# Can any explain to me why the Lagrangian is T *minus* V (i.e. KE - PE)?

The sum of them would be the total energy in the system, but I don't get the significance of the difference between them.

Read U for V

My question is why the sign is negative. So far I am thinking that Work by conservative forces is Delta U. If the total energy of the system is constant (conservative forces) Delta T + W = 0 (The work only comes from conservative forces, so the only way T changes is by Work. But Work is -Delta U (the change of potential) so Delta T - Delta U = 0 for a closed system. Hence it is T - U.

### 2 Answers

- Steve4PhysicsLv 75 years ago
It is because of the 'Principle of Least Action'.

'Action' is the integral of T - V over the path of a particle. Nature finds the path where this integral is minimised. Mathematically, this can be shown to correpond to Newton's laws of motion.

Watch the 10 minute video in the link which explains far better than anything which could fit here.

If you search for 'principle of least action' you'll find loads more information.

Source(s): https://www.youtube.com/watch?v=xz7jLnWcxMs - Anonymous5 years ago
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