Finding Potential Energy of a Particle Constrained to a Surface

[Jen6]

Theres a question in which I need to derive a langrangian, and then the equations of motion for a particle constrained onto a surface. I think I'll be able to do it, except for the fact that I have no idea how to formulate the POTENTIAL ENERGY of the particle. I'd like to have a go at the rest myself, but can someone help me with how to just find the potential energy?
The particle is constrained to a surface:
z(x,y)= x^2/a^2 + y^2/b^2,
and its moving in presence of a gravitational force in -ve z direction.

Obviously the gravitational force is F=-mg, and potential will be some function of x and y??

I'd really appreciate some help!

 

[gabbagabbahey]

Start with the gravitational potential; what is that?

 

[Jen6]

Thanks for replying. Gravitational potential is the potential energy per unit mass? Sorry, I still don't know where to go from there

 

[gabbagabbahey]

The gravitational potential is defined by $$\vec{F}_g=-\vec{\nabla}U_g$$, so for $$\vec{F}_g=-mg\hat{z}$$, one usually writes $$U_g=mgz$$

...Surely you've seen this before?

 

[Jen6]

Yes, I have. Sorry, I knew it would be something obvious like that; sometimes I just overcomplicate things in my head and I miss what I should really get straight away. Thankyou for your help, I should be able to do it now.