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In classical mechanics the Lagrangian and Hamiltonian formulations are mostly equivalent (though, confusingly the Lagrangian formulation is due to Hamilton, who

In classical mechanics the Lagrangian and Hamiltonian formulations are mostly equivalent (though, confusingly the Lagrangian formulation is due to Hamilton, who has shown that the Lagrange equations can be derived from a variational principle, while the complete Hamiltonian formulation is due to some later phycisists; for what has become the Hamiltonian formulation, Hamilton has shown only how to obtain the system of first order equations from the system of second order equations, but that had already been done before by Cauchy in 1831 in a journal that few were reading, so it was ignored). Still, in classical mechanics some consider the Hamiltonian formulation to be more fundamental, because the first order equations can be applicable to some problems that have discontinuities incompatible with second-order equations, though such problems are artificial (real systems are continuous enough, strong discontinuities appear only through approximations). However this changes completely in relativistic mechanics, where the Hamiltonian is not invariant, while the Lagrangian is a relativistic invariant quantity. This makes the Lagrangian formulation a far better choice in relativistic mechanics and it is a strong argument to consider the Lagrangian formulation as the fundamental one and the Hamiltonian formulation as only an approximation that can be used at small velocities or only as a mathematical trick for numeric solutions. When the Lagrangian formulation is used, after a coordinate system is chosen, it is always possible to use the Legendre transformation to obtain a Hamiltonian system of first order equations. However, in the relativistic case the system depends on the coordinate system. Therefore, if the coordinate system is changed, the Hamiltonian equations must be derived again from the invariant Lagrangian formulation. The reason why the Lagrangian is a relativistic invariant is that this scalar value is the projection of the energy-momentum 4-vector on the trajectory curve in space-time. The Hamiltonian is just the temporal component of the 4-vector, which is changed by any coordinate transformation. Therefore L is more fundamental than H, in the same sense that the magnitude of a vector is more fundamental than any of the components that the vector happens to have in some particular coordinate system. The traditional formulation of the quantum mechanics using H is a serious inconvenience for extending it to the relativistic case. Coherent formulations of the relativistic quantum mechanics must also use L instead of H.

oddthink 3y ago
Thank you! The Lagrangian as projection of energy-momentum actually makes sense, unlike the "let's just subtract potential from kinetic. No reason, it just works" story. I'd been idly wondering that for a while (and this is as someone with a physics degree, though in astro, which is a good bit more applied).
- adrian_b 3y ago
  Hamilton has introduced what he has called the "Principal Function S", which is used in his variational principle on which the Lagrangian formulation is based. Nowadays this function is frequently called "Hamilton's action", though this is not a good idea because it causes confusions with what Hamilton, like all his predecessors, called "action", which is the integral of the kinetic energy. The "Principal Function S", which is a scalar value, i.e. a relativistic invariant quantity, is the line integral of the Lagrangian over the trajectory in space-time, i.e. it is the line integral of the energy-momentum 4-vector over the trajectory in space-time. Like any line integral of a vector, the line integral of the energy-momentum 4-vector is equal to the line integral over the trajectory of its projection on that trajectory. This is why the Lagrangian is the projection of the energy-momentum 4-vector. Hamilton has found the correct form of this line integral in relativistic theory, even if that was about 3 quarters of century before the concept of 4-vectors became understood. The "Principal Function S", i.e. the integral of the energy-momentum, can be considered as a more fundamental quantity than the Lagrangian, which is its derivative (the energy-momentum vector is its gradient). In quantum mechanics the "Principal Function S" is the phase of the wave function, so it is even more obvious that it must be an invariant quantity.
- ElDji 3y ago
  This is the exact point that confused me a lot (and still confuses me) when I tried to read the "The Theoretical Minimum: What You Need to Know to Start Doing Physics" : "Hey, let's just fix/define the lagrangian as T - V and you'll see that after some magical math stuff in the following chapter, we'll find back newtonian equations. Trust me for now". If anyone has a reference/book/paper that allows you to learn this concept more intuitively, I'd be grateful.

oddthink 3y ago

Thank you! The Lagrangian as projection of energy-momentum actually makes sense, unlike the "let's just subtract potential from kinetic. No reason, it just works" story. I'd been idly wondering that for a while (and this is as someone with a physics degree, though in astro, which is a good bit more applied).

adrian_b 3y ago
Hamilton has introduced what he has called the "Principal Function S", which is used in his variational principle on which the Lagrangian formulation is based. Nowadays this function is frequently called "Hamilton's action", though this is not a good idea because it causes confusions with what Hamilton, like all his predecessors, called "action", which is the integral of the kinetic energy. The "Principal Function S", which is a scalar value, i.e. a relativistic invariant quantity, is the line integral of the Lagrangian over the trajectory in space-time, i.e. it is the line integral of the energy-momentum 4-vector over the trajectory in space-time. Like any line integral of a vector, the line integral of the energy-momentum 4-vector is equal to the line integral over the trajectory of its projection on that trajectory. This is why the Lagrangian is the projection of the energy-momentum 4-vector. Hamilton has found the correct form of this line integral in relativistic theory, even if that was about 3 quarters of century before the concept of 4-vectors became understood. The "Principal Function S", i.e. the integral of the energy-momentum, can be considered as a more fundamental quantity than the Lagrangian, which is its derivative (the energy-momentum vector is its gradient). In quantum mechanics the "Principal Function S" is the phase of the wave function, so it is even more obvious that it must be an invariant quantity.
ElDji 3y ago
This is the exact point that confused me a lot (and still confuses me) when I tried to read the "The Theoretical Minimum: What You Need to Know to Start Doing Physics" : "Hey, let's just fix/define the lagrangian as T - V and you'll see that after some magical math stuff in the following chapter, we'll find back newtonian equations. Trust me for now". If anyone has a reference/book/paper that allows you to learn this concept more intuitively, I'd be grateful.