Is always a Lagrangian L=T-V ?

[Karlisbad]

That's my question..although in more general cases $$L=T-V$$

$$H=T+V$$ however there're several important exceptions..for example:

a) Classically (Non relativisitc) the Gravitational "Energy" (=Hamiltonian for a time-independent Potential) is:

$$H=(1/2)\int_{V}\rho (\gra \phi)^{2}$$

b) Einstein-HIlbert Lagrangian $$L=\sqrt (-g) R$$ -g is the

determinant of the metric and R is Ricci scalar.

Is there always a kind of "transform" so you can always split te Lagrangian into a Kinetic and a potential terms...

 

[StatMechGuy]

First of all, it is not sufficient that the hamiltonian have a time-independent potential for it to equal the energy. That's only for the hamiltonian to be conserved in time (actually, it's that $$dH/dt = \partial_t L$$ that you need). Also, you might want to look up the situation of a charged particle in the magnetic field. the lagrangian is decidedly NOT T - V, but the hamiltonian does equal thte energy of the charged particle (does not include the energy of the field).

 

[charng]

The short answer is no, it is not always true that the Lagrangian is equal to T-V. While this is a common expression in classical mechanics, there are exceptions to this rule, such as in the cases mentioned in the question.

In general, the Lagrangian is a function that describes the dynamics of a system and can take different forms depending on the specific system and its interactions. In classical mechanics, the Lagrangian is typically written as T-V, where T represents the kinetic energy and V represents the potential energy. However, in other fields such as general relativity, the Lagrangian can take on a different form and may not necessarily be split into T and V terms.

In the examples provided, the Lagrangian for gravitational energy and the Einstein-Hilbert Lagrangian do not follow the T-V form. This is because they represent different physical systems and interactions. In the case of gravitational energy, the Lagrangian is written in terms of the gravitational potential and its derivative, while in general relativity, the Lagrangian is written in terms of the metric and its curvature.

So while it may be possible to transform the Lagrangian into a T-V form in some cases, it is not always the case and is not a general rule. The form of the Lagrangian depends on the specific system and its interactions, and cannot always be written as T-V.