Physics 68 Lagrangian Mechanics (5 of 25) Simple Harmonic Motion: Example 

Note that the general solution to the (d^2/(dt)^2)x + w^2*x = 0 equation is missing a phase factor phi. It should be: x = A*cos(w*t+phi) . This is the reason why it indeed does not matter if it is sine or cosine, since both functions only differ by a phase factor of pi/2 .

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One Detail which caused a bit ofconfusion is that v(0)=0, which isn't mentioned in the task, but is implied in your solution (v(0)=Awsin(0)=0). If you have this information you can go ahead and use the euler Ansatz x(t)=Ae^(Yt)+Be^-(Yt) with Y=sqrt(-w^2) thus v(0) = Y(Ae^(Y*0)-Be^-(Y*0)) = Y(A-B) = 0 Thus A=B! With Y=i*w and C = 2A we can write: x(t)=Ae^(Yt)+Ae^-(Yt) = C/2*(e^(iwt)+e^(-iwt)) = C*cos(wt) Still great video!

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I think the special solution might be x(t)=x0*cos(wt)+(vo/w)*sin(wt).. great lectures thank you:)

I appreciate your assistance, Now I understand Lagrangian .

Please also make a video on Newtonian and Hamiltonian Mechanics. I loved your videos Sir.

That was so beautiful!! Thank you for sharing it

🤩👌 your work is best

You are so incredible honestly

What I never understood is why don't we use the actual method of solving second order ODE's. Can't I just assume that e^rt is a solution and then use euler's identity to get a linear combination of both cos and sin? I felt like saying it was Acos(wt) was always cutting off info. If you could clear this up that'd be nice. I'm taking my first physics class this fall so I want to get rid of any misconceptions.

The way of teaching is so good,Thanks a lot

For SHM, the differential equation is solved by any function, whose 2nd derivative is negative of original function - so sin and cis work. But f(x)= e^(ix) also satisfies the requirement. Why use sin and cos and not e^ix? Is it because this is introductory and exponential is reserved for a future exciting episode?

thank you sir great effort keep it up God bless you sir

How do you handle situations where the energy in the Lagrangian is not conserved? For example, if you would add friction to the SHO in this example, how would you then approach the problem? How can you handle a damped harmonic oscillator using the Lagrangian framework?

awesome video professor

These videos are interesting but why not write the Lagrangian equations as dL/dx = d/dt(dL/dv) instead of their difference equals zero? Writing it that way in the first place goes more directly to the analogy of F = ma where F = dL/dx and ma = d/dt(dL/dv) (rather than, as in the video, having to constantly swap summands on either side of the equation to get it into the that aesthetic form.)

Is there a specific reason sine or cosine doesn't matter? Intuitively that would seem to give different values for position for the same time

why was the second term ( partial differential of L with respect to x) taken as a positive quantity in the last video, but as a negative quantity in this video? when, in fact, it is negative in both cases. (-mg and -kx respectively)

Hi sir. What if there is a friction force µmg ? How do we include it ?

Excellent lecture 🙏🙏🙏🙏

Helped a lot, thank you sir

after 2:28 didn't get how u said that, as its solution?

thanks

Thanks sir.

I didn't get how do we get equation for x from the former equation

Sir how do we handle the case when there is gravity? ( Vertical model's) In this way : T should be 1/2mxddot^2 and V= 1/2kx^2 + mgx. The problem is that in this way we will have mddotx + kx +mg that is not correct. Could you help me?

Sir, will it be different if the motion of the box is moving in the same direction and the spring is at an angle to the horizontal?

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Thank you sir