Lagrange's theorem describes fluid motion using a Lagrangian description that tracks individual fluid particles over time rather than describing the fluid properties at fixed spatial locations like the Eulerian description. The Lagrangian description follows Newton's laws of motion for individual particles, making it easier to apply concepts from solid mechanics. However, the Eulerian description is more commonly used in fluid mechanics problems because it is not practical to track every particle in complex flows. Lagrange's equations, derived using the calculus of variations, provide an alternative formulation of classical mechanics that has advantages over Newtonian mechanics such as applying to any coordinate system.

1. ASSALAM-O-ALAIKUM.

2. LAGRAGNG’S THEOREME
By:
GHULAM MURTAZA AWAN (15-ME-35)
murtazaawan1999@gmail.com
Swedish college of engineering and technology, Rahim Yar Khan

3. Contents:
 FLUID MOTION
 METHODS OF DESCRIBING FLUID MOTION
 LAGRANGE’S THEOREM
 LAGRANGE’S METHOD FOR FLUID MECHANICS
 COMPARISON BETWEEN LAGRANGE AND EULER’S THEOREM
 LAGRANGE’S EQUATION
 ADVANTAGES OVER NEWTONIAN MECHANICS
 CONNECTION TO EULER-LAGRANGE EQUATION
 PROPERTIES OF THE EULER–LAGRANGE EQUATION
 EXAMPLES AND APPLICATIONS

4. FLUID MOTION
 WHAT IS FLUID?
a substance, as a liquid or gas, that is capable of flowing and that changes its
shape at a steady rate when acted upon by a force tending to change its shape.
 The motion of fluid depend upon the fluid and shape of the passage
through which the fluid particles moves.
• A fluid consist of an innumerable number of particles,
• Whenever a fluid is in motion, these particles move along certain lines.

5. MOTION OF FLUID PARTICLES
 The molecule of liquid and gas have freedom to move.
 The motion of fluid depend upon the fluid and shape of the passage through
which the fluid particles moves.
 A fluid consist of an innumerable number of particles,
Whenever a fluid is in motion, these particles move along certain lines.

6. Velocity Field
 Different fluid particles in fluid flow ,which move at different velocities and
may be subjected to different accelerations.
 The velocity and acceleration of a fluid particle may change both w.r.t time
and space.
 In the study of fluid flow it is necessary to observe the motion of the fluid
particles at various points in space and at a successive instant at a time.

7. METHODS OF DESCRIBING FLUID MOTION
 There are two methods of motion of fluid paticles.
1) Lagrangian method:
 This method deals with the individual particles.
 Langrangian description of fluid flow tracks the position and velocity of
individual particles.
 (E.g.: Track the location of migrating bird.
 Motion is described based upon Newton's laws.
 Named after Italian mathematician Joseph Louis
 Lagrange (1736-1813).

8. 2) Eulerian method
 Describes the flow field (velocity,acceleration, pressure, temperature, etc.)
as functions of position and time.
 It deals with the flow pattern of all the particles.
 Count the birds passing a particular location
 If you were going to study water flowing in a pipeline
which approach would you use? Eulerian Description
now we are going to discuss the lagrange’s method.

9. LAGRANGE’S THEOREM
Lagrange theorem exists in many fields, respectively
 Lagrange theorem in fluid mechanics
 Lagrange theorem in calculus
 Lagrange theorem in number theory
 Lagrange's theorem in group theory

10. LAGRANGE’S METHOD FOR FLUID
MECHANICS
 Lagrangian mechanics is a reformulation of classical mechanics, introduced by the
Italian-French mathematician and astronomer Joseph-Louis Lagrangein 1788.
 The Lagrange method is based on the study of single fluid particle movement
process as the basis, all of the particle motion, constitute the entire fluid
movement.
 Coordinate position of each particle with a start time of (a, B, c), as a symbol of
the particle.
 Any time any particles in the space position (x, y, z) can be seen as (a, B, C and T)
function,
 The basic characteristics of Lagrange method: tracking the motion of the fluid
particles
 Advantages: can directly use the particle dynamics in solid mechanics analysis

11. COMPARISON BETWEEN LAGRANGE AND
EULER’S THEOREM
LAGRANGETHEOREM
 It is not possible to track each
"particle" in a complex flow field.
Thus, the Lagrangian description is
rarely used in fluid mechanics.
EULER’S THEOREM
 The Eulerian Description is one in
which a control volume is defined,
within which fluid flow properties
of interest are expressed as fields.

12. LAGRANGE’S EQUATION

13. LAGRANGE…

14. Advantages Over Newtonian Mechanics
 We are now going to use the ideas of the previous lecture to develop a new formalism for
mechanics, called Lagrangian mechanics, invented by Lagrange (1736-1813).
 There are two important advantages of the Lagrange formalism over that of Newton.
 First, as we have seen, Lagrange’s equations take the same form for any coordinate system, so
that the method of solution proceeds in the same way for any problem.
 Second, the Lagrangian approach eliminates the forces of constraint. This makes the Lagrangian
formalism easier to solve in constrained problems.
 This chapter is the heart of advanced classical mechanics, but it introduces some new methods
that will take getting used to. Once you master it, you will find it an extraordinarily powerful
way to solve mechanics problems.

15. CONNECTION TO EULER-LAGRANGE
EQUATION

16. Properties of the Euler–Lagrange equation
 Non uniqueness
The Lagrangian of a given system is not unique. A Lagrangian L can be multiplied
by a nonzero constant a, an arbitrary constant b can be added, and the new
Lagrangian aL + b will describe exactly the same motion as L. Each Lagrangian
will obtain exactly the same equations of motion.
 Invariance under point transformations
Given a set of generalized coordinates q, if we change these variables to a new
set of generalized coordinates s according to a point transformation q = q(s, t),
the new Lagrangian L′ is a function of the new coordinates. Similary…
 Energy conservation
 Mechanical similarity
 Interacting particles
 Cyclic coordinates and conserved momenta

17. EXAMPLES
 Conservative force
 Cartesian coordinates
 Polar coordinates in 2d and 3d
 Pendulum on a movable support
 Two-body central force problem
 Electromagnetism
 Extensions to include non-conservative forces

18. ALTERNATIVE FORMULATIONS OF CLASSICAL
MECHANICS
 Momentum space formulation
 Higher derivatives of generalized coordinates
 Optics
 Relativistic formulation
 Quantum mechanics
 Classical field theory
 Noether's theorem

19. APPLICATIONS…
 LAGRANGE’S APPLICATIONS
 Sprays particles
 bubble dynamics
 rarefied gasses etc.
 EULUERIAN APPLICATION
 Simulation of microscale airborne probes etc.