
**FIND THE GENERAL SOLUTION OF ‘XP+YQ=Z’
SOLVE- BECAUSE THE GIVEN EQUATION STRICTLY FOLLOWS THE TERM
Pp+Qq=R so the auxillary equation stands out as-
dx/x= dy/y= dz/z
From dx/x= dz/z, we get by integrating is lnx=lnz+lna or x/z=a
Again from dy/y= dz/z, we get by integrating is lny=lnz+lnb or y/z=b
Let u(x,y,z)=x/z=a and v(x,y,z)=y/z=b
Therefore the general solution stands out to be φ(x/z, y/z)=0

10. PRACTICAL FIELD OF LAGRANGE’S EQUATION
LAGRANGE’S EQUATION IS SPECIALLY USED IN MECHANICS.
THE EQUATION IS ALSO USED FOR HEAT AND THERMODYNAMICS AND
EVEN FOR GRAPHING (TAUTOCHRONE).
IN TERMS OF CLASSICAL MECHANICS, THE EQUATION IS EQUIVALENT TO
NEWTON’S LAWS OF MOTION. BUT IT HAS THE ADVANTAGE THAT IT
TAKES THE SAME FORM IN ANY SYSTEM OF GENERALIZED
COORDINATES, AND IS BETTER SUITED TO GENERALIZATIONS.

11. ANY
QUESTION??