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

 

[paranormal]

Dear student,

Thank you for reaching out for help with your Lagrangian Mechanics problems. It can be challenging to understand a new concept in physics, but with some practice and guidance, I am sure you will be able to grasp the concepts of Lagrangian Mechanics.

Firstly, let me explain what a Lagrangian and Action are in the context of Lagrangian Mechanics. A Lagrangian is a mathematical function that describes the dynamics of a system, given its position and velocity. It is defined as the difference between the kinetic and potential energy of the system. In simpler terms, it is a way to represent the motion of a system using only its position and velocity, without having to consider forces acting on it.

An Action, on the other hand, is a quantity that represents the total energy of a system over a certain period of time. It is given by the integral of the Lagrangian over time. In other words, it is the sum of the kinetic and potential energy of the system over a period of time.

Now, let's look at your first problem. You are given a particle moving on a parabola near the earth's surface. To find the Lagrangian of this particle, you need to first write down the kinetic and potential energy of the particle. The kinetic energy of a particle is given by ½mv^2, where m is the mass and v is the velocity. In this case, the kinetic energy would be ½m(x')^2, as the particle is moving along the x-axis. The potential energy, on the other hand, is given by the height of the particle above the ground, which is z = ax^2. So, the potential energy would be mgh, where g is the acceleration due to gravity and h is the height. Substituting the value of z, we get the potential energy as mgax^2. Therefore, the Lagrangian of the particle would be the difference between the kinetic and potential energy, which is L = ½m(x')^2 - mgax^2.

Moving on to your second problem, you are given an artificial action that involves acceleration. To find the equation of motion for this system, you need to apply the Euler-Lagrange equations. These equations are used to find the path that minimizes the action of a system. In this case, the action is given by S = (t1 to t2)∫(