Does mental satisfaction lead to physical satisfaction? Does mental satisfaction lead to physical satisfaction?
Does physical satisfaction lead to mental satisfaction?
The Canadian Mental Health Association of Ontario is pretty unambiguous about the bidirectionally causal relationships between physical and mental well-being:


  Mental and physical health is fundamentally linked. There are multiple associations between mental health and chronic physical conditions that significantly impact people’s quality of life, demands on health care and other publicly funded services, and generate consequences to society.  The World Health Organization (WHO) defines: health as a state of complete physical, mental and social well-being and not merely the absence of disease or infirmity.   The WHO states that “there is no health without mental health.”
  
  Nowhere is the relationship between mental and physical health more evident than in the area of chronic conditions. The associations between mental and physical health are:
  
  
  Poor mental health is a risk factor for chronic physical conditions.
  People with serious mental health conditions are at high risk of experiencing chronic physical conditions.
  People with chronic physical conditions are at risk of developing poor mental health.
  
  
  The social determinants of health impact both chronic physical conditions and mental health. Key aspects of prevention include increasing physical activity, access to nutritious foods, ensuring adequate income and fostering social inclusion and social support. This creates opportunities to enhance protective factors and reduce risk factors related to aspects of mental and physical health.


The Mental Health Foundation's report, "Let's get physical: The impact of physical activity on wellbeing" offers a good review of several connections between exercise and well-being of both kinds. It's very broad, yet accessible, so I'll let it speak for itself. A peer-reviewed research literature review on the same topic is also available (Penedo &amp; Dahn, 2005), as well as some original correlational research that linked physical and psychological health subjectively (Edwards, Ngcobo, Edwards, &amp; Palavar, 2005). These are just some quick results from a Google search for "physical mental well-being correlation"; plenty more out there.

References


- Edwards, S. D., Ngcobo, H. S., Edwards, D. J., &amp; Palavar, K. (2005). Exploring the relationship between physical activity, psychological well-being and physical self-perception in different exercise groups. South African Journal for Research in Sport, Physical Education and Recreation, 27(1), 75–90. Retrieved from http://www.sirc.ca/newsletters/mid-jan10/documents/S-996370.pdf.
- Penedo, F. J., &amp; Dahn, J. R. (2005). Exercise and well-being: A review of mental and physical health benefits associated with physical activity. Current Opinion in Psychiatry, 18(2), 189–193.