# Can you derive SUVAT equations using a Langrangian?

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1. Nov 2, 2015

### gracie

Please help, I've put this is true on my personal statement, but I'm now fairly sure that it is not, but a more expert opinion would be extremely helpful.

2. Nov 2, 2015

### BvU

Hello Gracie, welcome to PF !

Impressed you should be concerned with Lagrangians at 17 !

But, to answer your question: yes you can. Simply write down the Lagrangian and then the Lagrange equations of motion follow.

3. Nov 2, 2015

### Rescy

Of course! Lagrangian is equivalent to Newton's second law of motion in Cartesian coordinates.

$$L= {1\over 2}m(x^2+y^2)-mgy$$ apply Euler-Lagrange equation for x and y and you derive a differential equation whose solutions are SUVAT.

It is a waste for Lagrangian Mechanics to be applied on SUVAT!

4. Nov 2, 2015

### nasu

The first term in that Lagrangian should have the derivatives of x and y in respect to time.

5. Nov 3, 2015

### Rescy

Of course! Lagrangian is equivalent to Newton's second law of motion in Cartesian coordinates.

$$L={1\over 2}m({\dot x}^2+{\dot y}^2)-mgy$$

Now apply Euler-Lagrange equation for x and y, we get

$$m\ddot x=0 \tag{1}$$
$$m\ddot y=-mg \tag{2}$$

From (1),
$$\dot x =v_x= \text{constant}$$

From (2),
$$\dot y=v_y=\dot y_0-gt$$ and
$$y=y_0+\dot y_0t-{1\over 2}gt^2$$

So you see how these equations can be deduced in the case of a projectile.
Similarly you can apply it to a system of one degree of freedom and get the complete SUVAT equations.

However, it is a waste for Lagrangian Mechanics to be applied on SUVAT!

6. Nov 3, 2015

### Rescy

You are absolutely right, and it is embarrassing to know that I forgot to add the dots! As I punishment, I've written out the full procedures for Gracie.

7. Nov 3, 2015

### BvU

What about the opportunity for gacie to discover this for herself ? "Give it a try and PF will help you further..." Then it woudn't have been a waste either !
(Don't agree that it's a waste in the first place !)

8. Nov 3, 2015

### gracie

Thank you so much everybody, I thought I knew the answer but wanted proof from experts, I really appreciate it :)

9. Nov 3, 2015

### Rescy

May I ask if you are applying to UK or US?

10. Nov 3, 2015

### gracie

11. Nov 3, 2015

### gracie

UK

12. Nov 3, 2015

### gracie

I know it's far above my present skill level, but understanding these sorts of thing even at a basic level is interesting for me, and you're right, it's not a waste of time, even though SUVAT are not complex I personally think it's amazing that something is so provable it can be done so in multiple ways. Physics is really cool ;3

This site has been really helpful with things I've struggled with, so thank you :)

13. Nov 8, 2015

### vanhees71

Stupid question from a German: What's SUVAT? From the answers I get it's the motion of a particle in the constant gravitational field of the earth, but what means the acronym (it's good practice to write out any acronym once at its first appearance in the text).

Of course, the use of a Lagrangian is never a waste, because it's much more clear than using "naive mechanics" concerning the underlying principles and it immideately reveals the symmetries of the problem, if written down in the proper coordinates.

14. Nov 8, 2015

### BvU

s = distance (metres, m)
u = initial velocity (metres per second, ms^-1)
v = final velocity (metres per second, ms^-1)
a = acceleration (metres per second squared, ms^-2)
t = time (seconds, s)

15. Nov 8, 2015

### lightarrow

And how can one derive distance, initial velocity and time, knowing only the lagrangian?

--
lightarrow

16. Nov 8, 2015

### BvU

This was to help vanHees

17. Nov 8, 2015

### vanhees71

So SUVAT stands for initial-value problem of the equations of motion of a point particle. Why don't you say so and use strange acronymes which are even imprecise when you resolve their meaning? Science should be expressed in as clear a language as possible!

18. Nov 8, 2015

### BvU

Guess where I found it ? Here !