What Is a Lagrangian and How Do I Use It in Mechanics Problems?

[Momentous]

Homework Statement

So we have started Lagrangian Mechanics in my class, and I really don't understand it at all. My teacher keeps doing the math on the board, but he hasn't really said what a Lagrangian is, and what an Action is. I really am lost from the start with these problems. Any help would be appreciated.
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1. A particle of mass m is conned to move on the parabola
z = ax^2 near the earth’s surface, where z points “up”. What is the Lagrangian of this
particle as a function of x and x'?
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2. Consider an action that involves acceleration:
S = (t1 to t2)∫(a/2)(x'') - U(x)
This is (to my knowledge) an artificial action since most actions do not depend on
acceleration. However, it is useful for our purposes in understanding how the Euler-
Lagrange equations are derived. Find the equation of motion for this system by varying
the action δS = 0 with respect to different paths x(t). You may assume that x(t1), x(t2), x'(t1) and x'(t2) are all known so that the paths all have these initial and nal
conditions.
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The shortest distance between two points on a surface is called a ”geodesic”. In spirit,
a geodesic is locally a straight line on the surface, but globally it bends with the
surface. For example, the lines of constant longitude and lattitude are geodesics of the
earth’s surface.

a. Use spherical polar coordinates to show that the length of a path on the surface
of a sphere is

L = R (θ1 to θ2)∫√[1 + sin^2(θ)*φ'(θ^2)]dθ
viewing the path as the function φ(θ) where θ is the polar angle and φ the
azimuthal angle of a point on the sphere.

b. Prove that the shortest distance between two points on the sphere is a great
circle. Hint place your initial point on the north pole.

Homework Equations

I think it mostly involves the equations given, as well as the Euler-Lagrange equations.

The Attempt at a Solution

I honestly have no formal attempt at any of these. I'm staring at them and have no idea where to start. I know that you need to minimize the Lagrangian, but I still really don't know how to write one. Knowing that would easily solve the first question, and I imagine that it'd help me solve the other two as well.

 

[Momentous]

For the first one, my only guess is writing it as a function of x, x', and t.

So I'd think that

L = (ax^2, 2ax, t)

Is that right?

 

[Cdg8676]

The Langrangian is written as L=K-U
The kinetic energy is K=m/2(x')^2
And the potential energy in this case is U=mgz Where z=x^2 so it becomes
U=mgx^2

This help?

 

[Cdg8676]

I'm sorry for the kinetic energy isn't should be K=m/2[(x')^2+(z')^2] and remember that z=x^2