Hamiltonian Math: Understanding p-dot and q-dot Terms

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In summary, the speaker is watching a lecture on the Hamiltonian and is confused about differentiating a generic function G with respect to p and q. They are wondering where the terms p-dot and q-dot come into play in the equation, as it seems like it should just be the partial derivative of G with respect to p and q. The responder clarifies that the given equation is using the chain rule to find the derivative of G with respect to t, which involves the terms p-dot and q-dot.
  • #1
DiracPool
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I'm watching a lecture on the Hamiltonian and can't figure out something. Here it is. Take a generic function G, and differentiate it with respect to p and q. What you get is the partial of G with respect to p TIMES the derivative of p (or p-dot), plus the derivative of G with respect to q TIMES q-dot.

My question is, where does the p-dot and q-dot terms come into the equation here? Why isn't it just the partial of G over p plus the partial of G over q?
 
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  • #2
What you've described looks like taking the derivative of G(p,q) with respect to t, using the chain rule:

$$\frac{dG(p,q)}{dt} = \frac{\partial G}{\partial p} \frac{dp}{dt} + \frac{\partial G}{\partial q} \frac{dq}{dt}$$
 
  • #3
It certainly does, thanks jtbell.
 

1. What is Hamiltonian math?

Hamiltonian math is a mathematical framework used in classical mechanics to describe the dynamics of a system. It is based on the concept of energy and involves the use of equations known as Hamilton's equations to describe the evolution of a physical system over time.

2. What are the p-dot and q-dot terms in Hamiltonian math?

The p-dot and q-dot terms in Hamiltonian math refer to the time derivatives of the position and momentum variables, respectively. These terms are used in Hamilton's equations to describe the change in position and momentum of a system over time.

3. How do p-dot and q-dot terms relate to each other?

P-dot and q-dot terms are related through Hamilton's equations, which state that the time derivative of the position variable is equal to the negative of the partial derivative of the Hamiltonian with respect to the momentum variable, and vice versa. In other words, they describe how changes in position affect momentum and vice versa.

4. Why is Hamiltonian math important?

Hamiltonian math is important because it provides a powerful mathematical framework for understanding the dynamics of physical systems. It allows scientists to make predictions about the behavior of a system based on its initial conditions and the forces acting upon it. It is also a fundamental tool in many areas of physics, including quantum mechanics and statistical mechanics.

5. How is Hamiltonian math used in real-world applications?

Hamiltonian math has a wide range of applications, including in engineering, physics, and chemistry. It is used to model and predict the behavior of complex systems such as planetary motion, fluid dynamics, and chemical reactions. It is also used in the development of new technologies, such as in the design of control systems for spacecraft and robots.

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