Understanding Hamiltonian Conservation Laws

In summary, the hamiltonian is a conserved quantity that is generated by an arbitrary phase-space distribution function. It is easy to show that the infinitesimal canonical transformation is a symmetry of the Hamiltonian if and only if there is a one-to-one correspondence between the generators of symmetries and conserved quantities.
  • #1
Physgeek64
247
11
I'm a little confused about the hamiltonian.

Once you have the hamiltonian how can you find conserved quantities. I understand that if it has no explicit dependence on time then the hamiltonian itself is conserved, but how would you get specific conservation laws from this?

Many thanks
 
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  • #2
Unfortunately you do not tell us about your level. Do you know Poisson brackets? If so, you question is quite easy to answer. Suppose you have an arbitrary phase-space function ##f(t,q^k,p_j)## (##k,j \in \{1,\ldots,f \}##) then the total time derivative is
$$\frac{\mathrm{d}}{ \mathrm{d} t} f=\dot{q}^k \frac{\partial f}{\partial q^k}+\dot{p}_j \frac{\partial f}{\partial p_j} + \partial_t f,$$
where the latter partial time derivative refers to the explicit time dependence of ##f## only. Now use the Hamilton equations of motion
$$\dot{q}^k=\frac{\partial H}{\partial p_k}, \quad \dot{p}_j=-\frac{\partial H}{\partial q^j}.$$
Plugging this in the time derivative you get
$$\frac{\mathrm{d}}{ \mathrm{d} t} f=\frac{\partial f}{\partial q^k} \frac{\partial H}{\partial p_k} - \frac{\partial f}{\partial p_j} \frac{\partial H}{\partial q^j} + \partial_t f=\{f,H \}+\partial_t f.$$
A quantity is thus obviously conserved by definition if this expression vanishes.

Applying this to the Hamiltonian itself you get
$$\frac{\mathrm{d}}{\mathrm{d} t} H=\{H,H\}+\partial_t H=\partial_t H,$$
i.e., ##H## is conserved (along the trajectory of the system) if and only if it is not explicitly time dependent.

Now an infinitesimal canonical transformation is generated by an arbitrary phase-space distribution function ##G##,
$$\delta q^k=\frac{\partial G}{\partial p_k}\delta \alpha=\{q^k,G\} \delta \alpha, \quad \delta p_j=-\frac{\partial G}{\partial q^j} \delta \alpha =\{p_j,G \} \delta \alpha, \quad \delta H=\partial_t G \delta \alpha.$$
From this it is easy to show that
$$H'(t,q+\delta q,p+\delta p) = H(t,q,p),$$
i.e., that the infinitesimal canonical transformation is a symmetry of the Hamiltonian, if and only if
$$\{H,G \}+\partial_t G=0,$$
but that means that
$$\frac{\mathrm{d}}{\mathrm{d} t} G=0$$
along the trajectory of the system, i.e., the generator of a symmetry transformation is a conserved quantity, and also any conserved quantity is the generator of a symmetry transformation. That means that there's a one-to-one relation between the generators of symmetries and conserved quantities, which is one of Noether's famous theorems.
 
  • #3
vanhees71 said:
Unfortunately you do not tell us about your level. Do you know Poisson brackets? If so, you question is quite easy to answer. Suppose you have an arbitrary phase-space function ##f(t,q^k,p_j)## (##k,j \in \{1,\ldots,f \}##) then the total time derivative is
$$\frac{\mathrm{d}}{ \mathrm{d} t} f=\dot{q}^k \frac{\partial f}{\partial q^k}+\dot{p}_j \frac{\partial f}{\partial p_j} + \partial_t f,$$
where the latter partial time derivative refers to the explicit time dependence of ##f## only. Now use the Hamilton equations of motion
$$\dot{q}^k=\frac{\partial H}{\partial p_k}, \quad \dot{p}_j=-\frac{\partial H}{\partial q^j}.$$
Plugging this in the time derivative you get
$$\frac{\mathrm{d}}{ \mathrm{d} t} f=\frac{\partial f}{\partial q^k} \frac{\partial H}{\partial p_k} - \frac{\partial f}{\partial p_j} \frac{\partial H}{\partial q^j} + \partial_t f=\{f,H \}+\partial_t f.$$
A quantity is thus obviously conserved by definition if this expression vanishes.

Applying this to the Hamiltonian itself you get
$$\frac{\mathrm{d}}{\mathrm{d} t} H=\{H,H\}+\partial_t H=\partial_t H,$$
i.e., ##H## is conserved (along the trajectory of the system) if and only if it is not explicitly time dependent.

Now an infinitesimal canonical transformation is generated by an arbitrary phase-space distribution function ##G##,
$$\delta q^k=\frac{\partial G}{\partial p_k}\delta \alpha=\{q^k,G\} \delta \alpha, \quad \delta p_j=-\frac{\partial G}{\partial q^j} \delta \alpha =\{p_j,G \} \delta \alpha, \quad \delta H=\partial_t G \delta \alpha.$$
From this it is easy to show that
$$H'(t,q+\delta q,p+\delta p) = H(t,q,p),$$
i.e., that the infinitesimal canonical transformation is a symmetry of the Hamiltonian, if and only if
$$\{H,G \}+\partial_t G=0,$$
but that means that
$$\frac{\mathrm{d}}{\mathrm{d} t} G=0$$
along the trajectory of the system, i.e., the generator of a symmetry transformation is a conserved quantity, and also any conserved quantity is the generator of a symmetry transformation. That means that there's a one-to-one relation between the generators of symmetries and conserved quantities, which is one of Noether's famous theorems.

Sorry- no I don't know about Poisson brackets- I'm a complete novice. Haven't encountered hamiltonians before, nor do I know much about them.

Thank you for your response though! Unfortunately I can't see any of the maths you've included- For some reason my computer thinks its an error
 
  • #4
Physgeek64 said:
Unfortunately I can't see any of the maths you've included- For some reason my computer thinks its an error

Right-click on the math error, go to Math Settings -> Math Renderer and try e.g. HTML-CSS (or some other renderer).
 

1. What is a Hamiltonian conservation law?

A Hamiltonian conservation law is a fundamental principle in physics that states that the total energy of a closed system remains constant over time. This means that energy can neither be created nor destroyed, only transformed from one form to another.

2. How is Hamiltonian conservation law related to classical mechanics?

In classical mechanics, Hamiltonian conservation law is derived from the principle of least action, which states that a system will follow the path that minimizes the action, or the integral of the Lagrangian over time. The Hamiltonian is a function that represents the total energy of a system, and its conservation is a consequence of the principle of least action.

3. What are some examples of systems that follow Hamiltonian conservation law?

Many physical systems follow Hamiltonian conservation law, including planetary motion, pendulums, and simple harmonic oscillators. Other examples include the conservation of energy in chemical reactions and the conservation of angular momentum in rotational motion.

4. What is the role of symmetries in Hamiltonian conservation laws?

Symmetries play a crucial role in Hamiltonian conservation laws. In physics, symmetries refer to properties of a system that remain unchanged under certain transformations. These symmetries are closely related to the conserved quantities in the system, such as energy, momentum, and angular momentum.

5. How can Hamiltonian conservation laws be applied in real-world scenarios?

Hamiltonian conservation laws have a wide range of applications in various fields, including mechanics, thermodynamics, and quantum mechanics. They are used to understand and predict the behavior of physical systems, and can also be used in engineering and technology to design efficient and sustainable systems.

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