### Related Questions

#### Sponsored Content

#### 2 Answered Questions

### [SOLVED] Compute the Legendre transform for a singular Lagrangian

**2019-06-10 22:32:11****VoB****299**View**7**Score**2**Answer- Tags: homework-and-exercises classical-mechanics lagrangian-formalism hamiltonian-formalism constrained-dynamics

#### 1 Answered Questions

### Inconsistency? Lagrangian with its Euler–Lagrange equation as condition

**2019-05-27 21:08:15****BigFOX I****79**View**2**Score**1**Answer- Tags: lagrangian-formalism action constrained-dynamics

#### 1 Answered Questions

### [SOLVED] Can all canonical transformations be generated using a generating function?

**2019-02-23 14:51:25****jak****334**View**1**Score**1**Answer- Tags: classical-mechanics lagrangian-formalism coordinate-systems hamiltonian-formalism phase-space

#### 1 Answered Questions

### [SOLVED] Why is the Hamiltonian zero in relativity?

**2017-01-15 02:41:06****Cham****1242**View**9**Score**1**Answer- Tags: general-relativity energy lagrangian-formalism hamiltonian-formalism constrained-dynamics

#### 1 Answered Questions

### [SOLVED] Build rotational Hamiltonian based on Lagrangian of general form

**2016-03-24 22:16:29****artfin****167**View**0**Score**1**Answer- Tags: homework-and-exercises lagrangian-formalism rotational-dynamics hamiltonian-formalism constrained-dynamics

#### 2 Answered Questions

### [SOLVED] Finding geodesics: Lagrangian vs Hamiltonian

**2016-02-24 13:35:56****user46446****1184**View**6**Score**2**Answer- Tags: general-relativity lagrangian-formalism hamiltonian-formalism variational-principle geodesics

#### 4 Answered Questions

### [SOLVED] Help understanding what the Hamiltonian signifies for the action compared with the Euler-Lagrange equations for the Lagrangian?

**2015-03-03 05:45:36****Stan Shunpike****762**View**3**Score**4**Answer- Tags: classical-mechanics harmonic-oscillator hamiltonian-formalism action

## 1 comments

## @J.G. 2017-05-18 13:21:57

The problem here is that, because there exist constraints of the form $f(q,\,p)=0$, the phase space coordinates of the usual Hamiltonian formulation aren't independent. I'm not sure how you encountered this Lagrangian, but this issue is a common hiccup in electromagnetism and (if you'll pardon a more obscure example) BRST quantisation. The good news is you can still form a Hamiltonian description equivalent to the Lagrangian one. The trick is to append suitable terms to the "naïve" Hamiltonian, as explained here, and as a result the Poisson brackets are upgraded to what are called Dirac brackets.

For your problem the full Hamiltonian is $H=-2\theta^2+c_1 p_\eta+c_2( p_\theta-\eta)$, where the $c_i$ remain to be computed as functions of undifferentiated phase space coordinates. In fact $c_1=\frac{\partial H}{\partial p_\eta}=\dot{\eta}=4\theta$ while $c_2=\frac{\partial H}{\partial p_\theta}=\dot{\theta}=0$, so $H=-2\theta^2+4\theta p_\eta$. You can verify this gives you the right equations of motion.

## @donnydm 2017-05-18 14:39:36

Thanks! It's actually a part of perturbed NLS Lagrangian after an attempt of finding solution using variational method. Are the Dirac's procedure covers all Lagrangian of 'irregular' forms?

## @J.G. 2017-05-18 16:07:28

@donnydm Given any Lagrangian with consistent equations of motion, this technique will ensure an equivalent Hamiltonian formulation, although if there are higher-order derivatives you also need a trick due to Ostrogradski. (An example of an inconsistent Lagrangian is $L=q$.)