- #1
davidbenari
- 466
- 18
The Hamiltonian is not always equal to the total energy. In fact the Hamiltonian for a system of particles could be defined as
##H=L-\sum \dot{q_i}\frac{\partial L}{\partial \dot{q_i}}##
Which is the total energy only if the potential energy is a function of ##q_i## and if the kinetic energy is a homogeneous quadratic function of ##\dot{q_i}##.
I know how to show that the condition ##\frac{\partial L}{\partial t}=0## implies ##\frac{d}{dt}H=0##.
But I was left wondering: People always say time-translational symmetry implies conservation of energy, but I don't think this is the case. Time translational symmetry implies the conservation of the Hamiltonian, which may or may not be the total energy.
So which one is true? Does time translational symmetry imply conservation of the Hamiltonian or of the Energy?
In my opinion it could imply the energy too, given a good set of coordinates that aren't flying around in space w.r.t to an inertial frame such that it would involve time in your Lagrangian...
Thanks.
##H=L-\sum \dot{q_i}\frac{\partial L}{\partial \dot{q_i}}##
Which is the total energy only if the potential energy is a function of ##q_i## and if the kinetic energy is a homogeneous quadratic function of ##\dot{q_i}##.
I know how to show that the condition ##\frac{\partial L}{\partial t}=0## implies ##\frac{d}{dt}H=0##.
But I was left wondering: People always say time-translational symmetry implies conservation of energy, but I don't think this is the case. Time translational symmetry implies the conservation of the Hamiltonian, which may or may not be the total energy.
So which one is true? Does time translational symmetry imply conservation of the Hamiltonian or of the Energy?
In my opinion it could imply the energy too, given a good set of coordinates that aren't flying around in space w.r.t to an inertial frame such that it would involve time in your Lagrangian...
Thanks.
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